<%BANNER%>

Extending the Nadaraya-Waston Estimator for Data with Spatially Correlated Errors

MISSING IMAGE

Material Information

Title:
Extending the Nadaraya-Waston Estimator for Data with Spatially Correlated Errors
Physical Description:
1 online resource (99 p.)
Language:
english
Creator:
Wang, Shu
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Statistics
Committee Chair:
Young, Linda
Committee Members:
Bliznyuk, Nikolay A
Ghosh, Malay
Fik, Timothy J

Subjects

Subjects / Keywords:
additive -- bandwidth -- correlated -- errors -- model -- nonparametric -- regression -- selection -- spatial
Statistics -- Dissertations, Academic -- UF
Genre:
Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
Spatial data arise in a variety of fields, including econometrics, epidemiology, environmental science, and image analysis. As spatial data are increasingly prevalent, appropriate quantitative analyses of the data are needed to provide meaningful results. Typically, spatial data can be decomposed into two components: large-scale variation and small-scale variation. From the early development of spatial data analysis, large scale variation has often been assumed to have some parametric structures, e.g., linear structure. However, in practice, this assumption is often too idealized. Nonparametric regression techniques have become increasingly popular as tools for spatial data analysis. This thesis studies nonparametric regression models with spatially correlated data. For the large scale variation of the data, the nonparametric and additive nonparametric structures are adopted; while for the small scale variation, the errors are assumed to have spatial covariance structure. In this thesis, we consider the performance of the traditional Nadaraya-Watson (N-W) estimator under spatially correlated errors. We show that in the presence of spatial correlation, the cross-validation bandwidth selection technique, which is a commonly used criterion to choose optimal bandwidth, fails to give the best bandwidth. We give a more efficient criterion, called the C2 criterion, for choosing optimal bandwidth. The C2 criterion is based on the weighted mean average squared error (WMASE) and accounts for the spatial correlation. Moreover, under the general nonparametric model, we adopt the two-step procedure, which was first considered in time series analysis. We use that concept to extend the traditional N-W estimator to obtain the spatial N-W estimator for spatially correlated data. The spatial N-W estimator is shown to be more efficient than the traditional N-W estimator when the errors are spatially correlated. For the additive nonparametric regression model, the marginal integral technique is used to avoid the difficulty of iterative procedures. We propose the additive spatial N-W estimator based on the marginal integral technique and the N-W estimator. The asymptotic mean and variance for the traditional, spatial, additive spatial N-W estimators under spatially correlated errors are investigated. Asymptotic normality is also shown for traditional, spatial, additive spatial N-W estimators. And, for all models, simulations are conducted to assess the performances of the proposed estimators. Under the general nonparametric regression model, the spatial N-W estimator performs better than the traditional N-W estimator, using the C2 criterion, and the traditional N-W estimator performs better when using the C2 criterion compared to the cross-validation technique. When the true model has additive form, the additive spatial N-W estimator has the best performance. As an illustration of our estimators, a case study of the temperature of the state of Florida is given.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Shu Wang.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Young, Linda.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-08-31

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045722:00001

MISSING IMAGE

Material Information

Title:
Extending the Nadaraya-Waston Estimator for Data with Spatially Correlated Errors
Physical Description:
1 online resource (99 p.)
Language:
english
Creator:
Wang, Shu
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Statistics
Committee Chair:
Young, Linda
Committee Members:
Bliznyuk, Nikolay A
Ghosh, Malay
Fik, Timothy J

Subjects

Subjects / Keywords:
additive -- bandwidth -- correlated -- errors -- model -- nonparametric -- regression -- selection -- spatial
Statistics -- Dissertations, Academic -- UF
Genre:
Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
Spatial data arise in a variety of fields, including econometrics, epidemiology, environmental science, and image analysis. As spatial data are increasingly prevalent, appropriate quantitative analyses of the data are needed to provide meaningful results. Typically, spatial data can be decomposed into two components: large-scale variation and small-scale variation. From the early development of spatial data analysis, large scale variation has often been assumed to have some parametric structures, e.g., linear structure. However, in practice, this assumption is often too idealized. Nonparametric regression techniques have become increasingly popular as tools for spatial data analysis. This thesis studies nonparametric regression models with spatially correlated data. For the large scale variation of the data, the nonparametric and additive nonparametric structures are adopted; while for the small scale variation, the errors are assumed to have spatial covariance structure. In this thesis, we consider the performance of the traditional Nadaraya-Watson (N-W) estimator under spatially correlated errors. We show that in the presence of spatial correlation, the cross-validation bandwidth selection technique, which is a commonly used criterion to choose optimal bandwidth, fails to give the best bandwidth. We give a more efficient criterion, called the C2 criterion, for choosing optimal bandwidth. The C2 criterion is based on the weighted mean average squared error (WMASE) and accounts for the spatial correlation. Moreover, under the general nonparametric model, we adopt the two-step procedure, which was first considered in time series analysis. We use that concept to extend the traditional N-W estimator to obtain the spatial N-W estimator for spatially correlated data. The spatial N-W estimator is shown to be more efficient than the traditional N-W estimator when the errors are spatially correlated. For the additive nonparametric regression model, the marginal integral technique is used to avoid the difficulty of iterative procedures. We propose the additive spatial N-W estimator based on the marginal integral technique and the N-W estimator. The asymptotic mean and variance for the traditional, spatial, additive spatial N-W estimators under spatially correlated errors are investigated. Asymptotic normality is also shown for traditional, spatial, additive spatial N-W estimators. And, for all models, simulations are conducted to assess the performances of the proposed estimators. Under the general nonparametric regression model, the spatial N-W estimator performs better than the traditional N-W estimator, using the C2 criterion, and the traditional N-W estimator performs better when using the C2 criterion compared to the cross-validation technique. When the true model has additive form, the additive spatial N-W estimator has the best performance. As an illustration of our estimators, a case study of the temperature of the state of Florida is given.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Shu Wang.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Young, Linda.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-08-31

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045722:00001


This item has the following downloads:


Full Text

PAGE 1

EXTENDINGTHENADARAYA-WASTONESTIMATORFORDATAWITHSPATIALLYCORRELATEDERRORSBySHUWANGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

PAGE 2

c2013ShuWang 2

PAGE 3

Idedicatethistomybelovedhusbandandmyparents. 3

PAGE 4

ACKNOWLEDGMENTS Iwouldliketothankmyadviser,LindaJ.Young,forprovidingmewithinnumerableopportunitiesandlessons,bothinworkandinlife,andenthusiasticguidanceandsupportthroughoutmystudyofthePh.Ddegree.Fromthecountlessinsightfuldiscussionswithher,Ibenetgreatlyinboththeacademicresearchandthesiswriting.Iamgratefultomycommitteemembers,Dr.MalayGhosh,Dr.NikolayBliznyukandDr.TimothyFikfortheirparticipationandvaluablesuggestionsduringtheproposal,dissertationreview,anddefenseprocess.Iwouldliketothankallthefaculties,staffsintheDepartmentofStatisticsatUniversityofFloridaandIgreatlyappreciatethenancialsupportfromDr.LindaJ.YoungandtheStatisticsDepartment.Ishallexpressmysincerethankstoallgraduatestudentsforprovidingsuchchallengingandenrichingworkenvironment.Atlast,Iwillthankmyparents,myauntandmyuncle,theoneslovemeandIlove.ThankyouforhelpingmewhenIhaveneededitmost,andforyourendlesspatienceandsupport.Iamforeverindebtedtomyhusband,mylove,Lei,youhavestoodbymysidefromthebeginningandhelpedmetoseeitthroughtotheend.YoualwaysencouragesmewhenIfeltfrustratedduringmyresearchandneverallowingmetoquit.Withoutthecares,fun,encouragementandunderstanding,Iwillneverbeabletonishmystudy. 4

PAGE 5

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 6 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 11 2MULTIVARIATEN-WESTIMATORUNDERSPATIALLYCORRELATEDERRORS 17 2.1Background ................................... 17 2.2MultivariateN-WEstimator .......................... 20 2.3BandwidthSelectionUnderSpatialDependence .............. 25 2.4SpatialN-WEstimator ............................. 28 2.5SpatialDependenceParametersEstimation ................. 30 2.6SimulationResult ................................ 33 2.6.1SimulationStudy1 ........................... 33 2.6.2SimulationStudy2 ........................... 35 2.6.3SimulationStudy3 ........................... 36 3ADDITIVESPATIALN-WESTIMATORUNDERSPATIALLYCORRELATEDERRORS ....................................... 45 3.1Background ................................... 45 3.2ModelandEstimator .............................. 47 3.3SimulationResult ................................ 52 3.3.1SimulationStudy1 ........................... 52 3.3.2SimulationStudy2 ........................... 53 3.3.3SimulationStudy3 ........................... 54 4CASESTUDY:U.S.PRECIPITATIONDATA .................... 65 5CONCLUSION .................................... 71 APPENDIX AACCOMPANIMENTTOCHAPTER2 ........................ 74 BACCOMPANIMENTTOCHAPTER3 ........................ 86 REFERENCES ....................................... 96 BIOGRAPHICALSKETCH ................................ 99 5

PAGE 6

LISTOFTABLES Table page 2-1Estimatedspatialdependenceparameters .................... 38 2-2ComparisonoftheaveragemeansquarederrorforthreeN-Westimators ... 38 2-3ComparisonoftheaveragedbiasforthreeN-Westimators ........... 38 2-4ComparisonoftheaveragemeansquarederrorforthreeN-Westimatorswithdifferentsamplesizes ................................ 38 2-5ComparisonoftheaveragedbiasforthreeN-Westimatorswithdifferentsamplesizes .......................................... 38 2-6ComparisonoftheaveragemeansquarederrorandaveragebiasforthreeN-Westimatorswhenthecovariancestructureismisspecied ......... 38 2-7ComparisonoftheaveragemeansquarederrorandaveragebiasforthreeN-Westimatorswhendataareindependent .................... 38 3-1AveragemeansquarederrorandmeanbiasforadditivespatialN-Westimator 56 3-2AverageMSEandaveragebiasfordifferentestimatorswithsamplesize50 .. 56 3-3AverageMSEandaveragebiasfordifferentestimatorswithsamplesize100 56 3-4AverageMSEandaveragebiasfordifferentestimatorswithsamplesize150 56 3-5AverageMSEandaveragebiasforthreeestimatorswhenmodelismisspecied1 ............................................ 56 3-6AverageMSEandaveragebiasforthreeestimatorswhenmodelismisspecied2 ............................................ 56 4-1Comparisionofthesumofsquareddifferencebetweenobservedandleave-one-outpredictedprecipitationusingthreeN-Westimators ................ 67 6

PAGE 7

LISTOFFIGURES Figure page 2-1ThedifferencesbetweentruemeanandthreeN-Westimatorswhenerrorsarespatiallycorrelated ................................ 39 2-2ThedifferencesbetweentruemeanandthreeN-Westimatorswhenerrorsarespatiallycorrelatedwithsamplesize50 .................... 40 2-3ThedifferencesbetweentruemeanandthreeN-Westimatorswhenerrorsarespatiallycorrelatedwithsamplesize100 ................... 41 2-4ThedifferencesbetweentruemeanandthreeN-Westimatorswhenerrorsarespatiallycorrelatedwithsamplesize150 ................... 42 2-5ThedifferencesbetweenthetruemeanandthreeN-Westimatorswhenthecovariancestructureismisspecied ........................ 43 2-6ThedifferencesbetweenthetruemeanandthreeN-Westimatorswhendataareindenpendent ................................... 44 3-1Simulationresultform ................................ 57 3-2Simulationresultform1 ............................... 58 3-3Simulationresultform2 ............................... 59 3-4ThedeviationoftheadditivespatialN-Westimator,theN-WestimatorandthespatialN-Westimatorfromthetruevaluewithsamplesize50 ........ 60 3-5ThedeviationoftheadditivespatialN-Westimator,theN-WestimatorandthespatialN-Westimatorfromthetruevaluewithsamplesize100 ....... 61 3-6ThedeviationoftheadditivespatialN-Westimator,theN-WestimatorandthespatialN-Westimatorfromthetruevaluewithsamplesize150 ....... 62 3-7ThedeviationoftheadditivespatialN-Westimator,theN-WestimatorandthespatialN-Westimatorfromthetruevaluewhenthemodelismisspecied 63 3-8ThedeviationoftheadditivespatialN-Westimator,theN-WestimatorandthespatialN-Westimatorfromthetruevaluewhenthemodelismisspecied 64 4-1ComparisonofthethreeN-Westimatorsusingtheleave-one-outestimation .. 67 4-2PredictedprecipitationofthestateofFloridausingthetraditionalN-WestimatorusingC2criterion ................................... 68 4-3PredictedprecipitationoftheStateofFloridausingthespatialN-Westimator 69 7

PAGE 8

4-4PredictedprecipitationoftheStateofFloridausingtheadditivespatialN-Westimator ....................................... 70 8

PAGE 9

AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyEXTENDINGTHENADARAYA-WASTONESTIMATORFORDATAWITHSPATIALLYCORRELATEDERRORSByShuWangAugust2013Chair:LindaJ.YoungMajor:StatisticsSpatialdataariseinavarietyofelds,includingeconometrics,epidemiology,environmentalscience,andimageanalysis.Asspatialdataareincreasinglyprevalent,appropriatequantitativeanalysesofthedataareneededtoprovidemeaningfulresults.Typically,spatialdatacanbedecomposedintotwocomponents:large-scalevariationandsmall-scalevariation.Fromtheearlydevelopmentofspatialdataanalysis,largescalevariationhasoftenbeenassumedtohavesomeparametricstructures,e.g.,linearstructure.However,inpractice,thisassumptionisoftentooidealized.Nonparametricregressiontechniqueshavebecomeincreasinglypopularastoolsforspatialdataanalysis.Thisthesisstudiesnonparametricregressionmodelswithspatiallycorrelateddata.Forthelargescalevariationofthedata,thenonparametricandadditivenonparametricstructuresareadopted;whileforthesmallscalevariation,theerrorsareassumedtohavespatialcovariancestructure2n(kXi)]TJ /F8 11.955 Tf 11.95 0 Td[(Xjk).Inthisthesis,weconsidertheperformanceofthetraditionalNadaraya-Watson(N-W)estimatorunderspatiallycorrelatederrors.Weshowthatinthepresenceofspatialcorrelation,thecross-validationbandwidthselectiontechnique,whichisacommonlyusedcriteriontochooseoptimalbandwidth,failstogivethebestbandwidth.Wegiveamoreefcientcriterion,calledtheC2criterion,forchoosingoptimalbandwidth.TheC2criterionisbasedontheweightedmeanaveragesquared 9

PAGE 10

error(WMASE)andaccountsforthespatialcorrelation.Moreover,underthegeneralnonparametricmodel,weadoptthetwo-stepprocedure,whichwasrstconsideredintimeseriesanalysis.WeusethatconcepttoextendthetraditionalN-WestimatortoobtainthespatialN-Westimatorforspatiallycorrelateddata.ThespatialN-WestimatorisshowntobemoreefcientthanthetraditionalN-Westimatorwhentheerrorsarespatiallycorrelated.Fortheadditivenonparametricregressionmodel,themarginalintegraltechniqueisusedtoavoidthedifcultyofiterativeprocedures.WeproposetheadditivespatialN-WestimatorbasedonthemarginalintegraltechniqueandtheN-Westimator.Theasymptoticmeanandvarianceforthetraditional,spatial,additivespatialN-Westimatorsunderspatiallycorrelatederrorsareinvestigated.Asymptoticnormalityisalsoshownfortraditional,spatial,additivespatialN-Westimators.And,forallmodels,simulationsareconductedtoassesstheperformancesoftheproposedestimators.Underthegeneralnonparametricregressionmodel,thespatialN-WestimatorperformsbetterthanthetraditionalN-Westimator,usingtheC2criterion,andthetraditionalN-WestimatorperformsbetterwhenusingtheC2criterioncomparedtothecross-validationtechnique.Whenthetruemodelhasadditiveform,theadditivespatialN-Westimatorhasthebestperformance.Asanillustrationofourestimators,acasestudyofthetemperatureofthestateofFloridaisgiven. 10

PAGE 11

CHAPTER1INTRODUCTIONSpatialdataariseinavarietyofelds,includingeconometrics,epidemiology,environmentalscienceandimageanalysis.Asspatialdataareincreasinglyprevalent,appropriatequantitativeanalysesofthedataareneededtoprovidemeaningfulresults.Thestatisticaltreatmentofsuchdatahasbeenbroadlydiscussedinrecentdecades,andanextensiveliteraturehasdeveloped,asin[ 10 ]and[ 35 ].Typically,spatialdatacanbedecomposedintotwocomponents,large-scaleandsmall-scalevariation.Theyaremodeledgenerallyas Ys=m(Xs)+s(1)wheresisthelocationindex,Ysrepresentsthespatialobservationandisascalar,Xsistheexplanatoryrandomvector,andm(Xs)denotesthelarge-scalevariation,andcapturesthenon-stationarymeaneffect.Theresidualsdenotesthesmall-scalevariation,reectingthestationaryspatialdependenceofthedata.Researchersareofteninterestedinestimatingthemeanfunctionm(Xs)=E(YsjXs)foragivensetofobservations(X1,Y1),(X2,Y2),,(Xn,Yn)inspatialregressionanalysis.Intheearlydevelopmentofspatialdataanalysis,m(Xs)wasoftenassumedtohavesomeparametricstructure,e.g.,linearstructure,andaparametricstructureform(X)isstillwidelyused.However,thisassumptionisoftentooidealizedformanyareasofapplication.Nonparametricregressiontechniqueshavebecomeincreasinglypopularastoolsforspatialdataanalysis.Becausethesetechniquesimposefewassumptionsabouttheshapeofthemeanfunction,theyareexibletoolsforuncoveringnonlinearrelationshipsamongvariables.Roughlyspeaking,nonparametricregressiontechniquesconsistofbasicsmoothinganddimensionreductionmethods.KernelregressionisonesmoothingmethodforestimatingthemeanfunctioninthenonparametricregressionmodelYs=m(Xs)+s,wherem(Xs)isasmoothdeterministic 11

PAGE 12

meanfunctionandsisastationaryspatialprocesswithmeanzeroandwithsomespatialcorrelation.TheNadaraya-Wastson(N-W)estimator,thePriestley-Chaoestimatorandthelocalpolynomialestimatorarethreeexistingkernelsmoothingestimators,whicharecommonlyused.TheN-Westimatorisdenedas^m(x,h)=Pni=1Kh(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(x)Yi Pnj=1Kh(Xj)]TJ /F8 11.955 Tf 11.96 0 Td[(x)whereKisakernelfunctionandhistheunivariatebandwidth.Itcanbeusedwithxed-designsanduniformlyrandomdesigns,aswellasthenon-uniformlyrandomdesigns.Inthemultivariatecase,foradesignpointx,theN-Westimatorhastheform^m(x)=Pni=1KH(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(x)Yi Pnj=1KH(Xj)]TJ /F8 11.955 Tf 11.95 0 Td[(x)whereHistheweightmatrix.NotethattheN-Westimatorisalocallyconstantestimator,i.e.,thesolutionofmin0nXi=1[Yi)]TJ /F5 11.955 Tf 11.96 0 Td[(0]2KH(Xi)]TJ /F8 11.955 Tf 11.96 0 Td[(x)AssumethatthedatahavebeensortedaccordingtotheunivariateindependentvariableX.ThePriestley-Chaoestimatorisdenedas^mPC(x,h)=nXi=1(Xi)]TJ /F3 11.955 Tf 11.96 0 Td[(Xi)]TJ /F4 7.97 Tf 6.59 0 Td[(1)Kh(Xi)]TJ /F3 11.955 Tf 11.95 0 Td[(x)YiwithX0=0.Forequally-spacedxeddesignsanduniformlyrandomdesignsintheregion=[0,b],thePriestly-Chaoestimatorisdenedas^mPC(x,h)=b nnXi=1Kh(Xi)]TJ /F3 11.955 Tf 11.96 0 Td[(x)YiThus,forthesedesigns,thisestimatorhasasimpleexpressionandcanbeappliedtohigherdimensionalproblems.However,fornon-uniformlyrandomdesigns,thedesign 12

PAGE 13

pointsmustbesorted,makingthePriestly-Chaoestimatordifculttoextendtohigherdimensions.Theideaoflocalpolynomialregressionissimple.Ifthemeanfunctionm(x)issmoothenough,theninaneighborhoodofagivenpointx,m(x)canbewellapproximatedbyapolynomialofp-thorder.Thissuggestsusingalocallyweightedpolynomialregressionbysolvingtheweightedleastsquaresproblemminfjgpj=0nXi=1(Yi)]TJ /F6 7.97 Tf 18.3 14.95 Td[(nXj=0j(Xi)]TJ /F3 11.955 Tf 11.95 0 Td[(x)j)2Kh(Xi)]TJ /F3 11.955 Tf 11.95 0 Td[(x)Let^jdenotetheminimizer.Thenthelocalpolynomialregressionestimatorofm(x)is^mLP(x,h)=0.p=1givesthelocallinearregression.Inthisthesis,wefocusontheNadaraya-Wastson(N-W)estimator.TheN-Westimatoristhemostcommonlyusednonparametricregressionestimatorinmanyareas,suchaseconometrics,see[ 32 ].Ithasalsobeenusedinimageprocessing[ 41 ],toestimatethemagnitudeofearthquakes[ 15 ]andinfMRIanalysis[ 9 ].Althoughnonparametricmodelshavebeenstudiedinthelast20years,mostoftheresearchhasfocusedonthecasewithi.i.d.andtemporallycorrelatedresiduals,e.g.,[ 20 ],[ 43 ]and[ 29 ].Onlyafewscholarshaveconsideredthecaseofspatiallycorrelatederrors,e.g.[ 30 ].Inthisthesis,wefocusonthenonparametricestimationofmodel(1-1)withspatiallycorrelatederrors.Morespecically,weassumetheerrors=(1,,n)haveanormaldistributionwith E(i)=0 (1) 13

