Statistical Modelling and Regression Structures: Festschrift in Honour of Ludwig Fahrmeir

485 Pages·2010·16.558 MB·English

by  Paul H. C. Eilers (auth.), Thomas Kneib, Gerhard Tutz (eds.)

Statistical Modelling and Regression Structures
·
Thomas Kneib Gerhard Tutz
Editors
Statistical Modelling and
Regression Structures
Festschrift in Honour of Ludwig Fahrmeir
Editors
Prof.Dr.ThomasKneib Prof.Dr.GerhardTutz
Institutfu¨rMathematik Institutfu¨rStatistik
CarlvonOssietzkyUniversita¨tOldenburg Ludwig-Maximilians-Universita¨tMu¨nchen
26111Oldenburg Akademiestraße1
Germany 80799Mu¨nchen
[email protected] Germany
[email protected]
ISBN978-3-7908-2412-4 e-ISBN978-3-7908-2413-1
DOI10.1007/978-3-7908-2413-1
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2009943264
(cid:2)c Springer-VerlagBerlinHeidelberg2010
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Foreword
Thecollectedcontributionscontainedwithinthisbookhavebeenwrittenbyfriends
andcolleaguestoacknowledgeLudwigFahrmeir’swidespreadandimportantimpact
onStatisticsasascience,whilecelebratinghis65thBirthday.
Asayoungstudent,LudwigstartedhiscareerasaMathematician,buthequickly
turnedintoarisingandshiningstarwithintheGermanandinternationalStatistics
community. He soon obtained both his PhD and his Habilitation at the Technical
UniversityofMunich.AfterashortperiodasavisitingprofessorattheUniversity
ofDortmund,hereturnedtohishomelandBavariaandwasappointedFullProfessor
ofStatisticsattheUniversityofRegensburg,attheageof32.
Someyearslater,hemovedtothecapitalofBavariaandbecameProfessoratthe
DepartmentofStatisticsattheUniversityofMunich.Hisappointmenthadsignificant
impactontheDepartmentsince,soonafterhisarrival,Ludwigstartedaninitiative
to establish a collaborativeresearchcenteronthe “StatisticalAnalysisofDiscrete
Structures.” After a successful application for initial funding,further funding was
extendedseveral times, until the research center reached the maximumperiod for
fundingin2006.Duringthecompleteduration,Ludwigservedasaspeakerofthe
researchcenterand–tociteoneofthefinalreferees–“manageditinaneasyand
efficientwayandcontributedseveralimportantresults.”
Duringthelastfortyyears,Ludwig’sworkhashadtremendousimpactontheSta-
tisticscommunity.Hewasamongthefirstresearcherstorecognizetheimportance
ofgeneralizedlinearmodelsandcontributedinaseriesofpaperstothetheoretical
backgroundof that modelclass. His interest in statistical modelling led to the or-
ganizationofaworkshopon“StatisticalModellingandGeneralizedLinearModels
(GLIM)”inMunichin1992andculminatedinthehighlycitedmonographon“Mul-
tivariate Statistical ModellingBased on GeneralizedLinear Models”that saw two
printingsandremainstobeakeyreferenceonappliedstatisticalmodellingutilizing
generalized linear models. Ludwig also had great influence on the creation of the
Statistical Modelling Society, and is currentlyon the advisoryboard of the corre-
spondingjournalon“StatisticalModelling.”Boththesocietyandjournalemerged
outoftheearlyGLIMworkshopsandproceedings.
v
vi Foreword
Ofcourse,Ludwig’sworkisdefinitelynotrestrictedtogeneralizedlinearmodels
but–onthecontrary–spansawiderangeofmodernStatistics.Heco-authoredor
co-editedseveralmonographs,e.g.onMultivariateStatistics,StochasticProcesses,
Measurementof Credit Risks, as well as popular textbookson Regression and an
IntroductiontoStatistics.Hisrecentresearchcontributionsaremostlyconcentrated
insemiparametricregressionandspatialstatisticswithinaBayesianframework.
When first circulating the idea of a Festschrift for the celebration of Ludwig’s
65thbirthday,allpotentialcontributorswereextremelypositive,manyimmediately
agreeing to contribute. These reactions atest to Ludwig’s high personal and pro-
fessionalappreciationinthestatisticalcommunity.Thefarreachingandvarietyof
subjectscoveredwithinthesecontributionsalsorepresentsLudwig’sbroadinterest
andimpactinmanybranchesofmodernStatistics.