PAGE 14

and Cov(i,j)=2n(kXi)]TJ /F8 11.955 Tf 11.96 0 Td[(Xjk) (1) where2isthevarianceofiforalli.Forthecorrelationfunctionn,Euclideandistanceisacommonmeasureofthedistancebetweentwopoints.Thespatialcorrelationfunctionnincludesfrequentlyusedcorrelationstructures,suchastheexponentialorGaussianfunctions.Moreover,weassumethelargescalevariationm(Xs)cantaketheadditivestructure,andhencemodel(1-1)canbeseparatedintotwosub-models: Ys=m(Xs)+s(1) Ys=+dXi=1mi(Xis)+s(1)whereissomeconstant,andm()andmi(),fori=1,2,,d,areunknownfunctions.Model(1-4)and(1-5)arewellknownasthegeneralnonparametricandadditivenonparametricmodels,respectively.WeextendthetraditionalN-Westimatorformodel(1-4)withgeneralnonparametricstructuretoestimatethemeanfunctionm(X)inthepresenceofspatiallycorrelatederrors.Model(1-5)wasrstintroducedintheearly1980's,leadingtothedevelopmentofavarietyoftheoreticalresultsandandtoitspracticalimplementation,e.g.[ 42 ].Comparedtogeneralnonparametricmodels,theadditivemodelsareattractivebecausetheygeneralizethelinearregressionmodelsandallowinterpretationofmarginalchanges,suchastheeffectofasingleexplanatoryvariable.Theyarealsointerestingfromatheoreticalpointofviewbecausetheycombineexiblenonparametricmodelingofmanyvariableswiththestatisticalprecisionthatistypicalforjustoneexplanatoryvariable. 14

PAGE 15

Moreover,itwasshownin[ 38 ]thateachadditivecomponentinthemodelcanbeestimatedwiththeone-dimensionalrateofconvergence.Intheearlydevelopmentofadditivemodels,estimationoftheadditivecomponentswasbasedontheback-ttingtechnique,suchas[ 23 ],whichprojectsthedataontothespaceofadditivefunctionsviatheleastsquaresmethod.Iterativealgorithmsmustbeusedintheprocedure,andalthoughcalculationusuallyconvergequickly,noclosedformoftheestimatorsisavailable.Toavoidthedifcultyoftheiterativeprocedures,[ 28 ]proposedmarginalintegrationoftheregressionfunctionforestimatingtheadditivecomponents.Thekeypointofthemarginalintegrationapproachistoestimatethewholesurfaceoftheregressionfunctionbylocalpolynomialttingrst,andthentotakethemarginalaverageoftheestimatorsoftheregressionfunctiontoobtaintheestimatorofeachadditivecomponent.[ 8 ]consideredthesamemethodwiththeN-Westimator.[ 34 ]extendedthemethodtoestimateboththeregressionfunctionanditsderivativesimultaneouslywithlocalpolynomialtting.However,theaboveworkswereallbasedonthei.i.d.assumption.[ 14 ]consideredanadditive,partiallylinearmodel,whichwasfurthergeneralizedtothespatialcasein[ 18 ].Inthisthesis,weconsidertheadditivemodel(1-5)withspatiallycorrelatederrors.Werstremovetheindependenceassumptionandassumethatspatialcorrelationexists.Secondly,adirectestimatorbasedontheN-Westimatorandusingthemarginalintegrationprocedureof[ 28 ]isdeveloped,therebyavoidingiterativeapproximationasinback-tting.Third,theexplanatoryvariablesareallowedtobecorrelatedwithajointdensityfthatdoesnotfactorize.Finally,thedimensionofXisextendedtoadimensiond2asin[ 28 ].Therestofthethesisisorganizedasfollows.InChapter2andChapter3,wefocusontheanalysisofmodels(1-4)and(1-5)respectively,wheretheerrorsareassumedtohavetheformof(1-2)and(1-3).InChapter2,weconsiderthepropertiesofthe 15

PAGE 16

traditionalN-Westimatorformodel(1-4)underspatiallycorrelatederrors.And,weproposethespatialN-WestimatorwhichisamoreefcientthanthetraditionalN-Westimatorundermodel(1-4)withspatiallycorrelatederrors.AndinChapter3,weconsiderthepropertiesoftheadditivespatialN-Westimatorundermodel(1-5).Forbothchapters,weconductsimulationsanalysistoevaluatetheperformancesoftheestimators.InChapter4,wediscusstheresultsandavenuesforfuturework. 16

PAGE 17

CHAPTER2MULTIVARIATEN-WESTIMATORUNDERSPATIALLYCORRELATEDERRORS 2.1BackgroundSpatialdataariseinavarietyofelds,includingeconometrics,epidemiology,environmentalscience,imageanalysisandmanyothers.Asspatialdataareincreasinglyprevalent,appropriatequantitativeanalysesareneededtoprovidemeaningfulresults.Thestatisticaltreatmentofsuchdatahasbeenbroadlydiscussedinrecentdecades,andanextensiveliteraturehasdeveloped,asin[ 10 ]and[ 35 ].Typically,spatialdatacanbedecomposedintotwocomponents,large-scaleandsmall-scalevariation.TheyaremodeledgenerallyasYs=m(Xs)+swheresisthelocationindex,Ysrepresentsthespatialobservationandisascalar,andXsistheexplanatoryrandomvector.m(Xs)denotesthelarge-scalevariation,reectingthenon-stationarymeaneffect.Theresidualsdenotesthesmall-scalevariation,reectingthestationaryspatialdependenceofthedata.Intheearlydevelopmentofspatialdataanalysis,m(Xs)wasoftenassumedtohavesomeparametricstructure,e.g.,alinearstructure.However,thisassumptionistooidealizedformanyapplications.Nonparametricregressiontechniqueshavebecomeincreasinglypopularastoolsforspatialdataanalysis.Becausethesetechniquesimposefewassumptionsabouttheshapeofthemeanfunction,theyareexibletoolsforuncoveringnonlinearrelationshipsamongvariables.KernelregressionisacommonmethodofestimatingthemeanfunctioninthenonparametricregressionmodelYs=m(Xs)+s,wherem(Xs)isasmoothdeterministicmeanfunctionandsisastationaryspatialprocesswithmeanzeroandwithsomespatialcorrelation.Inthischapter,thetraditionalNadaraya-Wastson(N-W)estimatorisconsideredforestimatingthemean 17

PAGE 18

functionbecauseitisfrequentlyusedinspatialapplications.Forinstance,[ 41 ]useditinimageprocessing,and[ 9 ]applieditinfMRIanalysis.Foradesignpointt,thetraditionalN-Westimatorhastheform^m(t)=Pni=1KH(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t)Yi Pnj=1KH(Xj)]TJ /F8 11.955 Tf 11.95 0 Td[(t)whereKiscertainkernelfunctionandHistheweightmatrix.ThetraditionalN-Westimatorisbasedontheassumptionthattheelementsintheerrorvector=(1,2,,n)Tareindependent.Theasymptoticmeansquarederrorofthekernelestimatorwasstudiedby[ 33 ].AvastliteraturefocusesonthistraditionalN-Westimator,e.g.,[ 39 ],[ 12 ],[ 13 ],[ 20 ]and[ 24 ].However,theindependenceassumptionisusuallynotareasonableoneforspatialdata.Onthecontrary,spatialdataaregenerallycorrelatedandnotallofthecorrelationcanbeaccountedforbythemeanm(Xi),resultinginacorrelatederrorstructure.Typically,thisdependenceismodeledasafunctionofspatialdistance.However,unlikewithtimeseries,nonaturalorderingispresentinspatialdata.Moreover,formsofirregularspacingofdataaremorecommonwithspatialthantimeseriesdata,andthisconsiderablycomplicatesmodelinganddevelopingrulesofstatisticalinference.Substantialresearchonunivariatekernelsmoothingwithcorrelateddatahasbeenconsideredwithxedandequally-spaceddesigns(see[ 30 ]foragoodreviewofthistopic).And,inthecaseofmultivariatecovariates,localpolynomialregressionhasbeenused.Forinstance,[ 16 ]assumedtheerrorprocessfigtobestronglymixingandthexi,sxed.Anadvantageoftheirideaisthat,whenaspecicstructureoftheerrortermisknown,onecanmakeuseofadditionalinformationtoobtainsomeimprovementinestimation.Toillustratethis,[ 43 ]assumederrorstofollowtheAR(1)model,buttheirworkisonlysuitableforthecaseofaxeddesignandequally-spacedexplanatoryvariables.[ 29 ]consideredanothermodiedlocalpolynomialestimatorwithgeneralparametricerrorstructure.Intheirapproach,theexplanatoryvariablesareassumed 18

PAGE 19

randomaswellasidenticallyandindependentlydistributed.Unfortunately,inmanyapplications,spatialdatatypicallyarenotrecordedonalattice.Ifthelocationsofobservationsareirregularly-spacedpointsingeographicspace,thepreviousworksarenotsuitable.Anotherobjectiveoftheexistingliteratureistoproposeestimatorsthat,byincorporatingtheinformationcontainedintheerrorcovariancestructure,willleadtobetterperformanceasymptoticallyorinnitesamples,visavisthetraditionalestimators,suchasthetraditionalN-Westimator.Whentheregressionfunctionm()isparametricallyspecied,itisstandardtousegeneralizedleastsquaresthatreectthecorrelationstructureintheerrorprocesstoimprovetheefciencyoftheleastsquareestimator.Whenm()isnonparametricallyspecied,[ 45 ]assumedthattheerrorprocessiisstationaryandmeanzeroandhasaninvertiblelinearprocessrepresentation:i=P1j=0cjUi)]TJ /F6 7.97 Tf 6.59 0 Td[(j,whereUjarei.i.d.(0,2u).And[ 45 ]showedthattheautocorrelationfunctionoftheerrorprocesscanhelpimproveestimatorsoftheregressionfunction.Howtobestincorporatetheerrorcovariancematrixinformationintothenonparametricregressionestimatorsisstillanopenquestion.[ 26 ]showthatintypicalrandomeffectspaneldatamodels,whenastandardkernelbasedestimatorisused,itisbettertoestimatetheregressionbyignoringthecorrelationstructurewithinaclustertheworkingindependenceapproach.Aparticularlypromisingapproachhasbeenthepre-whitenmethodproposedby[ 31 ]and[ 45 ].However,asinthecaseofthelocallinearestimator,theasymptoticpropertiesofthispre-whitenestimatorhavebeenestablishedonlyforspecicparametricstructuresoftheerrorcovariance(randomeffectspaneldataandautocorrelatederrors).[ 40 ]proposedamoreefcientestimatorforthenonparametricmodelwhentheerrorsareautocorrelated.[ 29 ]proposedanewtwo-stepestimator,inspiredby[ 31 ],thatincorporatesinformationcontainedintheerror 19

PAGE 20

covariancestructureandisasymptoticallynormalunderfairlymildrestrictionsontheparametricstructureofthecovariance.Inthischapter,wefocusonthemultivariateN-Westimatorundercorrelatederrors.[ 4 ]foundtheasymptoticbias,varianceanddistributionoftheN-Westimator.However,theirworkonlyconsideredtheunivariatecaseandwaslimitedtothex-designcase.Here,weextendtheresultstoaccountformultiplecovariateswitharandomdesign.Anotherobjectiveofourpaperistoproposeestimatorsthat,byincorporatingtheinformationcontainedintheerrorcovariancestructure,havebetterperformancethanthetraditionalestimators.[ 29 ]consideredatwo-stepestimator.However,atwo-stepestimatorwithspatiallycorrelatederrorshasnotbeenconsidered.Therefore,inthispaper,wealsoproposeatwo-stepprocedureforthespatialN-WestimatorthatismoreefcientthanthetraditionalN-Westimatorwhenspatialcorrelationispresent.Therestofthischapterisorganizedasfollows.InChapter2.2,weinvestigatetheasymptoticbias,varianceanddistributionofthetraditionalN-Westimator.InChapter2.3,wehaveshownthatthetraditionalN-Westimatorusingtheleave-one-outcrossvalidationmethodtoselectbandwidthisnotsatisfactorywhentheerrorsarecorrelated,andanewmethodisproposedbasedonweightedmeanaveragesquarederror(WMASE).Anewtwo-stepprocedureisgiveninChapter2.4,andthemethodsofestimatingthespatialdependenceparametersareinChapter2.5.ThenthesimulationstudiesareinChapter2.6followedbyacasestudyinChapter2.7. 2.2MultivariateN-WEstimatorSupposetherearenobservationsY=(Y1,Y2,,Yn)T,X=(XT1,XT2,,XTn)T,(Xi,Yi)RdR,d>1,i=1,2,,n,considerthefollowingregressionmodel: Yi=m(Xi)+i (2) 20

PAGE 21

Fori=1,2,,n,ihasnormaldistributionwithE(i)=0Cov(i,j)=2n(kXi)]TJ /F8 11.955 Tf 11.95 0 Td[(Xjk)Inthecorrelationfunctionn,Euclideandistanceisacommonmeasureofthedistancebetweentwopoints.Thecommonspatialcorrelationfunctionnincludesfrequentlyusedcorrelationstructures,suchastheexponentialorGaussianfunctions.Theobjectiveistoestimatetheregressionfunctionm(X)atsomepointX2Rd.Avastliteratureexistsonhowtoproceedwithestimatingm(X).Here,wefocusourattentiononthetraditionalN-Westimatorduetoitspopularityinspatialdataanalysis.ThemultivariateN-Westimatorform(t)hastheform ^m(t)=Pni=1KH(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t)Yi Pnj=1KH(Xj)]TJ /F8 11.955 Tf 11.95 0 Td[(t) (2) whereHisaddweightmatrix,whichisassumedtobesymmetricandinvertible,andKH(x)=1 jHjK(H)]TJ /F4 7.97 Tf 6.58 0 Td[(1x)whereKisakernelfunction.Here,weassumeK(x)isnonzeroonlyontheregionkxk1.Andtherefore,thecorrespondinglocalneighborhoodusedtoestimatem(t)isfX2t:kH)]TJ /F4 7.97 Tf 6.59 0 Td[(1(X)]TJ /F8 11.955 Tf 11.95 0 Td[(t)k1gSometimes,HistakentobehId,wherehisascalarandIdisthed-dimensionidentitymatrix.BeforeweexplorethepropertiesofthetraditionalN-Westimatorwhenerrorsarecorrelated,wesummarizeherethemainassumptionsused.Assumption1:RK(u)du=1,RuK(u)du=0andRuuTK(u)du=2(K)Id,RK2(u)du=(K2),foru2Rd 21

PAGE 22

Assumption2:aK(u)bforsomeconstants00suchthat8u1,u2jK(u1)]TJ /F8 11.955 Tf 11.95 0 Td[(u2)jLku1)]TJ /F8 11.955 Tf 11.96 0 Td[(u2kAssumption3:ForweightmatrixH,njHj!1,H!0,andmax=minisbounded,wheremaxandminarethemaximumandminimumeigenvaluesofH.Assumption4:limn!1nZn(kXk)dX=R<1nZjn(kXk)jdX=O(1)limn!1nZI(kH)]TJ /F11 5.978 Tf 5.75 0 Td[(1Xk")jn(kXk)jdX=0(A1)-(A3)aresimilartotheassumptionsgivenby[ 33 ],[ 30 ]fortheunivariatecaseand[ 20 ]forthemultivariatecasewithi.i.d.errors.Inassumption(A3),H!0indicatesthateveryentryofHgoesto0.SinceherewealsoassumeHissymmetricandpositivedenite,H!0isequivalenttomax!0.Sincemax=minisboundedabove,alltheeigenvaluesofH!0atthesamerate.ThusjHjisaquantityoforderO(dmax).TheconditionnjHj!1requireseveryeigenvalueofHtoconvergeto0slowerthanO(1 n1=d).(A4),whichsetssomeconstraintsonthecovariancematrix,isageneralizationoftheassumptionfortheunivariatecasewithxeddomaindesignmentionedin[ 30 ]and[ 2 ].Anexampleofavalidcorrelationfunctionisn(x)=exp()]TJ /F5 11.955 Tf 9.3 0 Td[(n1=2kxk)WerstconsidertheasymptoticmeansquarederrorofthetraditionalN-Westimatorinthepresenceofcorrelatederrors.Tothisaim,inLemmas2.1and2.2,respectively,wederivetheasymptoticexpressionsforbothconditionalbiasandvariancetermsgivenXi. 22

PAGE 23

Lemma2.1:Giventheassumptionthatm(X)isLipschizcontinuouswithorderr,i.e,thereexitsaconstantC,s.t.jm(X1))]TJ /F3 11.955 Tf 11.96 0 Td[(m(X2)jCkX1)]TJ /F8 11.955 Tf 11.95 0 Td[(X2kr,fort2Rd,wehaveE(^m(t)))]TJ /F3 11.955 Tf 11.95 0 Td[(m(t)=O(kHkr)Proof:IntheappendixA.Lemma2.2:Letnt=totalnumberofpointsint,wheretistheneighborhoodregionwhenestimatingm(t),Sni=Pj2t,j6=ijn(kXi)]TJ /F8 11.955 Tf 11.95 0 Td[(Xjk)j,andSn=Pi2tSni.ThenVar(^m(t))=OSn n2tProof:IntheappendixATheorem2.1:MSE(^m(t))=O(kHkr)+OSn n2tProof:UnderbothAssumption3andAssumption4,wehaveSn n2t!0andkHkr!0.ThusTheorem2.1isadirectresultfromLemma2.1andLemma2.2.TheexactratesofconvergenceforthemeansquarederrorcanbeobtainedundersomeassumptionsonthedistributionofXi.SupposeX1,,Xnf(X)andsupposethatfiscontinuousandalsohasacontinuoussecondderivative.Letrf(t)representtherstorderderivativeoffattandHf(t)betheHessianmatrixoffatt.Lemma2.3([ 20 ]):Under(A1-A4),fort2RdE[^m(t))]TJ /F3 11.955 Tf 11.95 0 Td[(m(t)]=r0m(t)H2rf(t) f(t)2(K)+1 22(K)tr(HHmH)+o(tr(H2))NotethatthebiasforthetraditionalN-WestimatorundercorrelatederrorsisthesameasthetraditionalN-Westimatorunderani.i.derrorstructure.Therefore,thespatialdependencedoesnotaffectthebiasofthekernelestimator.However,wewill 23

PAGE 24

seethatthevarianceofthetraditionalN-Westimatorisaffectedbythecorrelationstructureinthefollowinglemma.Lemma2.4:Under(A1-A4),fort2Rd,Var(^m(t))=2(K2)(1+f(t)R) f(t)njHj+o1 njHjwhereR<1isasgiveninAssumption4.Proof:IntheappendixA.Ifthereisnospatialdependence,wehaveVar(^m(t))=2(K2) f(t)njHj+o(1 njHj)givenby[ 20 ].Theorem2.2:Under(A1-A4),fort2Rd,theasymptoticmeansquarederror(AMSE)forthetraditionalN-WkernelestimatorisAMSE(^m(t))=r0m(t)H2rf(t) f(t)2(K)+1 22(K)tr(HHmH)2+2(K2)(1+f(t)R) f(t)njHjProof:directlyfollowsfromLemma2.3andLemma2.4.Underassumption(A3),weknowthatmax=minisbounded;therefore,O(max)=O(min).UndertheassumptionthatHissymmetric,wecanwriteH=MMT,whereMisanorthonormalmatrixand)]TJ /F1 11.955 Tf 10.27 0 Td[(isthediagonalmatrixwitheigenvaluesofH.ItisknownthatjHj=Qdi=1i.ThenwehavendminnjHj=nQdi=1indmax.Meanwhile,sinceO(max)=O(min),O(njHj)=O(ndmax)=O(ndmin).Ontheotherhand,usingthedecompositionH=MMT,wecanwritetr(H2)=tr(M)]TJ /F4 7.97 Tf 18.63 5.14 Td[(2MT)=tr(MTM)]TJ /F4 7.97 Tf 18.63 5.15 Td[(2)=tr()]TJ /F4 7.97 Tf 6.94 5.15 Td[(2)=Xi2i.Therefore,d2mintr(H2)d2max.Thus,O(tr(H2))=O(2max)=O(2min). 24

PAGE 25

^m(t)issaidtobeaconsistentestimatorwhenitsMSEgoestozero.BasedontheresultsofLemma2.3andLemma2.4,MSEof^m(t)!0iftr(H2)2+1 njHj!0asn!1,whichimplies4max+1 ndmax!0.Therefore,maxmustconvergetozeroslowerthan1=(n1=d)for^m(t)tobeconsistent.AndsinceH!0andnjHj!1(Assumption3),weknowmaxconvergestozeroslowerthan1=(n1=d).Hence,underassumptions1-4,^m(t)isaconsistentestimatorofm(t)atpointt.Nextweconsidertheasymptoticdistributionof^m(t),whichrequiresanadditionalassumptionontheconvergenceratesofH.Theorem2.3:Undertheassumptionthatmax=O(1 n1=(4+d)),p njHj(^m(t))]TJ /F3 11.955 Tf 11.95 0 Td[(m(t))]TJ /F3 11.955 Tf 11.96 0 Td[(Bn)!N(0,2)where2=2(K2)[1+f(t)R] f(t)andBn=r0m(t)H2rf(t) f(t)2(K)+1 22(K)tr(H2Hm).Proof:Notethatgivenmax=O(1 n1=(4+d)),p njHjtr(H2)p ndmaxd2max=n1 2d4+d 2max=O(1).Therefore,E(p njHj(^m(t))]TJ /F3 11.955 Tf 11.96 0 Td[(m(t))]TJ /F3 11.955 Tf 11.95 0 Td[(Bn))=p njHjo(tr(H2))=o(1).FromTheorem2.3,wehaveVar(p njHj(^m(t))]TJ /F3 11.955 Tf 11.95 0 Td[(m(t))]TJ /F3 11.955 Tf 11.96 0 Td[(Bn))=2+o(1). 2.3BandwidthSelectionUnderSpatialDependenceThetraditionalN-WestimatorusesaweightmatrixH,andthereforethebandwidthHisanimportantpartofthetraditionalestimator.ThebandwidthHcontrolsthe 25

PAGE 26

smoothness,biasandvarianceoftheestimateofthemeanfunction.Whendataareuncorrelated,varioustechniqueshavebeendevelopedfordeterminingsuitablevaluesofthebandwidthfromdata.Amongthem,leave-one-outcrossvalidationisapopularchoice.However,whendataarecorrelated,thecorrelationamongerrorshasaneffect.Here,weassesstheeffectofcorrelationontheleave-one-outcrossvalidationmethod.Leave-one-outcrossvalidationchoosestheHthatminimizes LOOCV(H)=1 nXifYi)]TJ /F7 11.955 Tf 13.8 0 Td[(^mH,)]TJ /F6 7.97 Tf 6.58 0 Td[(i(Xi)g2 (2) where^mH,)]TJ /F6 7.97 Tf 6.59 0 Td[(i(Xi)istheN-Westimatorbasedonthedatawithouttheithobservation.Theorem2.4:FortraditionalN-Wkernelestimator,E(LOOCV(H))=1 nE[Xi(m(Xi))]TJ /F7 11.955 Tf 13.8 0 Td[(^mH,)]TJ /F6 7.97 Tf 6.59 0 Td[(i(Xi))2]+2)]TJ /F7 11.955 Tf 13.15 8.08 Td[(2K(0)[f(X)R)]TJ /F7 11.955 Tf 11.96 0 Td[(1]2 njHjf(X))]TJ /F3 11.955 Tf 11.95 0 Td[(K(0)+o(1)where2isthecommonvarianceoferrorsandRisgivenisassumption4.Proof:IntheappendixA.Notethat,ifthereisnocorrelation,E(LOOCV(H))=1 nE[Pi(m(Xi))]TJ /F7 11.955 Tf 13.88 0 Td[(^mH,)]TJ /F6 7.97 Tf 6.59 0 Td[(i(Xi))2]+2.Inthiscase,E(LOOCV(H)))]TJ /F5 11.955 Tf 13.1 0 Td[(2isanunbiasedestimatorof1 nE[Pi(m(Xi))]TJ /F7 11.955 Tf -416.42 -23.9 Td[(^mH,)]TJ /F6 7.97 Tf 6.59 0 Td[(i(Xi))2].However,undercorrelatederrors,thecorrelationhasanadditionaltermintheexpectation;consequently,theLOOCVmethodmaynotgiveadesirableresult.FollowingtheresultofTheorem2.4,weproposeacriterionofselectingtheoptimalbandwidthasminimizingC1=1 nXifYi)]TJ /F7 11.955 Tf 13.8 0 Td[(^mH,)]TJ /F6 7.97 Tf 6.58 0 Td[(i(Xi)g2+2K(0)[f(X)R)]TJ /F7 11.955 Tf 11.95 0 Td[(1]2 njHjf(X))]TJ /F3 11.955 Tf 11.96 0 Td[(K(0)Notethat,inpractice,weusuallydonotknowR;therefore,itisdifculttoapplythiscriterion. 26

PAGE 27

AnalternativeapproachtoidentifyingtheoptimalbandwidthischoosingtheonethatminimizestheMeanIntegratedSquaredError(MISE):MISE(H)=ZMSE(H)f(X)dX=ZE[^m(X,H))]TJ /F3 11.955 Tf 11.95 0 Td[(m(X)]2f(X)dXInthesimplecase,H=hnId,forascalerhn.Aftersometediouscalculation,theasymptoticMISE(AMISE)of^mH(t)isfoundtobeAMISE(hn)=1 422(K)h4nC1+2(K2) nhdn+2(K2)R nhdnwhereC1isaconstant,whichdoesnotdependonnorhn,butdependsontheunknownfunctionm(X).NotethatAMISEalsodependsonR.Therefore,weturntotheWMASE,whichisdenedasWMASE(H)=1 nXiE[^mH(Xi))]TJ /F3 11.955 Tf 11.96 0 Td[(m(Xi)]2w(Xi)wherew(Xi)fori=1,2,,naresomeweightfunctions.ItiseasytoseethatXi[Yi)]TJ /F7 11.955 Tf 13.8 0 Td[(^mH(Xi))2w(Xi)]=Xi[(Yi)]TJ /F3 11.955 Tf 11.96 0 Td[(m(Xi)+m(Xi))]TJ /F7 11.955 Tf 13.8 0 Td[(^mH(Xi))2w(Xi)]=Xiw(Xi)[(Yi)]TJ /F3 11.955 Tf 11.95 0 Td[(m(Xi))2+(m(Xi))]TJ /F7 11.955 Tf 13.8 0 Td[(^mH(Xi))2+2(Yi)]TJ /F3 11.955 Tf 11.96 0 Td[(m(Xi))(m(Xi))]TJ /F7 11.955 Tf 13.8 0 Td[(^mH(Xi))] (2)Takingtheexpectationonbothsidesofequation(2-4),wehaveE[Xi(Yi)]TJ /F7 11.955 Tf 13.8 0 Td[(^mH(Xi))2w(Xi)]=2+nWAMSE(H)+2Xiw(Xi)E[(Yi)]TJ /F3 11.955 Tf 11.96 0 Td[(m(Xi))(m(Xi))]TJ /F7 11.955 Tf 13.8 0 Td[(^mH(Xi))]=2+nWAMSE(H))]TJ /F7 11.955 Tf 11.96 0 Td[(2Xiw(Xi)Cov(Yi,^mH(Xi)) 27

PAGE 28

WritingY=(Y1,Y2,,Yn)TandLi=KH(X1)]TJ /F4 7.97 Tf 6.58 0 Td[(Xi) PjKH(Xj)]TJ /F4 7.97 Tf 6.59 0 Td[(Xi),,KH(Xn)]TJ /F4 7.97 Tf 6.59 0 Td[(Xi) PjKH(Xj)]TJ /F4 7.97 Tf 6.59 0 Td[(Xi)T,wehave^mH(Xi)=LTiY.Denoting(i)n=(n(Xi)]TJ /F8 11.955 Tf 11.96 0 Td[(X1),,n(Xi)]TJ /F8 11.955 Tf 11.96 0 Td[(Xn))T,weobtainCov(Yi,^mH(Xi))=2LTi(i)nTherefore,1 nE[Xi(Yi)]TJ /F7 11.955 Tf 13.8 0 Td[(^mH(Xi))2w(Xi)]=2 n+WAMSE(H))]TJ /F7 11.955 Tf 13.15 8.09 Td[(22 nXiw(Xi)LTi(i)nThus,1 nPi(Yi)]TJ /F7 11.955 Tf 14.84 0 Td[(^mH(Xi))2w(Xi)+22 nPiw(Xi)LTi(i)nisanunbiasedestimatorof2 n+WAMSE(H).Followingtheresultabove,wecanspecifythecriterionasselectingthebandwidthHthatminimizes C2=1 nXi(Yi)]TJ /F7 11.955 Tf 13.8 0 Td[(^mH(Xi))2w(Xi)+22 nXiw(Xi)LTi(i)n).(2) 2.4SpatialN-WEstimatorThetraditionalN-Westimatorwasoriginallyproposedundertheassumptionofindependence.Therefore,thefactthatnoneoftheinformationprovidedbytheerrorcovariancestructureisusedinitsconstructionsuggeststhatanalternativeestimatorthatincorporatesthecorrelationinformationmightprovideimprovedperformance.Heresuchanestimator,whichwecallthespatialN-Westimator,issuggested.DenoteVar()=2andassumeisnonsingular.LetP=(P1,P1,,Pn)T=)]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 2Y+(I)]TJ /F8 11.955 Tf 11.95 0 Td[()]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 2)m(X).ThenE(P)=m(X)Var(P)=2IFrom(2-1),weknowYisnormallydistributed,andthusP1,P2,,PnareindependentlydistributedandhavethesamemeanvectorasY=(Y1,Y2,,Yn)T.Therefore,weconsiderusingPtoestimatem(X). 28

PAGE 29

First,assumeHrepresentstheweightmatrixusedforthetraditionalN-WestimatorbasedontheC2criterionandHistheweightmatrixusedforthespatialN-Westimator,whichwillbegivenlaterinthissection.Similartotheassumptiongivenin[ 29 ],wegiveanadditionalassumptionforthefollowingwork.Assumption2.5:ForweightmatrixH,njHj!1,H!0,andmax=minisbounded,wheremaxandminarethemaximumandminimumeigenvaluesofH.Further,max max=min min=o(1),n=O()]TJ /F4 7.97 Tf 10.82 0 Td[((4+d)max)Letm(t)=PiKH(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t)Pi PjKH(Xj)]TJ /F8 11.955 Tf 11.95 0 Td[(t)SincePareindependent,wehavethat([ 20 ])E[m(t))]TJ /F3 11.955 Tf 11.96 0 Td[(m(t)]=r0m(t)H2rf(t) f(t)2(K)+1 22(K)tr(H2Hm)+o(tr(H2))Var[m(t)]=2(K2)R f(t)njHj+o(1 njHj)Theasymptoticdistributionofm(t)isgiveninthefollowinglemma.Moreover,wewillproposeaspatialN-Westimatorsimilartom(t)thathasthesameasymptoticdistributionasm(t).Lemma5:Underassumptions1-5,p njHj(m(t))]TJ /F3 11.955 Tf 11.96 0 Td[(m(t))]TJ /F3 11.955 Tf 11.96 0 Td[(Bn)!N(0,21)whereBn=r0m(t)H2rf(t) f(t)2(K)+1 22(K)tr(H2Hm)and21=2(K2) f(t)Note,sinceP1,P2,,Pnareindependent,lemma2.5istheresultin[ 20 ].Here,thebiasofthesmootherm(t)isthesameasthatoftraditionalN-Westimator;however,m(t)hasasymptoticvarianceproportionalto2.Mostofthecovariancestructuresofspatialdataarepositivelycorrelated,e.g.,Gaussiancovariancefunction,exponentialcovariancefunction.Hencem(t)ismoreefcientthanthetraditionalN-Westimator^m(t). 29

PAGE 30

However,inpractice,wedonotknowPbecauseitdependsontheunknownquantitym(X).Therefore,weconstruct ^P=)]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 2Y+(I)]TJ /F8 11.955 Tf 11.95 0 Td[()]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 2)^m(X)=(^P1,^P2,,^Pn)T(2)where^m(X)isthetraditionalN-Westimator.ThenthespatialN-Westimatoris~m(t)=PiKH(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t)^Pi PjKH(Xj)]TJ /F8 11.955 Tf 11.95 0 Td[(t)Actually,ourestimationprocedurehastwosteps: Step1:Obtainaninitialestimateofm(x)usingthetraditionalN-Westimator^m(x). Step2:Construct^Pusingtheformula(4.1)andthenconstructthespatialN-Westimator~m(x)Theorem2.5:Denote()]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 2)ij=vij.UnderAssumption1-5,andsupifPj(vij)g=O(1)p njHj(~m(t))]TJ /F3 11.955 Tf 11.96 0 Td[(m(t))]TJ /F3 11.955 Tf 11.96 0 Td[(Bn)!N(0,21)whereBnand21aredenedinlemma2.5.NotethatthespatialN-Westimator~m(t)isasymptoticallyequivalenttom(t)andhenceismoreefcientthan^m(t). 2.5SpatialDependenceParametersEstimationBoththebandwidthselectioncriterionC2inChapter2andthespatialN-Westimatoraredependentonthespatialcovariancestructure2n.Inpractice,thecovariancestructureisusuallyunknownandmustbeestimated.Here,weassumethecovariancematrixhasaparametricformanddependsontheparameter.Beforewediscussthemethodofestimatingtheparameter,werstconsiderestimatingthesemi-variogram,whichisanimportantcharacteristicofaspatialprocess.Forthestationaryerrorprocessf"(s),s2Rdg,thefunctionC(d)=Cov("(s+d),"(s)) 30

PAGE 31

iscalledthecovariogramoftheerrorprocess.Thesemi-variogramisdenedasr(d)=1 2Var("(s+d))]TJ /F5 11.955 Tf 11.95 0 Td[("(s))Forastationaryprocess,asimplerelationshipexistsbetweenthesemi-variogramandcovariogram:C(d)=2)]TJ /F3 11.955 Tf 11.96 0 Td[(r(d)where2=C(0)isthevarianceof"(s).Inaddition,weassumethesemi-variogramisisotropic,whichmeansr(d)=r(kdk).Thesemi-variogramisestimatedinthefollowingtwosteps: 1. Calculatetheempiricalsemi-variogram.Givenadistanced,wedeneS(d,)=f(i,j):d)]TJ /F5 11.955 Tf 11.95 0 Td[(ksi)]TJ /F8 11.955 Tf 11.95 0 Td[(sjk0isthetolerancevalue,whichisusuallysmall.Letn(d,)bethenumberofelementsinS(d,).Thentheempiricalsemi-variogramisgivenby^r(d)=1 2n(d,)X(i,j)2S(d,)("i)]TJ /F5 11.955 Tf 11.96 0 Td[("j)2And,ifthetolerancevalueissmall,weknowthat^r(d)isanunbiasedestimatorofr(d).Following[ 10 ],wecanselectasequenceoffdkgKk=1with0
PAGE 32

Supposeforeachdesignpoint,thereareJrepeatedmeasurementYi1,Yi2,,YiJ,i=1,,n.NotethatCov(Yi.,Yi.)=Cov("i.,"i.)=1 JC(ksi)]TJ /F8 11.955 Tf 11.96 0 Td[(sik).Inordertogettheunbiasedestimatorofthesemi-variogramr(d),werstconsiderthefollowingtwoequations:E"1 J)]TJ /F7 11.955 Tf 11.95 0 Td[(1Xj(Yij)]TJ /F7 11.955 Tf 13.89 2.66 Td[(Yi.)(Yij)]TJ /F7 11.955 Tf 13.89 2.66 Td[(Yi.)#=C(ksi)]TJ /F8 11.955 Tf 11.96 0 Td[(sik)andE"1 n(J)]TJ /F7 11.955 Tf 11.95 0 Td[(1)XiXj(Yij)]TJ /F7 11.955 Tf 13.89 2.65 Td[(Yi.)2#=2.Notethat,therstequationgivesanunbiasedestimatorforthecovariancefunction,andthesecondequationprovidesanunbiasedestimatorforthecommonvariance2.Hence,1 n(J)]TJ /F7 11.955 Tf 11.95 0 Td[(1)XiXj(Yij)]TJ /F7 11.955 Tf 13.89 2.65 Td[(Yi.)2)]TJ /F7 11.955 Tf 23.81 8.08 Td[(1 J)]TJ /F7 11.955 Tf 11.96 0 Td[(1Xj(Yij)]TJ /F7 11.955 Tf 13.89 2.65 Td[(Yi.)(Yij)]TJ /F7 11.955 Tf 13.89 2.65 Td[(Yi.)isanunbiasedestimatorofthesemi-variogramr(ksi)]TJ /F8 11.955 Tf 12.51 0 Td[(sik).Therefore,theempiricalsemi-variogramisgivenby^r(d)=1 n(d,)X(i,i)2S(d,)f1 n(J)]TJ /F7 11.955 Tf 11.95 0 Td[(1)XiXj(Yij)]TJ /F7 11.955 Tf 13.89 2.66 Td[(Yi.)2)]TJ /F7 11.955 Tf 23.81 8.09 Td[(1 J)]TJ /F7 11.955 Tf 11.96 0 Td[(1Xj(Yij)]TJ /F7 11.955 Tf 13.89 2.66 Td[(Yi.)(Yij)]TJ /F7 11.955 Tf 13.89 2.66 Td[(Yi.)g=1 n(J)]TJ /F7 11.955 Tf 11.95 0 Td[(1)XiXj(Yij)]TJ /F7 11.955 Tf 13.89 2.66 Td[(Yi.)2)]TJ /F7 11.955 Tf 27.36 8.09 Td[(1 n(d,)X(i,i)2S(d,)f1 J)]TJ /F7 11.955 Tf 11.96 0 Td[(1Xj(Yij)]TJ /F7 11.955 Tf 13.89 2.66 Td[(Yi.)(Yij)]TJ /F7 11.955 Tf 13.89 2.66 Td[(Yi.)gApplyingthetwo-stepproceduredenedinthissection,wecanobtainestimatesofthespatialdependenceparameters.Whenthereisonlyonerealizationateachpointinthedesign,estimationofthespatialcorrelationismoredifcult.Onewaytoestimateparametersinthecovariancemodelisusingresidualsfromaninitialestimateofthemeanfunction(theso-calledpilotmethod).Thegeneralalgorithmisasfollows([ 15 ]): 32

PAGE 33

1. ObtainapilotbandwidthmatrixH=Hpilot(forinstance,usingthestandardleave-one-outcrossvalidationcriterion). 2. UsingthetraditionalN-Westimator,withbandwidthmatrixHpilot,obtaintheresiduals:^"i=Yi)]TJ /F7 11.955 Tf 13.8 0 Td[(^mH(Xi)i=1,2,,n. 3. Computetheclassicalsemi-variogramestimatorfromtheresiduals:^r(dk)=1 2n(dk,)X(i,j)2S(dk,)(^"i)]TJ /F7 11.955 Tf 12.53 0 Td[(^"j)2foragivensequence0
PAGE 34

Theerrors"i,jhavezeromeanandsatisfyCov("i,j,"i,j)=8><>:220kxi)]TJ /F4 7.97 Tf 6.59 0 Td[(xikj=j0otherwiseThreepairsoff,g:(1.00,0.70),(1.00,0.50)and(0.50,0.30)weresimulated.Foreachpair,1000simulationswereconducted.TherststepistotestthespatialdependenceparameterestimationmethodforrepeatedmeasurementsmentionedinChpater2.5.Themeanand,inparentheses,thestandarderroroftheestimatedandbasedon1000simulationsareshowninTable 2-1 .Wecanseetheestimatedparametersareclosetothetruevaluesandhavesmallstandarderrors.Thenextstepistocomparethethreenonparametricestimators.Threetypesofestimatorshavebeencalculatedusingtheestimatedspatialdependenceparametersforthethreepairs.TheEpanechnikovkernelisusedhere,andtheweightmatrixisH=hI2,whereh=(h1,h2)T.TheaveragemeansquarederrorsandbiasesaregiveninTables 2-2 and 2-3 .LOOCVN-WstandsforthetraditionalN-Westimatorusingleave-one-outcrossvalidationmethodtochoosethebandwidth.C2N-WisthetraditionalN-WestimatorusingC2criterionweproposedinChapter2.3andSpatialN-WisthespatialN-Westimatorbasedonthetwo-stepprocedureproposedinChapter2.4.Fortheaveragemeansquarederrors,numbersinparenthesesaretheassociatedstandarderrors.Forallthreepairsof(,),thespatialN-Westimatorhasthesmallestbiasandmeansquarederror.TheN-Westimatorusingleave-one-outcrossvalidationcriterionhasthelargestbiasesandmeansquarederrorunderallthreeconditions,indicatingtheleave-one-outcriterionperformspoorlyunderspatiallycorrelatederrors.InFigure2-1,weplottheaveragedifferencesbetweenthetruemeanandthreeestimatorsbasedon1000simulations.Fromthegraph,wecanalsoseethat,ingeneral, 34