BotheditorsofthisFestschriftwereluckyenoughtoworkwithLudwigatseveral
occasionsandinparticularearlyintheircareersasPhDstudentsandPostDocs.His
personalandprofessionalmentorshipandhisstrongcommitmentmadehimaperfect
supervisorandhispatient,confidentandencouragingworkingstylewillalwaysbe
rememberedbyallofhisstudentsandcolleagues.Ludwigalwaysprovidedafriendly
workingenvironmentthatmadeitapleasureandanhonortobeapartofhisworking
group.We areproudtobeableto saythatLudwigismuchmorethana colleague
butturnedintoafriendforbothofus.
OldenburgandMunich,January2010 ThomasKneib,GerhardTutz
Acknowledgements
Theeditorswouldliketoexpresstheirgratitudeto
• allauthorsofthisvolumefortheiragreementtocontributeandtheireasycoop-
erationatseveralstagesofputtingtogetherthefinalversionoftheFestschrift.
• JohannaBrandt,JanGertheiss,AndreasGroll,FelixHeinzl,SebastianPetry,Jan
Ulbrichtand Stephanie Rubenbauerfor their invaluable contributionsin proof-
readingandcorrectionofthepapers,aswellasinsolvingseveralLATEX-related
problems.
• theSpringerVerlagforagreeingtopublishthisFestschriftandinparticularNils-
Peter Thomas, Alice Blanck and Frank Holzwarth for the smooth collabora-
tion in preparing th emanuscript.
vii
Contents
ListofContributors.............................................. xix
TheSmoothComplexLogarithmandQuasi-PeriodicModels .......... 1
PaulH.C.Eilers
1 Foreword ................................................ 1
2 Introduction.............................................. 1
3 DataandModels.......................................... 2
3.1 TheBasicModel.................................. 3
3.2 SplinesandPenalties .............................. 3
3.3 StartingValues.................................... 7
3.4 SimpleTrendCorrectionandPriorTransformation ..... 8
3.5 AComplexSignal................................. 8
3.6 Non-normalDataandCascadedLinks ................ 10
3.7 AddingHarmonics ................................ 11
4 MoretoExplore .......................................... 12
5 Discussion ............................................... 15
References..................................................... 17
P-splineVaryingCoefficientModelsforComplexData ................ 19
BrianD.Marx
1 Introduction ............................................. 19
2 “LargeScale”VCM,withoutBackfitting...................... 22
3 NotationandSnapshotofaSmoothingTool:B-splines.......... 24
3.1 GeneralKnotPlacement............................ 25
3.2 SmoothingtheKTBData........................... 25
4 UsingB-splinesforVaryingCoefficientModels................ 26
5 P-splineSnapshot:Equally-SpacedKnots&Penalization........ 28
5.1 P-splinesforAdditiveVCMs........................ 30
5.2 StandardErrorBands .............................. 30
6 OptimallyTuningP-splines ................................ 31
7 MoreKTBResults ........................................ 33
8 ExtendingP-VCMintotheGeneralizedLinearModel .......... 33
9 Two-dimensionalVaryingCoefficientModels ................. 36
ix
x Contents
9.1 Mechanicsof2D-VCMthroughExample ............. 37
9.2 VCMsandPenaltiesasArrays....................... 39
9.3 EfficientComputationUsingArrayRegression......... 40
10 DiscussionTowardMoreComplexVCMs..................... 41
References..................................................... 42
PenalizedSplines,MixedModelsandBayesianIdeas.................. 45
Go¨ranKauermann
1 Introduction.............................................. 45
2 NotationandPenalizedSplinesasLinearMixedModels ........ 46
3 ClassificationwithMixedModels ........................... 48
4 VariableSelectionwithSimplePriors ........................ 50
4.1 MarginalAkaikeInformationCriterion ............... 50
4.2 ComparisoninLinearModels ....................... 53
4.3 Simulation ....................................... 55
5 DiscussionandExtensions.................................. 56
References..................................................... 57
BayesianLinearRegression—DifferentConjugateModelsandTheir
(In)SensitivitytoPrior-DataConflict ............................... 59
GeroWalterandThomasAugustin
1 Introduction.............................................. 59
2 Prior-dataConflictinthei.i.d.Case .......................... 62
3 TheStandardApproachforBayesianLinearRegression(SCP) ... 64
3.1 Updateofβ|σ2 .................................. 65
3.2 Updateofσ2 ..................................... 66
3.3 Updateofβ ...................................... 67
4 An AlternativeApproachforConjugatePriorsin Bayesian
LinearRegression(CCCP) ................................. 68
4.1 Updateofβ|σ2 .................................. 71
4.2 Updateofσ2 ..................................... 71
4.3 Updateofβ ...................................... 75
5 DiscussionandOutlook.................................... 76
References..................................................... 77
AnEfficientModelAveragingProcedureforLogisticRegressionModels
UsingaBayesianEstimatorwithLaplacePrior ...................... 79
ChristianHeumannandMoritzGrenke
1 Introduction.............................................. 79
2 ModelAveraging ......................................... 80
2.1 Orthogonalization ................................. 81