PAGE 35

thespatialN-WestimatorhasthesmallestbiaswhiletheLOOCVN-Westimatorhasthelargestbias. 2.6.2SimulationStudy2Inthissimulationstudy,we'dliketoevaluatetheperformancesofthethreeestimatorsbyconductingsimulationstudieswithdifferentsamplesize.Werandomlygeneratedesignpointsxi=(xi,1,xi,2),wherexi,sN(i,1),andiN(2,2),s=1,2.Thenwegenerate5repeatedmeasurementsoftheresponsevariableatthesedesignedpointsfollowingthetestingmodel:Yi,j=xi,1+xi,2+xi,1xi,2+"i,j.Theerrors"i,jhavezeromeanandsatisfyCov("i,j,"i,j)=8><>:2exp()]TJ /F5 11.955 Tf 9.3 0 Td[(kxi)]TJ /F8 11.955 Tf 11.96 0 Td[(xik2)j=j0otherwiseWeusedthreesamplesizes,n1=50,n2=100andn3=150with2=1and=0.25.Threetypesofestimatorshavebeencalculatedforeachofthesamples.TheEpanechnikovkernelisusedhere,andtheweightmatrixisH=hI2,whereh=(h1,h2)T.Theaveragemeansquarederrorsandbiasesbasedon1000simulationsaregiveninTables 2-4 andTable 2-5 .Forallthreesizes,thespatialN-Westimatorhasthesmallestbiasandmeansquarederror.TheN-Westimatorusingleave-one-outcrossvalidationcriterionhasthelargestbiasesandmeansquarederrorsunderallthreeconditions,indicatingtheleave-one-outcriterionperformspoorlyunderspatiallycorrelatederrors.Thetablesalsoshowthattheaveragemeansquarederrorsandbiasesdecreasedwhenthesamplesizeincreased.InFigures2-2to2-4,weplottheaveragedifferencesbetweenthetruemeanandthreeestimatorsbasedon1000simulationsforthedifferentsizes.Fromthegraphs,we 35

PAGE 36

canalsoseethat,ingeneral,thespatialN-WestimatorhasthesmallestbiaswhiletheLOOCVN-Westimatorhasthelargestbias. 2.6.3SimulationStudy3Asecondsimulationstudywasconductedinwhichtheassumedcovariancestructurewasmisspecied.Thepurposewastoassesswhethertheefciencyoftheestimatorswoulddecrease.Intherstsetofsimulations,werandomlygenerate50designpointsxi=(xi,1,xi,2),wherexi,sN(i,1),andiN(2,2),s=1,2.Thenwegenerated5repeatedmeasurementsoftheresponsevariableatthesedesignpointsfollowingthetestingmodel:Yi,j=xi,1xi,2+"i,jfori=1,2,,50,j=1,2,5Theerrors"i,jhavezeromeanandsatisfyCov("i,j,"i,j)=8><>:2exp()]TJ /F5 11.955 Tf 9.3 0 Td[(kxi)]TJ /F8 11.955 Tf 11.96 0 Td[(xik2)j=j0otherwiseWeset2=1and=0.25.Inthesimulation,weassumethatthecovariancestructurehasthefollowingform:Cov("i,j,"i,j)=8><>:2exp()]TJ /F5 11.955 Tf 9.3 0 Td[(kxi)]TJ /F8 11.955 Tf 11.95 0 Td[(xik)j=j0otherwiseThesamethreeestimatorsasintherstsimulationstudywereconsidered.EachwascalculatedusingtheEpanechnikovkernel.TheweightmatrixwasassumedtobeH=hI2,whereh=(h1,h2)T.TheaveragemeansquarederrorsandbiasesaregiveninTable 2-6 .Fortheaveragemeansquarederrors,numbersinparenthesesaretheassociatedstandarderrors.Therefore,wecanseethat,althoughtheassumedcovariancematrixismisspecied,thetwo-stepestimatorstillworksbetterthantheothertwo.Figure2-5showsthe 36

PAGE 37

differencesbetweenthetruefunctionandthethreeestimators.Fromthisgraph,wecanalsoseethatthespatialN-Westimatorhastheleastbiasamongthethethreeestimators.Inthesecondsetofsimulations,wegenerateddataunderi.i.d.assumptionoftheerrors.Werandomlygenerate50designpointsxi=(xi,1,xi,2),wherexi,sN(i,1),andiN(2,2),s=1,2.Thenwegenerated5repeatedmeasurementsoftheresponsevariableatthesedesignedpointsfollowingthetestingmodel:Yi,j=xi,1+xi,2+xi,1xi,2+"i,jfori=1,2,,50,j=1,2,5.Theerrors"i,jweregeneratedfromi.i.d.N(0,2e).Inthissimulation,weset2e=1.2.Wefurtherassumedthatspatialcorrelationwasresent,andtheerrorshavethecovariancestructureasfollow:Cov("i,j,"i,j)=8><>:2exp()]TJ /F5 11.955 Tf 9.3 0 Td[(kxi)]TJ /F8 11.955 Tf 11.95 0 Td[(xik)j=j0otherwiseUsingthespatialdependenceparameterestimationmethodforrepeatedmeasurementsgiveninChapter2.5,wehavetheestimated2eandare^2e=1.34and^=3.67.Thesamethreeestimatorsasintherstsimulationstudywereconsidered.EachwascalculatedusingtheEpanechnikovkernel.TheweightmatrixwasassumedtobeH=hI2,whereh=(h1,h2)T.TheaveragemeansquarederrorsandbiasesaregiveninTable 2-7 .Fortheaveragemeansquarederrors,numbersinparenthesesaretheassociatedstandarderrors.Fromthetable,wecanseethatwhenthetruedataareindependent,eventhoughwemisspeciedthecorrelationfunction,thespatialN-WestimatorandtheN-WestimatorusingC2criterionperformasgoodasorevenbetterthanthetraditionalN-Westimatorusingleave-one-outcrossvalidationmethod.Thisisalsoseeninthegraphofthedifferencesbetweenthetruefunctionandthreeestimators(seeFigure2-6). 37

PAGE 38

Table2-1. Estimatedspatialdependenceparameters ^^ 1.000.700.984(0.12)0.672(0.07) 1.000.500.985(0.05)0.483(0.05) 0.50.30.497(0.03)0.287(0.08) Table2-2. ComparisonoftheaveragemeansquarederrorforthreeN-Westimators LOOCVN-WC2N-WSpatialN-W 1.000.700.641(0.23)0.387(0.19)0.323(0.19) 1.000.500.34(0.1)0.145(0.07)0.108(0.07) 0.50.30.08(0.005)0.04(0.01)0.03(0.01) Table2-3. ComparisonoftheaveragedbiasforthreeN-Westimators LOOCVN-WC2N-WSpatialN-W 1.000.700.150.040.006 1.000.500.220.080.02 0.50.30.110.050.03 Table2-4. ComparisonoftheaveragemeansquarederrorforthreeN-Westimatorswithdifferentsamplesizes SamplesizeLOOCVN-WC2N-WSpatialN-W 501.19(1.05)1.12(1.05)1.08(1.06) 1001.06(0.99)1.03(1.00)1.01(1.07) 1501.006(1.02)1.015(1.04)1.00(1.02) Table2-5. ComparisonoftheaveragedbiasforthreeN-Westimatorswithdifferentsamplesizes SamplesizeLOOCVN-WC2N-WSpatialN-W 50-0.017-0.015-0.013 1000.0090.0080.004 150-0.005-0.004-0.003 Table2-6. ComparisonoftheaveragemeansquarederrorandaveragebiasforthreeN-Westimatorswhenthecovariancestructureismisspecied EstimatorAveragesquarederrorAveragebias LOOCVN-W1.3(0.89)0.003(0.0075) N-WC21.1(0.89)0.0008(0.0007) SpatialN-W1.03(0.88)0.0003(0.0004) Table2-7. ComparisonoftheaveragemeansquarederrorandaveragebiasforthreeN-Westimatorswhendataareindependent EstimatorAveragesquarederrorAveragebias LOOCVN-W0.36(0.13)0.006(0.001) N-WC20.31(0.11)0.0057(0.001) SpatialN-W0.32(0.12)0.0057(0.001) 38

PAGE 39

Figure2-1. ThedifferencesbetweentruemeanandthreeN-Westimatorswhenerrorsarespatiallycorrelated 39

PAGE 40

Figure2-2. ThedifferencesbetweentruemeanandthreeN-Westimatorswhenerrorsarespatiallycorrelatedwithsamplesize50 40

PAGE 41

Figure2-3. ThedifferencesbetweentruemeanandthreeN-Westimatorswhenerrorsarespatiallycorrelatedwithsamplesize100 41

PAGE 42

Figure2-4. ThedifferencesbetweentruemeanandthreeN-Westimatorswhenerrorsarespatiallycorrelatedwithsamplesize150 42

PAGE 43

Figure2-5. ThedifferencesbetweenthetruemeanandthreeN-Westimatorswhenthecovariancestructureismisspecied 43

PAGE 44

Figure2-6. ThedifferencesbetweenthetruemeanandthreeN-Westimatorswhendataareindenpendent 44

PAGE 45

CHAPTER3ADDITIVESPATIALN-WESTIMATORUNDERSPATIALLYCORRELATEDERRORS 3.1BackgroundAnadditivenonparametricregressionmodelhastheform E(YsjXs)=m(Xs)=+dXi=1mi(Xis)(3)whereYsisascalardependentvariable,X=(X1s,,Xds)isavectorofexplanatoryvariables,isaconstantandfmj()gdj=1isasetofunknownfunctionssatisfyingEXj[mj(Xj)]=0,wherefj(Xj)isthemarginaldensityfunctionofXj.Toourknowledge,thenonparametricadditivemodelin(3-1)wasrstconsideredinanon-statisticalcontextby[ 25 ]whocalleditadditiveseparable.Inthestatisticalliterature,thenonparametricadditivemodelswererstintroducedintheearly1980's,leadingtothedevelopmentofavarietyoftheoreticalresultsandapplications,e.g.[ 42 ].Comparedtogeneralnonparametricmodels,theadditivemodelsareattractivebecausetheygeneralizethelinearregressionmodelsandallowinterpretationofmarginalchanges,suchastheeffectofasingleexplanatoryvariable.Theyarealsointerestingfromatheoreticalpointofviewbecausetheycombineexiblenonparametricmodelingofmanyvariableswiththestatisticalprecisionthatistypicalforjustoneexplanatoryvariable.Forthetwodimensionalcase,letm(x1,x2)beabivariateregressionfunction;anadditivesub-modelism(x1,x2)=+m1(x1)+m2(x2).[ 38 ]showedthatbothm1andm2canbeestimatedwiththeone-dimensionalconvergencerateofn2=5,wherenisthesamplesize.Therefore,theadditivemodelsavoidtheso-called'curseofdimensionality'thataffectsthegeneralnonparametricmodels.Intheearlydevelopmentofadditivemodels,estimationoftheadditivecomponentswasbasedontheback-ttingtechnique,whichprojectsthedataontothespaceofadditive 45

PAGE 46

functionsviatheleastsquaresmethod,anditerativealgorithms.Thewidely-usedHastie&Tibshiraniestimationprocedurein[ 23 ]involvesmultipleiterations,wheretheadditivestructureisusedineachsteptoobtainestimatesofm1andm2.Adisadvantageisthatitsstatisticalpropertiesarenotwellunderstood.Toavoidthedifcultyoftheiterativeprocedures,[ 28 ]proposedmarginalintegrationoftheregressionfunctionforestimatingtheadditivecomponents.Supposem(X)isoftheadditiveform,andf()]TJ /F6 7.97 Tf 6.58 0 Td[(j)isthejointdensityof(Xi1,,Xi(j)]TJ /F4 7.97 Tf 6.58 0 Td[(1),Xi(j+1),,Xid).Thenforanyxedx2R,theapproachin[ 28 ]isbasedonthefactthatmj(x)+=Zm(x1,,x,,xd)f()]TJ /F6 7.97 Tf 6.59 0 Td[(j)(x1,,xd)Ys6=jdxsprovidedEXj[mj(Xj)]=0,forj=1,,d.Thekeypointofthemarginalintegrationapproachistoestimatethewholesurfaceoftheregressionfunctionbylocalpolynomialttingrst,andthentotakethemarginalaverageoftheestimatorsoftheregressionfunctiontoobtaintheestimatorofeachadditivecomponent.Asthismethodprovidesclosed-formestimators,theasymptoticpropertiescanbeinvestigatedeasily,andhencehavebeenextensivelystudied.[ 21 ]usedmarginalsmoothingonindependentpredictorvariables.[ 8 ]consideredthesamemethodwiththeNadaraya-Watson(N-W)estimator.[ 34 ]extendedthemethodtoestimateboththeregressionfunctionanditsderivativesimultaneouslywiththelocalpolynomialtting.However,theaboveworkswereallbasedonthei.i.d.assumption.Inthischapter,weconsidertheadditivemodelwithspatiallycorrelatederrors.Thisproblemhasreceivedlittleattentionbefore.[ 7 ]consideredtheadditivemodelwithdependentdata,differentfromthei.i.d.assumption.Werstremovetheindependenceassumptionandassumethatspatialcorrelationexists.Secondly,adirectestimatorbasedontheN-Westimatorandusingthemarginalintegrationprocedureof[ 28 ]isdeveloped,therebyavoidingiterativeapproximationasinback-tting.Third,the 46

PAGE 47

explanatoryvariablesareallowedtobecorrelatedwithajointdensityfthatdoesnotfactorize.Finally,thedimensionofXisextendedtod2asin[ 28 ].Theoutlineofthischapterisasfollows.InChapter3.2,wepresentthemodelaswellastheestimationprocedure,andtheoreticalresultsoftheadditivespatialN-Westimatoraregiven.InChapter3.3,simulationresultswillbeshown. 3.2ModelandEstimatorAnadditivenonparametricregressionmodelwithspatialcorrelationhastheform Ys=m(Xs)+s (3) where=(1,,n)hasanormaldistributionwithE(i)=0andCov(i,j)=2n(kXi)]TJ /F8 11.955 Tf 11.96 0 Td[(Xjk).m(Xs)hasanadditivestructureoftheform m(Xs)=+dXi=1mi(Xis) (3) whereYisascalardependentvariableandXs=(X1s,X2s,,Xds)Tisavectorofexplanatoryvariableatlocations,fors=1,2,,n.Forthecorrelationfunctionn,Euclideandistanceisacommonmeasureofthedistancebetweentwopoints.Thespatialcorrelationfunctionnincludesfrequentlyusedcorrelationstructures,suchastheexponentialandGaussianfunctions.ThejointdensityofXsisgivenbyf(X),X=(X1,,Xd)T,andfi(Xi)isthemarginaldensityoftheithrandomvariable.Notethatwedonotassumethatf(X)=Qni=1fi(Xi).Therefore,allresultsapplytocovariates,whetherornottheyarecorrelated. 47

PAGE 48

Moreover,weassumetheunknownfunctionmi(xi)haszeroexpectation;thatis,Rmi(xi)fi(xi)dxi=0fori=1,2,,d.Foranyxedvaluexk,wedeneX(k)s=(X1s,,X(k)]TJ /F4 7.97 Tf 6.59 0 Td[(1)s,xk,X(k+1)s,,Xds)TandX()]TJ /F6 7.97 Tf 6.58 0 Td[(k)s=(X1s,,X(k)]TJ /F4 7.97 Tf 6.58 0 Td[(1)s,X(k+1)s,,Xds)withdensitydenotedbyf)]TJ /F6 7.97 Tf 6.59 0 Td[(k(X()]TJ /F6 7.97 Tf 6.59 0 Td[(k)s).E(m(X(k)s))=+mk(xk),foranyxedvaluexk.Therefore,theestimatorofmk(xk)isgivenby ~mk(xk)=1 nnXs=1^m(X(k)s))]TJ /F7 11.955 Tf 13.89 2.66 Td[(Y (3) where^m(X(k)s)isthemultivariateN-Westimatorofm(X(k)s).ThemultivariateN-Westimatorofm(X(k)s)is^m(X(k)s)=nXt=1KH(Xt)]TJ /F8 11.955 Tf 11.95 0 Td[(X(k)s)Yt Pnj=1KH(Xj)]TJ /F8 11.955 Tf 11.96 0 Td[(X(k)s)=nXt=1Khk(Xtk)]TJ /F3 11.955 Tf 11.96 0 Td[(xk)KH(X)]TJ /F6 7.97 Tf 6.59 0 Td[(kt)]TJ /F8 11.955 Tf 11.96 0 Td[(X)]TJ /F6 7.97 Tf 6.58 0 Td[(ks)Yt Pnj=1Khk(Xjk)]TJ /F3 11.955 Tf 11.95 0 Td[(xk)KH(X)]TJ /F6 7.97 Tf 6.59 0 Td[(kj)]TJ /F8 11.955 Tf 11.95 0 Td[(X)]TJ /F6 7.97 Tf 6.59 0 Td[(ks) (3)wherethemultivariatekernelfunctionK(X)istheproductoftheunivariatekernelfunctionK,i.e.,K(X)=Qdi=1K(Xi).Further,weassumetheweightmatrixH=hId,whereh=(h1,,hd)T.Notationally,hkisthebandwidthforthekthcovariate,andHisthe(d)]TJ /F7 11.955 Tf 11.95 0 Td[(1)(d)]TJ /F7 11.955 Tf 11.95 0 Td[(1)bandwidthmatrixwithoutthebandwidthforthekthcovariate.Substituting(3-5)in(3-4),wehavetheestimatorofmk(xk)basedonmultivariateN-Westimator.Beforeexploringthepropertiesoftheestimatorhavingform(3-4),wesummarizethemainassumptionshere.Assumption1:RK(u)du=1,RuK(u)du=0andRu2K(u)du=2(K),RK2(u)du=(K2) 48

PAGE 49

Assumption2:aK(u)bforsomeconstants00suchthat8u1,u2jK(u1)]TJ /F3 11.955 Tf 11.96 0 Td[(u2)jLju1)]TJ /F3 11.955 Tf 11.95 0 Td[(u2j.Assumption3:limn!1nZn(kXk)dX=R<1nZjn(kXk)jdX=O(1)limn!1nZI(kH)]TJ /F11 5.978 Tf 5.75 0 Td[(1Xk")jn(kXk)jdX=0Assumption4:Thedensitiesfj(),f)]TJ /F6 7.97 Tf 6.58 0 Td[(j()andf()arebounded,andcontinuouslytwicedifferentiable.Thed-dimensionalrandomvectorsXsarei.i.d.Assumption5:Allthesecondderivativesofm(x)existandarecontinuous.mj(xj)haszeromeanandboundedsecondmoment.Assumption1andassumption2arethebasicassumptionsforthekernelfunctions.Andassumption3setssomeconstraintsonthecovariancematrix.Anexampleofavalidcorrelationfunctionisn(x)=exp()]TJ /F5 11.955 Tf 9.3 0 Td[(n1=2kxk).Assumptions4and5arecommonassumptionsformarginalintegrationtechniques.InLemmas3.1and3.2,werstconsiderthemeanandvarianceof(3-4)basedonthemultivariateN-Westimatorin(3-5)undertheaboveassumptions.Lemma3.1:Under(A1)-(A5)statedabove,andforsomevaluexj,lethjbethebandwidthofthejthcovariate,Hbethe(d)]TJ /F7 11.955 Tf 12.12 0 Td[(1)(d)]TJ /F7 11.955 Tf 12.12 0 Td[(1)bandwidthmatrixwithoutthebandwidthforthejthcovariate.Ifthebandwidthsarechosensuchthathj!0,H!0, 49

PAGE 50

hj=O(n)]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 5)andlimn!1nhjjHj!1,thenE(~mj(xj))]TJ /F3 11.955 Tf 11.95 0 Td[(mj(xj))=bj+o(h2j)wherebj=bj1+bj2withbj1=h2j2(K) 2Z[dXs6=jms(Xls)f)]TJ /F6 7.97 Tf 6.58 0 Td[(j(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l) f(X(j)l)f00j(X(j)l)dX()]TJ /F6 7.97 Tf 6.58 0 Td[(j)l]andbj2=h2j2(K) 2[1 2m00j(xj)+m0j(xj)Zf)]TJ /F6 7.97 Tf 6.59 0 Td[(j(X()]TJ /F6 7.97 Tf 6.58 0 Td[(j)l) f(X(j)l)f0j(X(j)l)dX()]TJ /F6 7.97 Tf 6.58 0 Td[(j)l]Lemma3.2:Forsomevaluexj,andwiththesameassumptionsasinLemma3.1,Var(~mj(xj))=2 nhj(K2)Zf2)]TJ /F6 7.97 Tf 6.59 0 Td[(j(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l) f(X(j)l)dX()]TJ /F6 7.97 Tf 6.58 0 Td[(j)l+R nhj(K2)[Zf)]TJ /F6 7.97 Tf 6.58 0 Td[(j(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l)dX()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l]2+o(1 nhj)=2j+o(1 nhj)with2j=2 nhj(K2)Rf2)]TJ /F13 5.978 Tf 5.75 0 Td[(j(X()]TJ /F13 5.978 Tf 5.76 0 Td[(j)l) f(X(j)l)dX()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l+R nhj(K2)[Rf)]TJ /F6 7.97 Tf 6.59 0 Td[(j(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l)dX()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l]2,where2isthevarianceofifori=1,2,,nandRisdenedinassumption3.Lemmas3.1and3.2givetheexpectationandvarianceoftheadditivecomponentestimatorsbasedonthemarginalintegrationmethod.FromtheassumptionsinLemma3.1,weknowthathj=O(n)]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 5),andthereforeO((nhj)1 2h2j)=O(n1 2h(1 2+2)j)=O(n1 2(n)]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 5)5 2)=O(1).Thus,basedonLemma3.1andLemma3.2,wehavethefollowingtheorem. 50

PAGE 51

Theorem3.1:UnderthesameassumptionsinLemma3.1,p nhj(~mj(xj))]TJ /F3 11.955 Tf 11.95 0 Td[(mj(xj))]TJ /F3 11.955 Tf 11.95 0 Td[(bj)!N(0,2j)where2j=nhj2j=2(K2)Rf2)]TJ /F13 5.978 Tf 5.75 0 Td[(j(X()]TJ /F13 5.978 Tf 5.76 0 Td[(j)l) f(X(j)l)dX()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l+R nhj(K2)[Rf)]TJ /F6 7.97 Tf 6.59 0 Td[(j(X()]TJ /F6 7.97 Tf 6.58 0 Td[(j)l)dX()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l]2,forj=1,2,d.Fromtheorem3.1,wehavethatthesinglecomponentestimatorhasanasymptoticnormaldistribution.Infact,inthefollowingtheorem,weshowthattheestimatorsoftheadditivecomponentsarejointlyasymptoticallynormallydistributed;moreover,thecorrelationbetweeneverypairoftheestimatedcomponentsis0.Theorem3.2:Underassumptionsinlemma3.1andthesamedenitionofhj,forj=1,2,,d,wehavethefollowingasymptoticnormality:0BBBBBBB@p nh1(~m1(x1))]TJ /F3 11.955 Tf 11.96 0 Td[(m1(x1))]TJ /F3 11.955 Tf 11.96 0 Td[(b1)p nh2(~m2(x2))]TJ /F3 11.955 Tf 11.96 0 Td[(m2(x2))]TJ /F3 11.955 Tf 11.96 0 Td[(b2)...p nhd(~md(xd))]TJ /F3 11.955 Tf 11.96 0 Td[(md(xd))]TJ /F3 11.955 Tf 11.95 0 Td[(bd)1CCCCCCCA!N(0,)where=diag(21,,2d).bjisdenedinLemma3.1and2jisdenedintheorem3.1,forj=1,2,,d.ProofisintheappendixB.Finally,inthefollowingtheorem,weshowthatthemeanfunctionestimatorbasedontheadditivecomponentestimatorsalsohasanasymptoticnormaldistribution.Theorem3.3:Let^=YanddenetheadditivespatialN-Westimatorofm(Xs)as~m(Xs)=Pdi=1~mi(Xis)+Y.Thenn2 5(~m(Xs))]TJ /F3 11.955 Tf 11.95 0 Td[(m(Xs))]TJ /F3 11.955 Tf 11.95 0 Td[(b)!N(0,2)whereb=Pdi=1biand2=Pdi=12i.ProofisintheappendixB. 51

PAGE 52

3.3SimulationResult 3.3.1SimulationStudy1Intherstsimulationstudy,wefocusedonevaluatingtheperformanceoftheadditivespatialN-Westimatorandtheadditivecomponentestimators.Onehundreddesignpointsxi=(xi,1,xi,2),wherexi,sU(0,1),s=1,2,wererandomlygenerated.Thetestingmodelwastheadditivestructure, Yi=sin(xi,1)+cos(xi,2)+"i(3)fori=1,2,,100.Herem1(xi,1)=sin(xi,1),m2(xi,2)=cos(xi,2)andm(xi,1,xi,2)=m1(xi,1)+m2(xi,2).Theerrorterm"wasgeneratedfromanormaldistributionwithE(")=0andCov("i,"j)=220kxi)]TJ /F4 7.97 Tf 6.59 0 Td[(xjk.Inthesimulation,2=1and=0.5.Fivehundredssimulationswereconducted.Weestimatemj(xi,j),fori=1,2,,100,andj=1,2,usingformulas(3-4)and(3-5).ForthemultivariateN-Westimatorgivenby(3-5),theEpanechnikovkernelwasadopted,andtheC2criterionwasusedtondthebestbandwidth.ThenweusedtheadditivespatialN-Westimator,whichisgivenintheTheorem3.3,toestimatethetrendsurfacem(xi)fori=1,2,,100.TheaveragemeansquarederrorandthemeanbiasoftheadditivespatialN-Westimator~m(x)andthetwoadditivecomponentestimators~m1and~m2arelistedinTable 3-1 .Theaveragevaluesofthesethreeestimatorsbasedonthe500simulationsareplottedwiththetruevaluesinFigures3-1to3-3.Ineachgure,theblacklinesarethetruevaluesandthereddashedlinesaretheestimatedvalues.Figure3-1compares 52

PAGE 53

theestimatedwholesurfacemeanwithitstruevaluesandgures3-2and3-3havethecomparisonfortheadditivecomponentm1andm2,respectively. 3.3.2SimulationStudy2Inthesecondsimulationstudy,wecomparedtheadditivespatialN-WestimatorinTheorem3.3withtwoothernonparametricestimatorsgiveninChapter2:themultivariateN-WestimatorusingtheC2criterionandthespatialN-Westimator.FirstrecallthattheN-Westimatorhastheform ^m(t)=Pni=1KH(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t)Yi Pnj=1KH(Xj)]TJ /F8 11.955 Tf 11.95 0 Td[(t)(3)ForthespatialN-WestimatorgiveninChapter2,denoteVar()=2andassumeisnonsingular.Weconstruct^P=)]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 2Y+(I)]TJ /F8 11.955 Tf 11.95 0 Td[()]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 2)^m(X)=(^P1,^P2,,^Pn)Twhere^m(X)istheN-WestimatorbasedontheC2criterion.ThenthespatialN-Westimatorism(t)=PiKH(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t)^Pi PjKH(Xj)]TJ /F8 11.955 Tf 11.95 0 Td[(t)Inthissimulationstudy,weusedthesametestingmodelanderrordistributionintherstsimulation.Andthesimulationswereconductedforthreedifferentsamplesize:n1=50,n2=100andn3=150.TheaveragemeansquarederrorsandaveragebiasesforthesethreeestimatorsandthestandarderrorsaredisplayedinTable 3-2 3-3 and 3-4 fordifferentsamplesize.Fromthethreetables,wecanseethatwhenthetruemodelhasanadditiveform,theadditivespatialN-Westimatorperformsbetterthantheothertwomethods.AndthespatialN-WestimatorisbetterthantheN-Westimator.Moreover,inFigure3-4,3-5and3-6,wecomparedtheaveragedifferencesbetweenthetruemeanfunctionandthethreeestimatorswithdifferentsamplesizes.ItcanbeseenthattheadditivespatialN-W 53

PAGE 54

estimatorhastheleastbiasingeneral,whiletheN-WestimatorusingC2criterionhasthelargestbias. 3.3.3SimulationStudy3Inthethirdsimulationstudy,wefocusedoncomparingtheperformanceoftheadditivespatialN-Westimator,multivariateN-WestimatorusingC2criterionandspatialN-Westimatorwhenthetruemodeldoesnothaveanadditiveform.WeaimattestingtherobustnessoftheadditivespatialN-Westimator.Intherstsetting,wegenerateonehundreddesignpointsxi=(xi,1,xi,2),wherexi,sN(i,1),andiN(2,2),s=1,2.Thenwegeneratetheresponsevaluesfollowingthetestingmodel: Yi=+xi,1+xi,1xi,2+"i(3)fori=1,2,,100.Theerrors"ihavezeromeanandsatisfyCov("i,"j)=2exp()]TJ /F5 11.955 Tf 9.3 0 Td[(kxi)]TJ /F8 11.955 Tf 11.95 0 Td[(xjk)fori6=j.Inthesimulation,2=1.5and=0.25Weestimatedthemeanbyassumingthemeanfunctionm(x1,x2)=m1(x1)+m2(x2)andusedtheformulas(3-4)and(3-5).TheaveragemeansquarederrorsandaveragebiasesofthesethreeestimatorsandtheirstandarderrorsareshowninTable 3-5 basedon1000simulationruns.Whenthetruemodelhasform(3-8),theadditivespatialN-WestimatorandthespatialN-Westimatorperformsimilarly,andbothworkbetterthantheN-Westimator.ThisisalsoreectedinFigure3-7,whichcomparesthedifferencesbetweenthetruemeanvalueandthethreeestimators.Inthesecondsetting,wechangethetestingmodelfrom(3-8)to Yi=+xi,1+xi,1xi,2+x2i,2+"i(3)fori=1,2,,100. 54

PAGE 55

Weusedthesamemethodstoestimatethemeanfunction.TheaveragemeansquarederrorsandaveragebiasesofthesethreeestimatorsandtheirstandarderrorsareshowninTable 3-6 basedon1000simulationruns.Fromthetablewecanseethatthethreeestimatorshavethesimilarperformance,buttheN-Westimatorhastheleastfavorableperformance. 55

PAGE 56

Table3-1. AveragemeansquarederrorandmeanbiasforadditivespatialN-Westimator Ave.MSEAve.Bias SpatialN-Westimator~m()0.0180.014 Additivecomponentestimator~m1()0.011-0.006 Additivecomponentestimator~m2()0.0540.06 Table3-2. AverageMSEandaveragebiasfordifferentestimatorswithsamplesize50 Ave.MSEAve.bias AdditiveN-Westimator0.91(1.05)-0.002(0.006) N-Westimator1.005(1.04)-0.007(0.008) SpatialN-Westimator0.96(1.05)-0.005(0.0008) Table3-3. AverageMSEandaveragebiasfordifferentestimatorswithsamplesize100 Ave.MSEAve.bias AdditiveN-Westimator0.018(0.05)0.014(0.018) N-Westimator0.04(0.09)-0.02(0.037) SpatialN-Westimator0.034(0.017)0.014(0.037) Table3-4. AverageMSEandaveragebiasfordifferentestimatorswithsamplesize150 Ave.MSEAve.bias AdditiveN-Westimator0.89(1.05)-0.035(0.04) N-Westimator0.96(1.05)-0.05(0.037) SpatialN-Westimator0.94(1.04)-0.04(0.04) Table3-5. AverageMSEandaveragebiasforthreeestimatorswhenmodelismisspecied1 Ave.MSEAve.bias AdditivespatialN-Westimator1.41(1.40)-0.02(0.011) N-Westimator1.66(1.32)-0.038(0.01) SpatialN-Westimator1.45(1.31)-0.017(0.01) Table3-6. AverageMSEandaveragebiasforthreeestimatorswhenmodelismisspecied2 Ave.MSEAve.bias AdditivespatialN-Westimator1.03(0.89)0.012(0.019) N-Westimator1.06(0.83)-0.019(0.006) SpatialN-Westimator1.02(0.82)0.013(0.006) 56

PAGE 57

Figure3-1. Simulationresultform 57

PAGE 58

Figure3-2. Simulationresultform1 58

PAGE 59

Figure3-3. Simulationresultform2 59

PAGE 60

Figure3-4. ThedeviationoftheadditivespatialN-Westimator,theN-WestimatorandthespatialN-Westimatorfromthetruevaluewithsamplesize50 60

PAGE 61

Figure3-5. ThedeviationoftheadditivespatialN-Westimator,theN-WestimatorandthespatialN-Westimatorfromthetruevaluewithsamplesize100 61

PAGE 62

Figure3-6. ThedeviationoftheadditivespatialN-Westimator,theN-WestimatorandthespatialN-Westimatorfromthetruevaluewithsamplesize150 62

PAGE 63

Figure3-7. ThedeviationoftheadditivespatialN-Westimator,theN-WestimatorandthespatialN-Westimatorfromthetruevaluewhenthemodelismisspecied 63

PAGE 64

Figure3-8. ThedeviationoftheadditivespatialN-Westimator,theN-WestimatorandthespatialN-Westimatorfromthetruevaluewhenthemodelismisspecied 64

PAGE 65

CHAPTER4CASESTUDY:U.S.PRECIPITATIONDATAInrelatingpublichealthoutcomestoenvironmentalfactors,thefactorsofinterest,e.g.,precipitation,PM2.5,ozone,etc.,mustbepredictedacrossthegeographicalareaofinterest.Inthissection,weconsiderthedatasetofU.S.precipitationinApril1948.ThedatasetisavailableaspartofthespampackageforRandhasbeenusedinmanyotherpapers,see[ 17 ],[ 5 ].Thedataconsistofmonthlyaveragesofanomaly,denedasthestandardizedsquarerootofprecipitationreadings,recordedbyn=5906weatherstationsthatarescatteredirregularlyoverthecontinentalU.S.Inthiscasestudy,wefocusonthepredictionoftheU.S.precipitationatstationlocationsusingthemethodsdiscussedinChapter2andChapter3.Notethatthestationsarenotonagrid;therefore,ourdataarenotequallyspaced.Precipitationdatavaryacrossthestate.Thus,weassumetheprecipitationdependsonthelocationofstations.Moreover,weassumethecovariancebetweentwostationsdependsontheparameters,2andthedistancedbetweentwostations.And,ithastheformCov(d)=2exp()]TJ /F5 11.955 Tf 9.3 0 Td[(d).Inourwork,weusetheparameterestimatorsgivenby[ 5 ].Finally,weassumethattheregressionmodelhasthegeneralnonparametricregressionformin(1-1)or,equivalently,theregressionmodelisYs=m(Xs)+sAnEpanechnikovkernelhasbeenusedforthethreedifferentestimators:thetraditionalN-WestimatorbasedontheC2criterion,thespatialN-WestimatorandtheadditivespatialN-Westimator.Foreachestimator,wecomputedthesumofsquareddifferencesbetweentheobservedprecipitationandtheleave-one-outpredictionofthe 65

PAGE 66

precipitationforthe5906stations(seeTable 4-1 ).Fromthetable,wecanseethatthespatialN-WestimatorandtheadditivespatialN-Westimatorperformsimilarly,andtheybothperformbetterthanthetraditionalN-WestimatorbasedontheC2criterion.InFigure4-1,thedeviationoftheleave-one-outpredictionfromtheobservedprecipitationisdisplayedforeachofthethreeN-Westimators.Atlast,WeextracttheprecipitationrecordsofthestateofFlorida(60stations)andfocusonthespatialpredicitonoftheprecipitationacrossthestateofFloridausingthetraditionalN-WestimatorbasedonC2criterion,thespatialN-WestimatorandtheadditivespatialN-WestimatoraredisplayedinFigures4-2,4-3and4-4,respectively.Fromthesethreegures,wecanseethatthespatialN-WestimatorandadditivespatialN-Westimatorprovidesimilarpredicitonresults. 66

PAGE 67

Table4-1. Comparisionofthesumofsquareddifferencebetweenobservedandleave-one-outpredictedprecipitationusingthreeN-Westimators EstimatorSumofsquareddifference C2N-Westimator271.3911 SpatialN-Westimator230.2244 AdditivespatialN-Westimator237.1825 Figure4-1. ComparisonofthethreeN-Westimatorsusingtheleave-one-outestimation 67

PAGE 68

Figure4-2. PredictedprecipitationofthestateofFloridausingthetraditionalN-WestimatorusingC2criterion 68

PAGE 69

Figure4-3. PredictedprecipitationoftheStateofFloridausingthespatialN-Westimator 69

PAGE 70

Figure4-4. PredictedprecipitationoftheStateofFloridausingtheadditivespatialN-Westimator 70

PAGE 71

CHAPTER5CONCLUSIONInthisthesis,wefocusonnonparametricregressionestimationwithspatiallycorrelateddata.Aswithallspatialdata,themajorchallengeishowtoseparatethemeanfunctionthatdescribestherelationshipbetweentheresponsevariableandasetofpredictorsfromspatiallycorrelatederrors.Forthelarge-scalevariationofthemodelm(X),weconsiderthenonparametricandadditivenonparametricstructure;forthesmallscalevariationofthemodel,weassumetheerrorshavethecovariancestructure2n(kXi)]TJ /F8 11.955 Tf 11.95 0 Td[(Xjk)withunknownparameter2,whichisthevariancecommontoallerrors,andknowncorrelationfunctionn.TheNadaraya-Watson(N-W)estimatorisextendedtoaccountforthespatialcorrelation.Underthegeneralnonparametricregressionmodel(1-1),wehaveshownthatthetraditionalcross-validationbandwidthselectionmethodfailstoprovidegoodbandwidthinthepresenceofspatiallycorrelatederrors.Weproposethedata-drivenbandwidthselectionmethod,calledtheC2criterion,whichaccountsforthepresenceofspatialcorrelation.Moreover,wealsoadoptthetwo-stepestimationprocedureconsideredin[ 29 ].Thisprocedurerstlyformsanewprocesswiththesameconditionalastheresponsevectorinthemodelandi.i.d.errors,andthenappliesthetraditionalN-Wttingtothenewprocess.[ 29 ]consideredsuchatwo-stepestimationintimeseriesanalysis.Usingthistwo-stepidea,weproposethespatialN-WestimatorbasedonthetraditionalN-Westimatorforspatiallycorrelateddata.Undermodel(1-1)witherrorshavingdistribution(1-2)and(1-3),wederivetheformulasfortheasymptoticbias,varianceofthetraditionalN-WestimatorundertheC2criterionandthespatialN-Westimator.Bothestimatorsarebiasedbutconsistentunder(A1)to(A5).ThebiasofthespatialN-WestimatorhasthesameorderasthebiasofthetraditionalN-Westimator,buttheasymptoticvarianceofthespatialN-Westimatorissmaller.Therefore,thespatialN-WestimatorismoreefcientthanthetraditionalN-W 71

PAGE 72

estimatorunderspatiallycorrelatederrors.Theasymptoticnormalityofbothestimatorsisestablishedinthethesis.Theadditivenonparametricmodel(1-5)isattractivebecauseitgeneralizesthelinearregressionmodelsandallowsinterpretationofmarginalchanges.Weusethethemarginalintegrationapproach,whichwasrstproposedby[ 28 ],toavoidthedifcultyoftheiterativeprocedures,suchastheHastie&Tibshiraniestimationprocedure.Underthenonparametricadditivemodel(1-5)withspatiallycorrelatederrors,weproposetheadditivespatialN-WestimatorbasedonthemarginalintegrationtechniqueandthetraditionalN-Westimator.TheasymptoticbiasesandvariancesoftheadditivespatialN-Westimatorisderived.Moreover,theasymptoticnormalityoftheadditivespatialN-Westimatorisproved.Simulationsareconductedtoassesstheperformanceofourestimators.Theresultsshowthatrst,underspatiallycorrelatederrors,thespatialN-Westimatorbasedonthetwo-stepprocedureperformsbetterthanthetraditionalN-WestimatorusingeithertheC2criterionorthecross-validationbandwidthselectiontechnique.Further,theC2criterionperformsbetterthanthecross-validationbandwidthselectiontechnique.Second,whenthetruemodelhasadditiveform,theadditivespatialN-WestimatorperformsbetterthaneitherthetraditionalorspatialN-Westimator.Third,whenthetruemodeldoesnothavetheadditiveform,theadditivespatialN-WestimatorhasaperformancecomparabletothespatialN-Westimator,andtheybothperformbetterthanthetraditionalN-Westimator.Asanillustrationofourapproaches,acasestudyofprecipitationdatainthecontinentalU.S.isgiven.Inthecasestudy,weestimatetheprecipitationof5906weatherstationsinthecontinentalU.S.inApril1948,andthenwepredicttheprecipitationacrosstheStateofFloridabasedonthe60weatherstations'data.ThetraditionalN-WestimatorbasedontheC2criterion,thespatialN-WestimatorandtheadditivespatialN-Westimatorareused.Bycomparingthesumofsquareddifferencesbetweentheobservedprecipitationandtheleave-one-outpredictionofthe 72

PAGE 73

precipitationofeachestimator,wendthatthespatialN-WestimatorandtheadditivespatialN-Westimatorperformsimilarly,andtheyarebothbetterthanthetraditionalN-WestimatorbasedontheC2criterion.Moreremainstobedone.First,wehavenotcarefullyconsideredestimationofthecovariancestructuresinthemodel.Inourwork,weassumethatthecovariancefunctionisknowngivensomeparameters,andestimatetheparametersusingthepilotmethodwhenthereisnorepeatedmeasurements.Others,suchas[ 37 ],[ 5 ]and[ 11 ],havefocusedontheestimationofcovariancefunction.[ 37 ]consideredtheleast-squaresapproachforestimatingparametersofaspatialvariogramandestablishedconsistencyandasymptoticnormalityoftheestimators.[ 5 ]obtainedanestimatorofthecovariancefunctionbyregressingsquareddifferencesoftheresponseontheirexpectation,whichequalsthevariogramplusanoffsetterminducedbythetrend.[ 11 ]usedthemethodofweightedleastsquarestoestimatethevariogramandalsoproposedtwosetsofweights.Second,ourworkonlyfocusesonthespatialtrend.Futureworkmayconsiderthespatio-temporalproblemsusingthenonparametricanalysistechnique.Thenonparametricestimatorsdevelopedherecouldprovideafoundationformethodstocomparemultipleimagesormapsofthesameareaorobjectsatdifferenttimepoints.Detectingtemporaltrends,suchasthetemperaturetrendinthestateofFloridaoveraspecictimeperiod,byappropriatelycombiningthenonparametricanalysisforspatialdataandtimeseriesdataisanotheravenueforresearch. 73