2.2 UnrestrictedMaximumLikelihoodEstimation ......... 82
2.3 RestrictedApproximateMaximumLikelihoodEstimation 83
2.4 ModelAveraging.................................. 84
2.5 Algorithm........................................ 86
3 SimulationStudy ......................................... 86
Contents xi
4 ConclusionandOutlook.................................... 88
References..................................................... 89
PosteriorandCross-validatoryPredictiveChecks:A Comparisonof
MCMCandINLA............................................... 91
LeonhardHeld,BirgitSchro¨dleandHa˚vardRue
1 Introduction.............................................. 91
2 TheINLAApproach ...................................... 92
2.1 ParameterEstimationwithINLA .................... 92
2.2 PosteriorPredictiveModelCheckswithINLA ......... 94
2.3 Leave-one-outCross-validationwithINLA............ 95
3 PredictiveModelCheckswithMCMC ....................... 96
3.1 PosteriorPredictiveModelCheckswithMCMC........ 97
3.2 Leave-one-outCross-validationwithMCMC .......... 97
3.3 ApproximateCross-validationwithMCMC ........... 98
4 Application .............................................. 99
4.1 AComparisonofPosteriorPredictiveModelChecks....101
4.2 A Comparison of Leave-one-outCross-validated
PredictiveChecks .................................103
4.3 AComparisonofApproximateCross-validationwith
PosteriorandLeave-one-outPredictiveChecksusing
MCMC..........................................106
5 Discussion ...............................................107
References.....................................................109
DataAugmentationandMCMC forBinaryandMultinomialLogit
Models ........................................................ 111
SylviaFru¨hwirth-SchnatterandRudolfFru¨hwirth
1 Introduction..............................................111
2 MCMCEstimationBasedonDataAugmentationforBinary
LogitRegressionModels...................................113
2.1 WritingtheLogitModelasaRandomUtilityModel ....113
2.2 DataAugmentationBasedontheRandomUtilityModel.114
2.3 TwoNewSamplersBasedonthedRUMRepresentation.116
2.4 Finite Mixture Approximationsto the Logistic
Distribution ......................................118
3 MCMC Estimation Based on Data Augmentationfor the
MultinomialLogitRegressionModel.........................120
3.1 DataAugmentationintheRUM .....................121
3.2 DataAugmentationinthedRUM ....................121
4 MCMCSamplingwithoutDataAugmentation.................123
5 ComparisonoftheVariousMCMCAlgorithms ................125
6 ConcludingRemarks ......................................130
References.....................................................131
xii Contents
GeneralizedSemiparametricRegressionwithCovariatesMeasuredwith
Error.......................................................... 133
ThomasKneib,AndreasBrezgerandCiprianM.Crainiceanu
1 Introduction..............................................133
2 SemiparametricRegressionModelswithMeasurementError.....135
2.1 ObservationModel ................................135
2.2 MeasurementErrorModel..........................136
2.3 PriorDistributions.................................136
3 BayesianInference........................................139
3.1 Posterior&FullConditionals .......................139
3.2 ImplementationalDetails&Software.................141
4 Simulations ..............................................143
4.1 SimulationSetup..................................143
4.2 SimulationResults ................................144
5 IncidentHeartFailureintheARICStudy .....................150
6 Summary ................................................153
References.....................................................153
DeterminantsoftheSocioeconomicandSpatialPatternofUndernutrition
bySexinIndia:AGeoadditiveSemi-parametricRegressionApproach .. 155
ChristianeBelitz,JudithHu¨bner,StephanKlasenandStefanLang
1 Introduction..............................................155
2 TheData ................................................158
3 MeasurementandDeterminantsofUndernutrition..............160
3.1 Measurement.....................................160
3.2 DeterminantsofUndernutrition......................161
4 VariablesIncludedintheRegressionModel ...................162
5 StatisticalMethodology-SemiparametricRegressionAnalysis...167
6 Results ..................................................170
7 Conclusion...............................................177
References.....................................................178
BoostingforEstimatingSpatiallyStructuredAdditiveModels.......... 181
NikolayRobinzonovandTorstenHothorn
1 Introduction..............................................181
2 Methods.................................................183
2.1 Spatio-TemporalStructuredAdditiveModels ..........183
2.2 TreeBasedLearners ...............................187
2.3 GeneralizedAdditiveModel ........................188
3 Results ..................................................189
3.1 ModelIllustrations ................................189
3.2 ModelComparison ................................193
4 Discussion ...............................................194
References.....................................................195