PAGE 74

APPENDIXAACCOMPANIMENTTOCHAPTER2ProofofLemma2.1:WriteWi(t)=KH(Xi)]TJ /F4 7.97 Tf 6.59 0 Td[(t) Pnj=1KH(Xj)]TJ /F4 7.97 Tf 6.58 0 Td[(t),thenE(^m(t))=E[nXi=1Wi(t)Yi]=nXi=1Wi(t)E(Yi)=nXi=1Wi(t)m(Xi)Therefore,E(^m(t)))]TJ /F3 11.955 Tf 11.96 0 Td[(m(t)=nXi=1Wi(t)(m(Xi))]TJ /F3 11.955 Tf 11.95 0 Td[(m(t))SincethekernelKhasnonzerovaluesonlyint,andkXi)]TJ /F8 11.955 Tf 11.96 0 Td[(tk=kHH)]TJ /F4 7.97 Tf 6.59 0 Td[(1(Xi)]TJ /F8 11.955 Tf 11.96 0 Td[(t)kkHkkH)]TJ /F4 7.97 Tf 6.58 0 Td[(1(Xi)]TJ /F8 11.955 Tf 11.96 0 Td[(t)kkHkThen,jE(^m(t)))]TJ /F3 11.955 Tf 11.96 0 Td[(m(t)jnXi=1Wi(t)CkXi)]TJ /F8 11.955 Tf 11.95 0 Td[(tknXi=1Wi(t)CkHkr=CkHkrProofofLemma2.2: 74

PAGE 75

Var(^m(t))=Var(PiKH(Xi)]TJ /F8 11.955 Tf 11.96 0 Td[(t)Yi PjKH(Xj)]TJ /F8 11.955 Tf 11.95 0 Td[(t))=1 (PjKH(Xj)]TJ /F8 11.955 Tf 11.95 0 Td[(t))2Var(nXi=1KH(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t)Yi)=1 (PjKH(Xj)]TJ /F8 11.955 Tf 11.96 0 Td[(t))2fXiK2H(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t)Var(Yi)+XiXj6=iKH(Xi)]TJ /F8 11.955 Tf 11.96 0 Td[(t)KH(Xj)]TJ /F8 11.955 Tf 11.96 0 Td[(t)Cov(Yi,Yj)g=1 (PjKH(Xj)]TJ /F8 11.955 Tf 11.95 0 Td[(t))2fXiK2H(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t)2+2XiXj6=iKH(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t)KH(Xj)]TJ /F8 11.955 Tf 11.95 0 Td[(t)n(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(Xj)gAndbasedonassumption2,wewillhave1 (PjKH(Xj)]TJ /F8 11.955 Tf 11.96 0 Td[(t))2=1 (Pj2tKH(Xj)]TJ /F8 11.955 Tf 11.96 0 Td[(t))21 a2n2tAnd,XiK2H(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t)2b22ntThen,letusconsiderXiXj6=iKH(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t)KH(Xj)]TJ /F8 11.955 Tf 11.96 0 Td[(t)n(kXi)]TJ /F8 11.955 Tf 11.96 0 Td[(Xjk)XiXj6=iKH(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t)KH(Xj)]TJ /F8 11.955 Tf 11.96 0 Td[(t)jn(kXi)]TJ /F8 11.955 Tf 11.95 0 Td[(Xjk)j=Xi2tKH(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t)Xj6=i,j2tKH(Xj)]TJ /F8 11.955 Tf 11.95 0 Td[(t)jn(kXi)]TJ /F8 11.955 Tf 11.95 0 Td[(Xjk)jbXi2tKH(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t)Snib2SnNotethatbothSniandSndependonweightmatrixH. 75

PAGE 76

Combiningtheaboveworkwithassumption4,wehaveVar(^m(t))=OSn n2tInordertoprooflemma2.4,weneedtoprovethefollowingthreepropositionsrst.Proposition1:1 nnXi=1KH(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t)=f(t)+o(1)1 nnXi=1KH(Xi)]TJ /F8 11.955 Tf 11.96 0 Td[(t)(Xi)]TJ /F8 11.955 Tf 11.96 0 Td[(t)=2(K)HHTrf(t)+o(HHT1d)1 nnXi=1KH(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t)(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t)(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t)T=2(K)f(t)HHT+o(HHT)ProofofProposition1:Thisisgivenby[ 20 ].Proposition2:limn!1njHjRI(ktk")jn(Ht)jdt=0Proof:njHjZktk"jn(t)jdt=njHjZkH)]TJ /F11 5.978 Tf 5.75 0 Td[(1uk"jn(u)j1 jHjdu=nZkH)]TJ /F11 5.978 Tf 5.75 0 Td[(1uk"jn(u)jdu)166(!0Notethattheapproximationinthelaststepisbasedonassumption4.Proposition3:1 n2nXi=1K2H(Xi)]TJ /F8 11.955 Tf 11.96 0 Td[(t)=(K2)f(t) njHj+o(1 njHj)1 n2nXi=1Xj6=iKH(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t)KH(Xj)]TJ /F8 11.955 Tf 11.95 0 Td[(t)n(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(Xj)=f2(t)(K2)R njHj+o(1 njHj)Proof:Letusprovetherststatementofproposition3.DenoteWn=1 n2Pni=1K2H(Xi)]TJ /F8 11.955 Tf 12.06 0 Td[(t).Inordertoprovetherststatementofproposition3,weneedtoprovethat E(njHjWn)=(K2)f(t)+o(1) (A) 76

PAGE 77

and Var(njHjWn)!0 (A) Inordertoprove(A-1)and(A-2),letusrstlookatRK2H(X)]TJ /F8 11.955 Tf 11.96 0 Td[(t)f(X)dX.ItcanbeshownthatZK2H(X)]TJ /F8 11.955 Tf 11.96 0 Td[(t)f(X)dX=Z1 jHj2K2(H)]TJ /F4 7.97 Tf 6.58 0 Td[(1X)]TJ /F8 11.955 Tf 11.95 0 Td[(t)f(X)dX=1 jHj2ZK2(u)f(t+Hu)jHjdu=1 jHjZK2(u)f(t+Hu)duUsingTaylorseriesapproximation,wewillhaveZK2(u)f(t+Hu)du=(K2)f(t)+o(1)Then,ZK2H(X)]TJ /F8 11.955 Tf 11.96 0 Td[(t)f(X)dX=1 jHj[(K2)f(t)+o(1)]=1 jHj(K2)f(t)+o(1 jHj)Therefore,E(njHjWn)=E(njHj1 n2nXi=1K2H(Xi)]TJ /F8 11.955 Tf 11.96 0 Td[(t))=jHjE(K2H(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t))=(K2)f(t)+o(1) 77

PAGE 78

AndVar(njHjWn)=Var(njHj1 n2nXi=1K2H(Xi)]TJ /F8 11.955 Tf 11.96 0 Td[(t)))=jHj2 n2Var(nXi=1K2H(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t))jHj2 nE(K4H(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t))jHj2b4 n!0Therefore,wehaveshownthat1 n2nXi=1K2H(Xi)]TJ /F8 11.955 Tf 11.96 0 Td[(t)=(K2)f(t) njHj+o(1 njHj)Inordertoprovethesecondstatement,weconsiderRK(u)n(H(u)]TJ /F8 11.955 Tf 11.95 0 Td[(v))durst.ZK(u)n(H(u)]TJ /F8 11.955 Tf 11.95 0 Td[(v))du=Zku)]TJ /F4 7.97 Tf 6.59 0 Td[(vk"K(u)n(H(u)]TJ /F8 11.955 Tf 11.95 0 Td[(v))du+Zku)]TJ /F4 7.97 Tf 6.59 0 Td[(vk<"K(u)n(H(u)]TJ /F8 11.955 Tf 11.95 0 Td[(v))du=P1+P2ConsiderP1andP2,respectively.First,byproposition2,wecanhave,njHjP1=njHjZku)]TJ /F4 7.97 Tf 6.59 0 Td[(vk"K(u)n(H(u)]TJ /F8 11.955 Tf 11.95 0 Td[(v))dunjHjZku)]TJ /F4 7.97 Tf 6.59 0 Td[(vk"K(u)jn(H(u)]TJ /F8 11.955 Tf 11.96 0 Td[(v))jdunjHjbZku)]TJ /F4 7.97 Tf 6.59 0 Td[(vk"jn(H(u)]TJ /F8 11.955 Tf 11.95 0 Td[(v))jdu)166(!0Therefore,wehaveP1=o(1 njHj) 78

PAGE 79

Then,P2=Zku)]TJ /F4 7.97 Tf 6.58 0 Td[(vk<"K(u)n(H(u)]TJ /F8 11.955 Tf 11.96 0 Td[(v))du=Zku)]TJ /F4 7.97 Tf 6.58 0 Td[(vk<"K(v)n(H(u)]TJ /F8 11.955 Tf 11.96 0 Td[(v))du+Zku)]TJ /F4 7.97 Tf 6.58 0 Td[(vk<"(K(u))]TJ /F3 11.955 Tf 11.96 0 Td[(K(v))n(H(u)]TJ /F8 11.955 Tf 11.96 0 Td[(v))du=C1+C2Usingassumption4andproposition2,wecanshowthatnjHjC1=njHjZK(v)n(H(u)]TJ /F8 11.955 Tf 11.96 0 Td[(v))du)]TJ /F3 11.955 Tf 11.96 0 Td[(njHjK(v)Zku)]TJ /F4 7.97 Tf 6.58 0 Td[(vk"n(H(u)]TJ /F8 11.955 Tf 11.95 0 Td[(v))du=K(v)nZn(t)dt)]TJ /F3 11.955 Tf 11.95 0 Td[(K(v)njHjZku)]TJ /F4 7.97 Tf 6.59 0 Td[(vk"n(H(u)]TJ /F8 11.955 Tf 11.96 0 Td[(v))du)166(!K(v)RwhereRisdenedinassumption4.And,sinceKisLipschitzcontinuous,jnjHjC2jL"njHjZku)]TJ /F4 7.97 Tf 6.58 0 Td[(vk"jn(H(u)]TJ /F8 11.955 Tf 11.95 0 Td[(v))jdu=o(1)Combiningtheresultsabove,wecanhavethatnjHjP2=K(v)R+o(1)Therefore, ZK(u)n(H(u)]TJ /F8 11.955 Tf 11.95 0 Td[(v))du=K(v)R njHj+o(1 njHj) (A) Noteherebothuandvareint,andwecannd"smallenoughtomakesure(u)]TJ /F8 11.955 Tf 11.96 0 Td[(v)alsoint.Now,considerproposition3. 79

PAGE 80

Usingtheequation(A-3)aboveandaTaylorseriesapproximation,wehavenjHjZZKH(u)]TJ /F8 11.955 Tf 11.96 0 Td[(t)KH(v)]TJ /F8 11.955 Tf 11.96 0 Td[(t)n(u)]TJ /F8 11.955 Tf 11.96 0 Td[(v)f(u)f(v)dudv=njHj1 jHj2ZZjHj2K(w)K(q)n(H(w)]TJ /F8 11.955 Tf 11.95 0 Td[(q))f(Hw+t)f(Hq+t)dwdq=njHjZZK(w)K(q)n(H(w)]TJ /F8 11.955 Tf 11.96 0 Td[(q))f2(t)dwdq+o(1)=f2(t)njHjZK(q)ZK(w)n(H(w)]TJ /F8 11.955 Tf 11.96 0 Td[(q))dwdq+o(1)=f2(t)ZK(q)(K(q)R+o(1))+o(1)=f2(t)(K2)R+o(1)ItfollowsthatZZKH(u)]TJ /F8 11.955 Tf 11.96 0 Td[(t)KH(v)]TJ /F8 11.955 Tf 11.96 0 Td[(t)n(u)]TJ /F8 11.955 Tf 11.96 0 Td[(v)f(u)f(v)dudv=f2(t)(K2)R njHj+o(1 njHj)Thethesecondstatementofproposition3canbegoteasily.Proofoflemma2.4:Var(^m(t))=Var(1 PjKH(Xj)]TJ /F8 11.955 Tf 11.96 0 Td[(t)XiKH(Xi)]TJ /F8 11.955 Tf 11.96 0 Td[(t)Yi)=1 PjK2H(Xj)]TJ /F8 11.955 Tf 11.96 0 Td[(t)Var(XiKH(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t)Yi))=2 PjK2H(Xj)]TJ /F8 11.955 Tf 11.96 0 Td[(t)fnXi=1XjKH(Xi)]TJ /F8 11.955 Tf 11.96 0 Td[(t)KH(Xj)]TJ /F8 11.955 Tf 11.95 0 Td[(t)n(Xi)]TJ /F8 11.955 Tf 11.96 0 Td[(Xj)g=2 PjK2H(Xj)]TJ /F8 11.955 Tf 11.96 0 Td[(t)fnXi=1K2H(Xi)]TJ /F8 11.955 Tf 11.96 0 Td[(t)+nXi=1Xj6=iKH(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(t)KH(Xj)]TJ /F8 11.955 Tf 11.96 0 Td[(t)n(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(Xj)gThen,Lemma2.4istheresultgivenproposition1and3.Proofoftheorem2.4: 80

PAGE 81

Firstly,theleave-one-outfunctioncanbewrittenas1 nnXi=1[Yi)]TJ /F7 11.955 Tf 13.8 0 Td[(^mH,)]TJ /F6 7.97 Tf 6.59 0 Td[(i(Xi)]2=1 nnXi=1[m(Xi))]TJ /F5 11.955 Tf 11.95 0 Td[(i)]TJ /F7 11.955 Tf 13.8 0 Td[(^mH,)]TJ /F6 7.97 Tf 6.58 0 Td[(i(Xi)]2=1 nnXi=1[(m(Xi))]TJ /F7 11.955 Tf 13.8 0 Td[(^mH,)]TJ /F6 7.97 Tf 6.58 0 Td[(i(Xi))2+2i+2i(m(Xi))]TJ /F7 11.955 Tf 13.8 0 Td[(^mH,)]TJ /F6 7.97 Tf 6.58 0 Td[(i(Xi))]=1 n[2Xi=1(m(Xi))]TJ /F7 11.955 Tf 13.8 0 Td[(^mH,)]TJ /F6 7.97 Tf 6.59 0 Td[(i(Xi))2+nXi=12i+2Xi=12i(m(Xi))]TJ /F7 11.955 Tf 13.8 0 Td[(^mH,)]TJ /F6 7.97 Tf 6.58 0 Td[(i(Xi))]Therefore,takingexpectationonbothsidesoftheequationabovegives1 nEfnXi=1[Yi)]TJ /F7 11.955 Tf 13.8 0 Td[(^mH,)]TJ /F6 7.97 Tf 6.58 0 Td[(i(Xi)]2g=1 nnXi=1E(m(Xi))]TJ /F7 11.955 Tf 13.8 0 Td[(^mH,)]TJ /F6 7.97 Tf 6.59 0 Td[(i(Xi))2+2)]TJ /F7 11.955 Tf 13.25 8.08 Td[(2 nnXi=1Cov(^mH,)]TJ /F6 7.97 Tf 6.58 0 Td[(i(Xi),Yi)Recallthat^mH,)]TJ /F6 7.97 Tf 6.59 0 Td[(i(Xi)=Pj6=iKH(Xj)]TJ /F8 11.955 Tf 11.96 0 Td[(Xi)Yj Pk6=iKH(Xk)]TJ /F8 11.955 Tf 11.95 0 Td[(Xi)ThenwecanrewriteCov(^mH,)]TJ /F6 7.97 Tf 6.59 0 Td[(i(Xi),Yi)=Cov(Pj6=iKH(Xj)]TJ /F8 11.955 Tf 11.95 0 Td[(Xi)Yj Pk6=iKH(Xk)]TJ /F8 11.955 Tf 11.96 0 Td[(Xi),Yi)=1 Pk6=iKH(Xk)]TJ /F8 11.955 Tf 11.95 0 Td[(Xi)Cov(Xj6=iKH(Xj)]TJ /F8 11.955 Tf 11.96 0 Td[(Xi)Yj,Yi)=Pj6=iKH(Xj)]TJ /F8 11.955 Tf 11.95 0 Td[(Xi)Cov(Yj,Yi) Pk6=iKH(Xk)]TJ /F8 11.955 Tf 11.95 0 Td[(Xi)=2Pj6=iKH(Xj)]TJ /F8 11.955 Tf 11.95 0 Td[(Xi)n(kXj)]TJ /F8 11.955 Tf 11.96 0 Td[(Xik) Pk6=iKH(Xk)]TJ /F8 11.955 Tf 11.96 0 Td[(Xi)=21 n[P2j=1KH(Xj)]TJ /F8 11.955 Tf 11.95 0 Td[(Xi)n(kXj)]TJ /F8 11.955 Tf 11.96 0 Td[(Xik))]TJ /F3 11.955 Tf 11.95 0 Td[(KH(0)] 1 n[Pnk=1KH(Xk)]TJ /F8 11.955 Tf 11.96 0 Td[(Xi))]TJ /F3 11.955 Tf 11.95 0 Td[(KH(0)] (A)Letuslookatthedenominatorin(A-4)rst.Inproposition1,wealreadyshowedthat 1 nnXk=1KH(Xk)]TJ /F8 11.955 Tf 11.95 0 Td[(Xi)=f(Xi)+o(1) (A) 81

PAGE 82

Inordertosimplifythenumerator,werstlookatRKH(u)]TJ /F8 11.955 Tf 11.83 0 Td[(Xi)n(ku)]TJ /F8 11.955 Tf 11.84 0 Td[(Xik)f(u)du.ItcanbeshownthatZKH(u)]TJ /F8 11.955 Tf 11.95 0 Td[(Xi)n(ku)]TJ /F8 11.955 Tf 11.96 0 Td[(Xik)f(u)du=Z1 jHjK(H)]TJ /F4 7.97 Tf 6.59 0 Td[(1(u)]TJ /F8 11.955 Tf 11.95 0 Td[(Xi))n(ku)]TJ /F8 11.955 Tf 11.95 0 Td[(Xik)f(u)du=ZK(vi)n(kHvik)f(Xi+Hvi)dvi=f(Xi)ZK(vi)n(kHvik)dvi+o(1)Usingequation(A-3),wehavenjHjZK(vi)n(kHvik)dvi=njHjZK(vi)n(kH(vi)]TJ /F8 11.955 Tf 11.95 0 Td[(0)k)dvi!K(0)RwhereRisdenedinassumption4.Therefore,RK(vi)n(kHvik)dvi=K(0)R njHj+o(1 njHj).SinceXiarei.i.d.,wewillhave 1 nnXj=1KH(Xj)]TJ /F8 11.955 Tf 11.96 0 Td[(Xi)n(kXj)]TJ /F8 11.955 Tf 11.96 0 Td[(Xik)=f(Xi)K(0)R njHj+o(1) (A) Combiningequations(A-4),(A-5)and(A-6),wehaveCov(^mH,)]TJ /F6 7.97 Tf 6.59 0 Td[(i(Xi),Yi)=2[1 nf(Xi)KH(0)R)]TJ /F4 7.97 Tf 13.22 4.71 Td[(1 nKH(0)] f(Xi))]TJ /F4 7.97 Tf 13.21 4.71 Td[(1 nKH(0)+o(1)=21 nKH(0)[f(Xi)R)]TJ /F7 11.955 Tf 11.95 0 Td[(1] f(Xi))]TJ /F4 7.97 Tf 13.22 4.7 Td[(1 nKH(0)+o(1)Then,2 nnXi=1Cov(^mH,)]TJ /F6 7.97 Tf 6.59 0 Td[(i(Xi),Yi)=22 nnXi=1[1 nKH(0)[f(Xi)R)]TJ /F7 11.955 Tf 11.95 0 Td[(1] f(Xi)]TJ /F4 7.97 Tf 13.22 4.71 Td[(1 nKH(0))+o(1)]=2KH(0)[f(X)R)]TJ /F7 11.955 Tf 11.95 0 Td[(1]2 nf(X))]TJ /F3 11.955 Tf 11.95 0 Td[(KH(0)+o(1)Therefore,theorem2.4isshown.Proofoftheorem2.5:Recallthat~m(t)andm(t)aretwoN-Westimatorsofm(t)thatuse^PandP,and~m(t))]TJ /F3 11.955 Tf 11.95 0 Td[(m(t)=~m(t))]TJ /F7 11.955 Tf 13.8 0 Td[(m(t)+m(t))]TJ /F3 11.955 Tf 11.96 0 Td[(m(t) 82

PAGE 83

Inlemma3.1,weshowedthatm(t))]TJ /F3 11.955 Tf 12.26 0 Td[(m(t)hastheasymptoticnormaldistribution.Hereweconsidertheasymptoticdistributionof~m(t))]TJ /F7 11.955 Tf 13.93 0 Td[(m(t).Differencesbetweenthesetwoestimatorscanberewrittenas~m(t))]TJ /F7 11.955 Tf 13.8 0 Td[(m(t)=PjKH(Xj)]TJ /F8 11.955 Tf 11.96 0 Td[(Xi)^Pj PkKH(Xk)]TJ /F8 11.955 Tf 11.95 0 Td[(Xi))]TJ /F9 11.955 Tf 13.15 18.44 Td[(PjKH(Xj)]TJ /F8 11.955 Tf 11.95 0 Td[(Xi)Pj PkKH(Xk)]TJ /F8 11.955 Tf 11.96 0 Td[(Xi)=1 PkKH(Xk)]TJ /F8 11.955 Tf 11.95 0 Td[(Xi)fXjKH(Xj)]TJ /F8 11.955 Tf 11.95 0 Td[(Xi)(^Pj)]TJ /F3 11.955 Tf 11.95 0 Td[(Pj)g=1 PkKH(Xk)]TJ /F8 11.955 Tf 11.95 0 Td[(Xi)fXjKH(Xj)]TJ /F8 11.955 Tf 11.95 0 Td[(Xi)[(^mj)]TJ /F3 11.955 Tf 11.96 0 Td[(mj))]TJ /F9 11.955 Tf 11.96 11.36 Td[(Xlvjl(^ml)]TJ /F3 11.955 Tf 11.95 0 Td[(ml)]g=1 PkKH(Xk)]TJ /F8 11.955 Tf 11.95 0 Td[(Xi)=nfXjKH(Xj)]TJ /F8 11.955 Tf 11.96 0 Td[(Xi)(^mj)]TJ /F3 11.955 Tf 11.96 0 Td[(mj)=n)]TJ /F9 11.955 Tf 11.29 11.36 Td[(XjKH(Xj)]TJ /F8 11.955 Tf 11.96 0 Td[(Xi)Xlvjl(^ml)]TJ /F3 11.955 Tf 11.95 0 Td[(ml)=ng (A)=1 C(T1)]TJ /F3 11.955 Tf 11.95 0 Td[(T2)BeforeconsideringT1andT2,werstrewrite^mj)]TJ /F3 11.955 Tf 11.96 0 Td[(mjin(A-7)^mj)]TJ /F3 11.955 Tf 11.96 0 Td[(mj=PtKH(Xt)]TJ /F8 11.955 Tf 11.95 0 Td[(Xj)Yt PmKH(Xm)]TJ /F8 11.955 Tf 11.95 0 Td[(Xj))]TJ /F3 11.955 Tf 11.95 0 Td[(mj=1 PmKH(Xm)]TJ /F8 11.955 Tf 11.95 0 Td[(Xj)XtKH(Xt)]TJ /F8 11.955 Tf 11.96 0 Td[(Xj)[Yt)]TJ /F3 11.955 Tf 11.95 0 Td[(mj]=1 PmKH(Xm)]TJ /F8 11.955 Tf 11.95 0 Td[(Xj)=nXtKH(Xt)]TJ /F8 11.955 Tf 11.96 0 Td[(Xj)[mt)]TJ /F3 11.955 Tf 11.96 0 Td[(mj+t]=n=1 PmKH(Xm)]TJ /F8 11.955 Tf 11.95 0 Td[(Xj)=n[XtKH(Xt)]TJ /F8 11.955 Tf 11.96 0 Td[(Xj)(mt)]TJ /F3 11.955 Tf 11.96 0 Td[(mj)=n+XtKH(Xt)]TJ /F8 11.955 Tf 11.96 0 Td[(Xj)t=n] 83

PAGE 84

Therefore,T1=XjKH(Xj)]TJ /F8 11.955 Tf 11.96 0 Td[(Xi)(^mj)]TJ /F3 11.955 Tf 11.95 0 Td[(mj)=n=1 PmKH(Xm)]TJ /F8 11.955 Tf 11.95 0 Td[(Xj)=nXjKH(Xj)]TJ /F8 11.955 Tf 11.96 0 Td[(Xi)fXtKH(Xt)]TJ /F8 11.955 Tf 11.95 0 Td[(Xj)(mt)]TJ /F3 11.955 Tf 11.96 0 Td[(mj)=n2+XtKH(Xt)]TJ /F8 11.955 Tf 11.95 0 Td[(Xj)t=n2g=1 PmKH(Xm)]TJ /F4 7.97 Tf 6.59 0 Td[(Xj) nfPjPtKH(Xj)]TJ /F8 11.955 Tf 11.96 0 Td[(Xi)KH(Xt)]TJ /F8 11.955 Tf 11.96 0 Td[(Xj)(mt)]TJ /F3 11.955 Tf 11.96 0 Td[(mj) n2+PjPtKH(Xj)]TJ /F8 11.955 Tf 11.95 0 Td[(Xi)KH(Xt)]TJ /F8 11.955 Tf 11.95 0 Td[(Xj)t n2g=S1+S2 SLetuslookatS1 SandS2 S,respectively.Basedonthepreviousworkandsomeworkby[ 29 ],wehave1 nKH(Xt)]TJ /F8 11.955 Tf 11.96 0 Td[(Xj)(mt)]TJ /F3 11.955 Tf 11.96 0 Td[(mj)=O(tr(H2)))1 n2XjXtKH(Xj)]TJ /F8 11.955 Tf 11.95 0 Td[(Xi)KH(Xt)]TJ /F8 11.955 Tf 11.95 0 Td[(Xj)(mt)]TJ /F3 11.955 Tf 11.95 0 Td[(mj)=O(tr(H2))AndPmKH(Xm)]TJ /F8 11.955 Tf 11.95 0 Td[(Xj) n=O(1)Therefore,S1 S=O(tr(H2))NotethatE(S2)=0.Andafteralongtediouscalculation,wewillhaveVar(p njHjS2)=o(1)Therefore,S2 S=o(1 p njHj) 84

PAGE 85

andthenT1 C=o(1 p njHj)+O(tr(H2))BasedontheassumptionsupifPj(vij)g=O(1),withsimilarcalculation,wewillalsohaveT2 C=o(1 p njHj)+O(tr(H2))Therefore,~m(t))]TJ /F7 11.955 Tf 13.8 0 Td[(m(t)=o(1 p njHj)+O(tr(H2))Underassumption5,itcanbeshownthatO(tr(H2))=o(1 p njHj)Therefore, p njHj(~m(t))]TJ /F7 11.955 Tf 13.8 0 Td[(m(t))=o(1) (A) Combining(A-8)withlemma2.5,weshowedtheorem2.5 85

PAGE 86

APPENDIXBACCOMPANIMENTTOCHAPTER3ProofofLemma3.1:FirstconsidertheN-Westimatorofm(X(j)i).WriteW(j)i=0BBBBBBB@KH(X1)]TJ /F8 11.955 Tf 11.95 0 Td[(X(j)i)000KH(X2)]TJ /F8 11.955 Tf 11.96 0 Td[(X(j)i)0.........000KH(Xn)]TJ /F8 11.955 Tf 11.96 0 Td[(X(j)i)1CCCCCCCAwherenisthenumberofdesignpoints.ThentheN-Westimatorofm(X(j)i)canbewrittenas ^m(X(j)i)=(1TW(j)i1))]TJ /F4 7.97 Tf 6.59 0 Td[(11TW(j)iY (B) where1isthevectorofall1's.Combining(3-4)and(B-1),wehave ~mj(xj)=1 nnXi=1(1TW(j)i1))]TJ /F4 7.97 Tf 6.59 0 Td[(11TW(j)iY)]TJ /F7 11.955 Tf 13.89 2.66 Td[(Y (B) Therefore,thedifferencebetween~mj(xj)andmj(xj)is~mj(xj))]TJ /F3 11.955 Tf 11.96 0 Td[(mj(xj)=1 nnXi=1(1TW(j)i1))]TJ /F4 7.97 Tf 6.58 0 Td[(11TW(j)iY)]TJ /F7 11.955 Tf 13.89 2.66 Td[(Y)]TJ /F3 11.955 Tf 11.96 0 Td[(mj(xj)=1 nnXi=1(1TW(j)i1))]TJ /F4 7.97 Tf 6.58 0 Td[(11TW(j)i[Y)]TJ /F7 11.955 Tf 11.96 0 Td[((Y+mj(xj))1]=1 nnXi=1(1TW(j)i1))]TJ /F4 7.97 Tf 6.58 0 Td[(11TW(j)i[m(X)+)]TJ /F7 11.955 Tf 13.89 2.65 Td[(Y1)]TJ /F3 11.955 Tf 11.96 0 Td[(mj(xj)1]=1 nnXi=1(1TW(j)i1))]TJ /F4 7.97 Tf 6.58 0 Td[(11TW(j)i)]TJ /F7 11.955 Tf 13.25 8.09 Td[(1 nnXi=1(1TW(j)i1))]TJ /F4 7.97 Tf 6.58 0 Td[(11TW(j)i1Y+1 nnXi=1(1TW(j)i1))]TJ /F4 7.97 Tf 6.58 0 Td[(11TW(j)i[m(X))]TJ /F3 11.955 Tf 11.95 0 Td[(mj(xj)1] (B)where=(1,2,,n)Tandm(X)=(m(X1),m(X2),,m(Xn))T 86

PAGE 87

Firstconsiderm(X))]TJ /F3 11.955 Tf 11.95 0 Td[(mj(xj)1in(C-3).Notethatm(Xs))]TJ /F3 11.955 Tf 11.96 0 Td[(mj(xj)=+dXu6=jmu(Xsu)+(mj(Xsj))]TJ /F3 11.955 Tf 11.95 0 Td[(mj(xj))Thenm(X))]TJ /F3 11.955 Tf 11.96 0 Td[(mj(xj)1=1+0BBBBBBB@mj(X1j))]TJ /F3 11.955 Tf 11.95 0 Td[(mj(xj)mj(X2j))]TJ /F3 11.955 Tf 11.95 0 Td[(mj(xj)...mj(Xnj))]TJ /F3 11.955 Tf 11.96 0 Td[(mj(xj))1CCCCCCCA+0BBBBBBB@Pdu6=jmu(X1u)Pdu6=jmu(X2u)...Pdu6=jmu(Xnu)1CCCCCCCATherefore,(B-3)maybeexpressedas~mj(xj))]TJ /F3 11.955 Tf 11.96 0 Td[(mj(xj)=1 nnXi=1(1TW(j)i1))]TJ /F4 7.97 Tf 6.58 0 Td[(11TW(j)i+1 nnXi=1(1TW(j)i1))]TJ /F4 7.97 Tf 6.59 0 Td[(11TW(j)i1()]TJ /F7 11.955 Tf 13.89 2.65 Td[(Y)+1 nnXi=1(1TW(j)i1))]TJ /F4 7.97 Tf 6.58 0 Td[(1nXl=1KH(Xl)]TJ /F8 11.955 Tf 11.95 0 Td[(X(j)i)(dXs6=jms(Xls))+1 nnXi=1(1TW(j)i1))]TJ /F4 7.97 Tf 6.58 0 Td[(1nXl=1KH(Xl)]TJ /F8 11.955 Tf 11.95 0 Td[(X(j)i)[mj(Xlj))]TJ /F3 11.955 Tf 11.96 0 Td[(mj(xj)]=T1+T2+T3+T4 (B)Inordertoprovelemma3.1,welookatT1,T2,T3,T4separately.Firstnotethat E(T1)=0(B)LetusthenlookatT2.NotethatT2=1 nnXi=1(1TW(j)i1))]TJ /F4 7.97 Tf 6.59 0 Td[(11TW(j)i1()]TJ /F7 11.955 Tf 13.89 2.65 Td[(Y)=)]TJ /F7 11.955 Tf 13.89 2.65 Td[(Yand)]TJ /F7 11.955 Tf 13.89 2.66 Td[(Y=1 nnXi=1(Yi)]TJ /F5 11.955 Tf 11.95 0 Td[()=1 n(m(Xi)+i)Apparently,p n()]TJ /F7 11.955 Tf 13.89 2.65 Td[(Y)=O(1) 87

PAGE 88

therefore,T2=O(n)]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 2)Andundertheassumptioninlemma3.1thathj=O(n)]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 5),wehave T2=o(h2j)(B)Next,weworkonT3andT4.First,recallthathjisthebandwidthforthejthcovariate;andHisthe(d)]TJ /F7 11.955 Tf 12.46 0 Td[(1)(d)]TJ /F7 11.955 Tf 12.46 0 Td[(1)bandwidthmatrixfortheremaining(d)]TJ /F7 11.955 Tf 12.47 0 Td[(1)covariates.From[ 8 ],wehave 1 nnXi=1KH(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l)]TJ /F8 11.955 Tf 11.96 0 Td[(X()]TJ /F6 7.97 Tf 6.58 0 Td[(j)i) f(X(j)i)=f)]TJ /F6 7.97 Tf 6.59 0 Td[(j(X()]TJ /F6 7.97 Tf 6.58 0 Td[(j)l) f(X(j)l)+o(1) (B) Further,recallproposition1intheappendixofthe[ 44 ], 1 n1TW(j)i1=f(X(j)i)+o(1) (B) using(B-7)and(B-8),wehaveT3=1 nnXi=1(1TW(j)i1))]TJ /F4 7.97 Tf 6.58 0 Td[(1nXl=1KH(Xl)]TJ /F8 11.955 Tf 11.96 0 Td[(X(j)i)(dXs6=jms(Xls))=1 n2nXi=1[1 f(X(j)i)+o(1)]nXl=1KH(Xl)]TJ /F8 11.955 Tf 11.96 0 Td[(X(j)i)(dXs6=jms(Xls))=1 n2nXl=1Khj(Xlj)]TJ /F3 11.955 Tf 11.96 0 Td[(xj)(dXs6=jms(Xls))nXi=1[1 f(X(j)i)+o(1)]KH(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l)]TJ /F8 11.955 Tf 11.96 0 Td[(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)i)=1 nnXl=1Khj(Xlj)]TJ /F3 11.955 Tf 11.95 0 Td[(xj)(dXs6=jms(Xls))[1 nnXi=1KH(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l)]TJ /F8 11.955 Tf 11.95 0 Td[(X()]TJ /F6 7.97 Tf 6.58 0 Td[(j)i) f(X(j)i)+1 nnXi=1KH(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l)]TJ /F8 11.955 Tf 11.95 0 Td[(X()]TJ /F6 7.97 Tf 6.58 0 Td[(j)i)o(1)]=1 nnXl=1Khj(Xlj)]TJ /F3 11.955 Tf 11.95 0 Td[(xj)(dXs6=jms(Xls))[f)]TJ /F6 7.97 Tf 6.58 0 Td[(j(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l) f(X(j)l)+o(1)]=T31+o(1)T32withT31=1 nPnl=1Khj(Xlj)]TJ /F3 11.955 Tf 12.95 0 Td[(xj)(Pds6=jms(Xls))f)]TJ /F13 5.978 Tf 5.75 0 Td[(j(X()]TJ /F13 5.978 Tf 5.76 0 Td[(j)l) f(X(j)l)andT32=1 nPnl=1Khj(Xlj)]TJ /F3 11.955 Tf -423.75 -23.91 Td[(xj)(Pds6=jms(Xls)). 88

PAGE 89

TondtheexpectationofT3,weconsiderT31andT32separately.UsingTaylorapproximationaboutxjandunderassumptions,wewillhaveE(T31)=E[1 nnXl=1Khj(Xlj)]TJ /F3 11.955 Tf 11.96 0 Td[(xj)(dXs6=jms(Xls))f)]TJ /F6 7.97 Tf 6.59 0 Td[(j(X()]TJ /F6 7.97 Tf 6.58 0 Td[(j)l) f(X(j)l)]=E[Khj(Xlj)]TJ /F3 11.955 Tf 11.96 0 Td[(xj)[dXs6=jms(Xls)]f)]TJ /F6 7.97 Tf 6.58 0 Td[(j(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l) f(X(j)l)]=ZKhj(Xlj)]TJ /F3 11.955 Tf 11.95 0 Td[(xj)[dXs6=jms(Xls)]f)]TJ /F6 7.97 Tf 6.59 0 Td[(j(X()]TJ /F6 7.97 Tf 6.58 0 Td[(j)l) f(X(j)l)f(Xl)dXl=ZK(u)[dXs6=jms(Xls)]f)]TJ /F6 7.97 Tf 6.59 0 Td[(j(X()]TJ /F6 7.97 Tf 6.58 0 Td[(j)l) f(X(j)l)f(Xl1,,xj+hju,,Xld)dudX)]TJ /F6 7.97 Tf 6.59 0 Td[(jl=bj1+o(h2j)AndusingTaylorapproximationandsimilarmethodusedtoobtain(B-7),wecangeto(1)T32=o(h2j)Therefore, E(T3)=bj1+o(h2j)(B)Next,weworkongettingtheexpectationofT4.Substituting(B-7)and(B-8)intotheexpressionofT4,wehaveT4=1 n2nXi=1[1 f(X(j)i)+o(1)]nXl=1KH(Xl)]TJ /F8 11.955 Tf 11.95 0 Td[(X(j)i)[mj(Xlj))]TJ /F3 11.955 Tf 11.96 0 Td[(mj(xj)]=1 n2nXl=1Khj(Xlj)]TJ /F3 11.955 Tf 11.95 0 Td[(xj)[mj(xlj))]TJ /F3 11.955 Tf 11.96 0 Td[(mj(xj)]nXi=1[(f)]TJ /F4 7.97 Tf 6.59 0 Td[(1(X(j)i)+o(1))KH(X()]TJ /F6 7.97 Tf 6.58 0 Td[(j)l)]TJ /F8 11.955 Tf 11.96 0 Td[(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)i)]=1 nnXl=1Khj(Xlj)]TJ /F3 11.955 Tf 11.96 0 Td[(xj)[mj(Xlj))]TJ /F3 11.955 Tf 11.96 0 Td[(mj(xj)][f)]TJ /F6 7.97 Tf 6.59 0 Td[(j(X()]TJ /F6 7.97 Tf 6.58 0 Td[(j)l) f(X(j)l)+o(1)]=T41+o(1)T42whereT41=1 nPnl=1Khj(Xlj)]TJ /F3 11.955 Tf 12.36 0 Td[(xj)[mj(Xlj))]TJ /F3 11.955 Tf 12.37 0 Td[(mj(xj)]f)]TJ /F13 5.978 Tf 5.76 0 Td[(j(X()]TJ /F13 5.978 Tf 5.75 0 Td[(j)l) f(X(j)l),T42=1 nPnl=1Khj(Xlj)]TJ /F3 11.955 Tf -444.4 -23.91 Td[(xj)[mj(Xlj))]TJ /F3 11.955 Tf 11.96 0 Td[(mj(xj)],andhj,Hhavethesamedenitionsasbefore. 89

PAGE 90

Now,welookatT41andT42separately.First,inanapproachsimilartothatusedtoobtaintheexpectationofT31,wehaveE[T41]=E[Khj(Xlj)]TJ /F3 11.955 Tf 11.95 0 Td[(xj)(mj(Xlj))]TJ /F3 11.955 Tf 11.95 0 Td[(mj(xj))f)]TJ /F6 7.97 Tf 6.58 0 Td[(j(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l) f(X(j)l)]=ZK(u)[mj(xj+hju))]TJ /F3 11.955 Tf 11.96 0 Td[(mj(xj)]f)]TJ /F6 7.97 Tf 6.58 0 Td[(j(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l) f(X(j)l)f(Xl)dudX()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l=bj2+o(h2j)SimilarlytoT32,wehaveo(1)T42=o(h2j)Therefore, E(T4)=bj2+o(h2j)(B)Therefore,combining(B-5),(B-6),(B-9),(B-10)together,wehaveE(~mj(xj))]TJ /F3 11.955 Tf 11.96 0 Td[(mj(xj))=E(T1+T2+T3+T4)=0+bj1+bj2+o(h2j)whichcompletestheproofoflemma3.1.ProofofLemma3.2:First,itisclearthatVar(~mj(xj))=Var(~mj(xj))]TJ /F3 11.955 Tf 12.94 0 Td[(mj(xj))andplugginginthedecompositiongivenby(B-4)andusing(B-6),wehaveVar(~mj(xj))=Var(T1+T2+T3+T4)=Var(T1)Therefore,inordertogetthevarianceof~mj(xj),weonlyneedtodeterminethevarianceT1.WebeginbyrewritingtheexpressionforT1. 90

PAGE 91

Using(B-7),(B-8),denitionofW(j)i,andtheassumptionsinLemma3.2,weobtainT1=1 nnXi=1(1TW(j)i1))]TJ /F4 7.97 Tf 6.59 0 Td[(11TW(j)i=1 n2nXi=1(f()]TJ /F4 7.97 Tf 6.59 0 Td[(1)(X(j)i)+o(1))nXl=1KH(Xl)]TJ /F8 11.955 Tf 11.96 0 Td[(X(j)i)l=1 n2nXl=1Khj(Xlj)]TJ /F3 11.955 Tf 11.96 0 Td[(xj)jnXi=1[(f()]TJ /F4 7.97 Tf 6.59 0 Td[(1)(X(j)i)+o(1))KH(X()]TJ /F6 7.97 Tf 6.58 0 Td[(j)l)]TJ /F8 11.955 Tf 11.96 0 Td[(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)i)]=1 n2nXl=1Khj(Xlj)]TJ /F3 11.955 Tf 11.96 0 Td[(xj)j[nXi=1KH(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l)]TJ /F8 11.955 Tf 11.95 0 Td[(X()]TJ /F6 7.97 Tf 6.58 0 Td[(j)i) f(X(j)i)+o(1)nXi=1KH(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l)]TJ /F8 11.955 Tf 11.96 0 Td[(X()]TJ /F6 7.97 Tf 6.58 0 Td[(j)i)]=1 nnXl=1Khj(Xlj)]TJ /F3 11.955 Tf 11.96 0 Td[(xj)j[f)]TJ /F6 7.97 Tf 6.59 0 Td[(j(X()]TJ /F6 7.97 Tf 6.58 0 Td[(j)l) f(X(j)l)+o(1)] (B)Fromtheproofinlemma4.1,weknowE(T1)=0.Therefore,weonlyneedtoworkonE(T21)togetthevarianceofT1.By(B-11),wecanwriteE[T21]=E[1 nnXl=1Khj(Xlj)]TJ /F3 11.955 Tf 11.95 0 Td[(xj)j[f)]TJ /F6 7.97 Tf 6.59 0 Td[(j(X()]TJ /F6 7.97 Tf 6.58 0 Td[(j)l) f(X(j)l)+o(1)]]2=E[1 nnXl=1Khj(Xlj)]TJ /F3 11.955 Tf 11.95 0 Td[(xj)jf)]TJ /F6 7.97 Tf 6.59 0 Td[(j(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l) f(X(j)l)+1 nnXl=1Khj(Xlj)]TJ /F3 11.955 Tf 11.95 0 Td[(xj)jo(1)]2=E[V1+V2o(1)]2=E[V21+V22o(1)+2V1V2o(1)]whereV1=1 nPnl=1Khj(Xlj)]TJ /F3 11.955 Tf 11.95 0 Td[(xj)jf)]TJ /F13 5.978 Tf 5.76 0 Td[(j(X()]TJ /F13 5.978 Tf 5.75 0 Td[(j)l) f(X(j)l)andV2=1 nPnl=1Khj(Xlj)]TJ /F3 11.955 Tf 11.95 0 Td[(xj)j.Now,rstconsiderV22.ItiseasilyseenthatV22=1 n2nXl=1K2hj(Xlj)]TJ /F3 11.955 Tf 11.96 0 Td[(xj)2j+1 n2nXl=1nXk6=lKhj(Xlj)]TJ /F3 11.955 Tf 11.96 0 Td[(xj)Khj(Xkj)]TJ /F3 11.955 Tf 11.96 0 Td[(xj)jkThen E(V22)=2 n2nXl=1K2hj(Xlj)]TJ /F3 11.955 Tf 11.96 0 Td[(xj)+1 n2nXl=1nXk6=lKhj(Xlj)]TJ /F3 11.955 Tf 11.96 0 Td[(xj)Khj(Xkj)]TJ /F3 11.955 Tf 11.96 0 Td[(xj)n(kXk)]TJ /F8 11.955 Tf 11.96 0 Td[(Xlk)(B)where2iscommonvarianceofifori=1,2,,n.RecalltheresultsinProposition3intheappendixof[ 44 ], 91

PAGE 92

1 n2nXl=1K2hj(Xlj)]TJ /F3 11.955 Tf 11.96 0 Td[(xj)=O(1 nhj) (B) and 1 n2nXl=1nXk6=lKhj(Xlj)]TJ /F3 11.955 Tf 11.95 0 Td[(xj)Khj(Xkj)]TJ /F3 11.955 Tf 11.95 0 Td[(xj)n(kXk)]TJ /F8 11.955 Tf 11.95 0 Td[(Xlk)=O(1 nhj) (B) inproofoflemma4in[ 44 ].Therefore,substituting(B-13)and(B-14)in(B-12),wewillhave E(V22o(1))=o(1 nhj)(B)Now,forE(V21),wehaveE[V21]=2 n2nXl=1K2hj(Xlj)]TJ /F3 11.955 Tf 11.96 0 Td[(xj)(f)]TJ /F6 7.97 Tf 6.59 0 Td[(j(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l))2 f2(X(j)l)+1 n2nXl=1nXk6=lKhj(Xlj)]TJ /F3 11.955 Tf 11.96 0 Td[(xj)Khj(Xkj)]TJ /F3 11.955 Tf 11.96 0 Td[(xj)f)]TJ /F6 7.97 Tf 6.59 0 Td[(j(X()]TJ /F6 7.97 Tf 6.58 0 Td[(j)l) f(X(j)l)f)]TJ /F6 7.97 Tf 6.59 0 Td[(j(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)k) f(X(j)k)n(kXk)]TJ /F8 11.955 Tf 11.96 0 Td[(Xlk)=V11+V12LetuslookatV11andV12separately.Firstly,intheproofofproposition3in[ 44 ],ithasbeenshownthat ZK2(u)f(t+Hu)du=(K2)f(t)+o(1) (B) Using(B-16),methodssimilartothoseusedtodevelop(B-16),and[ 8 ],wehaveE[K2hj(Xlj)]TJ /F3 11.955 Tf 11.95 0 Td[(xj)f2)]TJ /F6 7.97 Tf 6.58 0 Td[(j(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l) f2(X(j)l)]=ZK2hj(Xlj)]TJ /F3 11.955 Tf 11.95 0 Td[(xj)f2)]TJ /F6 7.97 Tf 6.59 0 Td[(j(X()]TJ /F6 7.97 Tf 6.58 0 Td[(j)l) f(X(j)l)f(Xl)dXl=1 hjZZK2(u)f2)]TJ /F6 7.97 Tf 6.58 0 Td[(j(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l) f2(X(j)l)f(Xl1,,xj+hju,,Xld)dudX()]TJ /F6 7.97 Tf 6.58 0 Td[(j)l=1 hjZf2)]TJ /F6 7.97 Tf 6.58 0 Td[(j(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l) f2(X(j)l)dX()]TJ /F6 7.97 Tf 6.59 0 Td[(j)lZK2(u)f(Xl1,,xj+hju,,Xld)du=1 hj[Zf2)]TJ /F6 7.97 Tf 6.58 0 Td[(j(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l) f(X(j)l)(K2)dX()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l+o(1)] (B) 92

PAGE 93

Then,using(B-17)wehaveV11=2 n1 hj[(K2)Zf2)]TJ /F6 7.97 Tf 6.59 0 Td[(j(X()]TJ /F6 7.97 Tf 6.58 0 Td[(j)l) f(X(j)l)dX()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l+o(1)]=2 nhj(K2)Zf2)]TJ /F6 7.97 Tf 6.59 0 Td[(j(X()]TJ /F6 7.97 Tf 6.58 0 Td[(j)l) f(X(j)l)dX()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l+o(1 nhj)Next,usingsimilarcalculation,wehaveV12=R(K2) nhj[Zf)]TJ /F6 7.97 Tf 6.59 0 Td[(j(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l)dX()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l]2+o(1 nhj)whereRisdenedinassumption3inChapter3.ItfollowsthatE[V21]=V11+V12=2 nhj(K2)Zf2)]TJ /F6 7.97 Tf 6.59 0 Td[(j(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l) f(X(j)l)dX()]TJ /F6 7.97 Tf 6.58 0 Td[(j)l+R(K2) nhj[Zf)]TJ /F6 7.97 Tf 6.58 0 Td[(j(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l)dX()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l]2+o(1 nhj)=2j+o(1 nhj) (B)where2jisdenedinlemma3.2.Finally,considerV1V2.First,basedonthedenitionofV1,itisclearthatE[p nhjV1]=0andusing,wehaveVar(p nhjV1)=E(nhjV21)=O(1)Thus,V1=O(1 p nhj)Usingthesameargument,wehaveV2=O(1 p nhj) 93

PAGE 94

Thus, E[o(1)V1V2]=o(1 nhj)(B)Therefore,combining(C-15),(C-18)and(C-19),wehaveshownlemma4.2.ProofofTheorem3.1:Theorem3.1followsdirectlyfromlemma3.1andlemma3.2.ProofofTheorem3.2:Giventheresultsoftheorem3.1,inordertoshowtheorem3.2,itissufcienttoshowthatthecovariancebetween~mj(xj)and~mk(xk)isasymptoticallyzero.foranyj,k.Usingtechniquessimilartothoseusedtoprovelemmas3.1and3.2,wehaveCov(~mj(xj),~mk(xk))=Cov(~mj(xj))]TJ /F3 11.955 Tf 11.95 0 Td[(mj(xj),~mk(xk))]TJ /F3 11.955 Tf 11.95 0 Td[(mk(xk))=Cov(1 nnXi=1Pnl=1KH(Xl)]TJ /F8 11.955 Tf 11.96 0 Td[(X(j)i)l Pnt=1KH(Xt)]TJ /F8 11.955 Tf 11.95 0 Td[(X(j)i),1 nnXi=1Pnl=1KH(Xl)]TJ /F8 11.955 Tf 11.96 0 Td[(X(k)i)l Pnt=1KH(Xt)]TJ /F8 11.955 Tf 11.95 0 Td[(X(k)i))=1 n2Cov(nXl=1Khj(Xlj)]TJ /F3 11.955 Tf 11.95 0 Td[(xj)lf)]TJ /F6 7.97 Tf 6.59 0 Td[(j(X()]TJ /F6 7.97 Tf 6.58 0 Td[(j)l) f(X(j)l),nXl=1Khk(Xlk)]TJ /F3 11.955 Tf 11.96 0 Td[(xk)lf)]TJ /F6 7.97 Tf 6.58 0 Td[(k(X()]TJ /F6 7.97 Tf 6.58 0 Td[(k)l) f(X(k)l))=1 n2E[nXl=1Khj(Xlj)]TJ /F3 11.955 Tf 11.96 0 Td[(xj)lf)]TJ /F6 7.97 Tf 6.58 0 Td[(j(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l) f(X(j)l)][nXm=1Khk(Xmk)]TJ /F3 11.955 Tf 11.95 0 Td[(xk)mf)]TJ /F6 7.97 Tf 6.59 0 Td[(k(X()]TJ /F6 7.97 Tf 6.59 0 Td[(k)m) f(X(k)m)]=1 n2E[Khj(Xlj)]TJ /F3 11.955 Tf 11.96 0 Td[(xj)Khk(Xlk)]TJ /F3 11.955 Tf 11.96 0 Td[(xk)f)]TJ /F6 7.97 Tf 6.59 0 Td[(j(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l) f(X(j)l)f)]TJ /F6 7.97 Tf 6.58 0 Td[(k(X()]TJ /F6 7.97 Tf 6.58 0 Td[(k)l) f(X(k)l)2l]+1 n2nXl=1nXm6=lE[Khj(Xlj)]TJ /F3 11.955 Tf 11.95 0 Td[(xj)Khk(Xmk)]TJ /F3 11.955 Tf 11.96 0 Td[(xk)f)]TJ /F6 7.97 Tf 6.58 0 Td[(j(X()]TJ /F6 7.97 Tf 6.59 0 Td[(j)l) f(X(j)l)f)]TJ /F6 7.97 Tf 6.58 0 Td[(k(X()]TJ /F6 7.97 Tf 6.58 0 Td[(k)m) f(X(k)m)lm] (B)By[ 7 ],(B-20)isO(n)]TJ /F4 7.97 Tf 6.58 0 Td[(1).Therefore,thecovariancebetween~mj(xj)and~mk(xk)convergestozero.ProofofTheorem3.3:Letusrewrite~m(Xs))]TJ /F3 11.955 Tf 11.95 0 Td[(m(Xs)rst.~m(Xs))]TJ /F3 11.955 Tf 11.95 0 Td[(m(Xs)=^+dXj=1~mj(Xsj))]TJ /F5 11.955 Tf 11.96 0 Td[()]TJ /F6 7.97 Tf 18.02 14.95 Td[(dXj=1mj(Xsj)=(Y)]TJ /F5 11.955 Tf 11.95 0 Td[()+dXj=1[~mj(Xsj))]TJ /F3 11.955 Tf 11.96 0 Td[(mj(Xsj)] (B) 94

PAGE 95

Weknowthat)]TJ /F7 11.955 Tf 13.89 2.66 Td[(Y=O(n)]TJ /F11 5.978 Tf 7.79 3.26 Td[(1 2)andE[~mj(Xsj))]TJ /F3 11.955 Tf 11.95 0 Td[(mj(Xsj)]=bj+o(h2j)Therefore,E[~m(Xs))]TJ /F3 11.955 Tf 11.95 0 Td[(m(Xs)]=O(n)]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 2)+dXj=1bj+o(dXj=1h2j)Andunderassumptioninlemma3.2,weknowthathj=O(n)]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 5),forj=1,2,,d,thus,wehaven2 5E[~m(Xs))]TJ /F3 11.955 Tf 11.96 0 Td[(m(Xs)]=o(1)Next,considerthevarianceof~m(Xs).Usingtheresultfromtheorem3.2andunderassumptionhj=O(n)]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 5),forj=1,2,,d,wewillhaveVar[n2 5(~m(Xs))]TJ /F3 11.955 Tf 11.96 0 Td[(m(Xs))]=Var[n2 5dXj=1(~mj(Xsj))]TJ /F3 11.955 Tf 11.96 0 Td[(mj(Xsj))]=n4 5dXj=1Var(~mj(Xsj))=dXj=1n4 5 nhj2j+dXj=1o(n4 5 nhj)!2where2isdenedintheorem3.3.Therefore,theorem3.3isproven. 95

PAGE 96

REFERENCES [1] Ali,M.M.(1979),Analysisofstationaryspatial-temporalprocesses:estimationandprediction.Biometrika. [2] Altman,N.S.(1990),KernelSmoothingofDatawithCorrelatedErrors.JournaloftheAmericanStatisticalAssociation. [3] Basu,S&Reinsel,G.C.(1994),Regressionmodelswithspatiallycorrelatederrors.JournaloftheAmericanStatisticalAssociation. [4] Benhenni,K.,Hedli-Griche,S.&Rachdi,M.(2010),Estimationoftheregressionoperatorfromfunctionalxed-designwithcorrelatederrors.JournalofMultivariateAnalysis. [5] Bliznyuk,N.,Carrol,R.J.,Genton,M.G.(2012),Variogramestimationinthepresenceoftrend. [6] Brabanter,K.D.Brabanter,J.D.,Suykens,J.A.K,&Moor,B.D.(2011),Kernelregressioninpresenceofcorrelatederrors.ournalofMachineLearningResearch. [7] Cai,Z&Fan,J.(2000),AverageregressionsurfacefordependentdataJournalofMultivariateAnalysis. [8] Chen,R.&Hardle,W.(1995),Estimationandvariableselectioninadditivenonparametricregressionmodels.DiscussionPaper. [9] Chu,C.,Ni,Y.,Tan,G.,Saunders,C.J.&Ashburner,J.(2011),KernelregressionforfMRIpatternprediction.NeuroImage. [10] Cressie,N.A.C.(1993),Statisticsforspatialdata,Wiley,NewYork. [11] DasS.,Rao,T.S.&Boshnakov,G.N.(2012),Ontheestimationofparametersofvariogramsofspatialstationaryisotropicrandomprocess. [12] Fan,J.(1992),Design-adaptivenonparametricregression.JournaloftheAmeri-canStatisticalAssociation. [13] Fan,J.&Gijbels,I.Localpolynomialmodellinganditsapplications.Chapman&Hall,London1996 [14] Fan,J.&Li,Q.(2003),Akernel-basedmethodforestimatingadditivepartiallylinearmodels.StatisticaSinica. [15] Francisco-Fernandez,M.&Quintela-del-Ro,A.(2011),Anonparametricanalysisofthespatialdistributionoftheearthquakesmagnitude.SeismologicalSocietyofAmerica. 96

PAGE 97

[16] Francisco-Fernandez,M.&Vilar-Fernandez,J.M.(2001),Localpolynomialregressionestimationwithcorrelatederrors.CommunicationinStatistics,A.30,1271-1293. [17] Furrer,R.,Genton,M.G.&Nychka,D.(2006),Covariancetaperingforinterpolationoflargespatialdatasets.JournalofComputationalandGraphi-calStatistics,15,502523. [18] Gao,J.,Lu,Z.,&Tjstheim,D.(2006),Estimationinsemiparametricspatialregression.TheAnnalsofStatistics. [19] Guan,Y.(2011),Bias-correctedvarianceestimationandhypothesistestingforspatialandmarkedpointprocessesusingsubsampling.Biometrics. [20] Hardle&Muller(1997),Multivariateandsemiparametrickernelregression. [21] Hardle,W.&Tsybakov,A.B.(1995),Additivenonparametricregressiononprincipalcomponents.Journalofnonparametricstatsitics. [22] Hallin,M.,Lu,X.&Tran,L.T.(2004),Locallinearspatialregression.TheAnnalsofStatistics. [23] Hastie,T.J.&Tibshirani,R.J.(1990),Generalizedadditivemodels.Chapman&Hall. [24] Konishi,S.&Kitagawa,G.(2008),InformationCriteriaandStatisticalModeling.Springer. [25] Leontief,W.(1947),Introductiontoatheoryofaninternalstructureoffunctionalrelationships.Econometrika. [26] Lin,X.&Carroll,R.J.(2000),Nonparametricfunctionestimationforclustereddatawhenthepredictorismeasuredwithout/witherror.JournaloftheAmericanStatisticalAssociation. [27] Linton,O.(1997),Efcientestimationofadditivenonparametricregressionmodels.Biometrika. [28] Linton,O.&NielsenJ.P.(1995),Kernelmethodofestimatingstructurednonparametricregressionbasedonmarginalintegration.Biometrika. [29] Martins-Filho,C.&Yao,F.(2009),Nonparametricreressionestimationwithgeneralparametricerrorvariance.JournalofMultivariateAnalysis. [30] Opsomer,J.D.,Wang,Y.&Yang,Y.(2001),Nonparametricregressionwithcorrelatederrors.Statist.Sci.. [31] Ruckstuhl,A.F.,Welsh,A.H.&Carroll,R.J.(2000),Nonparametricfunctionestimationoftherelationshipbetweentworepeatedlymeasuredvariables.StatisticaSinica. 97

PAGE 98

[32] Robinson,P.M.(2008),NonparametricRegressionwithSpatialData. [33] Ruppert,D.&Wand,M.P.(1994),Multivariateweightedleastsquaresregression.TheAnnalsofStatistics. [34] Severance-Lossin,E.&Sperlich,S.(1999),Estimationofderivativesforadditiveseparablemodels.AJournalofTheoreticalandAppliedStatistics. [35] Schabenberger,O.&GotwayC.A.(2005),Statisticalmethodsforspatialdataanalysis.Chapman&Hall/CRC. [36] Schimek,M.G.(1996),SmoothingandRegression:approaches,computationandapplication.Wiley,NewYork. [37] Lahiri,SN,Lee,Y.,&Cressie,N.(2002),Onasymptoticdistributionandasymptoticeffciencyofleastsquaresestimatorsorspatialvariogramparameters.JournalofStatisticalPlanningandInference. [38] Stone,C.J.(1985),Additiveregressionandothernonparametricmodels.TheAnnalsofStatistics. [39] Stone,C.J.(1977),ConsistentnonparametricregressionTheAnnalsofStatistics. [40] Su,L.&Ullah,A.(2006),Moreefcientestimationinnonparametricregressionwithnonparametricautocorrelatederrors.EconometricTheory. [41] Takeda,H.,Farsiu,S.&Milanfar,P.(2007),Kernelregressionforimageprocessingandreconstruction.IEEEtransactionsonimageprocessing. [42] Venables,W.N.&Ripley,B.(1994),ModernappliedstatisticswithS-plus.SpringerVerlag,NewYork. [43] Vilar-Fernandez,J.M&Francisco-Fernandez,M.(2002),LocalpolynomialregressionsmootherswithAR-errorstructure.Test. [44] Wang,S&Young,L.J.(2013),Multivariatekernelregressionwithspatiallycorrelatederrors. [45] Xiao,Z.,Linton,O.B.,Carroll,R.J.&Mammen,E.(2003),Moreefcientlocalpolynomialestimationinnonparametricregressionwithautocorrelatederrors.JournaloftheAmericanStatisticalAssociation. 98

PAGE 99

BIOGRAPHICALSKETCH ShuWangwasbornandraisedinXi'an,Shaanxi,China.Shuwastheonlychild.SheearnedherB.S.degreeinstatisticsfromBeijingNormalUniversityin2008.In2008shejoinedtheDepartmentofStatisticsatNorthCarolinaStateUniversity.Andin2009,shetransferredtotheDepartmentofStatisticsattheUniversityofFlorida.ShereceivedherPh.D.fromtheDepartmentofStatisticsattheUniversityofFloridainthesummerof2013.Whilepursuingherdegree,ShuworkedasaresearchassistantfortheDepartmentofStatisticsandNationalAgricultureStatisticsService.SheinternedatJohnson&JohnsonPharmaceutical,R&Dinthesummerof2011.ShehaspresentedherresearchatJSMin2012. 99