Introduction to robust estimation and hypothesis testing

609 Pages·2005·2.62 MB·English

by  Rand R. Wilcox

Introduction
to Robust Estimation
and Hypothesis Testing
Second Edition
Rand R. Wilcox
Department of Psychology
University of Southern California
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Contents
Preface................................................................. xvii
Chapter1 Introduction .............................................. 1
1.1 ProblemswithAssumingNormality................................ 2
1.2 Transformations.................................................... 6
1.3 TheInfluenceCurve................................................ 7
1.4 TheCentralLimitTheorem......................................... 8
1.5 IstheANOVAFRobust?........................................... 9
1.6 Regression.......................................................... 10
1.7 MoreRemarks...................................................... 10
1.8 UsingtheComputer:RandS-PLUS................................. 11
1.9 SomeData-ManagmentIssues...................................... 13
1.9.1 EliminatingMissingValues................................ 17
Chapter2 AFoundationforRobustMethods........................ 19
2.1 BasicToolsforJudgingRobustness................................. 20
2.1.1 QualitativeRobustness.................................... 21
2.1.2 InfinitesimalRobustness................................... 23
2.1.3 QuantitativeRobustness................................... 25
2.2 SomeMeasuresofLocationandTheirInfluenceFunction........... 26
2.2.1 Quantiles.................................................. 26
2.2.2 TheWinsorizedMean..................................... 27
iii
iv Contents
2.2.3 TheTrimmedMean....................................... 29
2.2.4 M-MeasuresofLocation................................... 30
2.2.5 R-MeasuresofLocation.................................... 33
2.3 MeasuresofScale................................................... 33
2.3.1 MeanDeviationfromtheMean............................ 34
2.3.2 MeanDeviationfromtheMedian.......................... 35
2.3.3 MedianAbsoluteDeviation................................ 35
2.3.4 Theq-QuantileRange...................................... 35
2.3.5 TheWinsorizedVariance.................................. 36
2.4 Scale-EquivariantM-MeasuresofLocation.......................... 36
2.5 WinsorizedExpectedValues........................................ 38
Chapter3 EstimatingMeasuresofLocationandScale .............. 43
3.1 ABootstrapEstimateofaStandardError........................... 44
3.1.1 RandS-PLUSFunctionbootse............................ 46
3.2 DensityEstimators................................................. 47
3.2.1 NormalKernel............................................ 47
3.2.2 Rosenblatt’sShiftedHistogram............................ 48
3.2.3 TheExpectedFrequencyCurve............................ 48
3.2.4 AnAdaptiveKernelEstimator............................. 49
3.2.5 RandS-PLUSFunctionsskerd,kerden,kdplot,rdplot,
akerd,andsplot.......................................... 50
3.3 TheSampleTrimmedMean......................................... 56
3.3.1 RandS-PLUSFunctiontmean............................. 59
3.3.2 EstimatingtheStandardErroroftheTrimmedMean....... 59
3.3.3 RandS-PLUSFunctionswin,winvar,andtrimse.......... 64
3.3.4 EstimatingtheStandardErroroftheSampleMedian,M... 64
3.3.5 RandS-PLUSFunctionmsmedse........................... 65
3.4 TheFinite-SampleBreakdownPoint................................ 65
3.5 EstimatingQuantiles............................................... 66
3.5.1 EstimatingtheStandardErroroftheSampleQuantile...... 67
3.5.2 RandS-PLUSFunctionqse................................ 69
3.5.3 TheMaritz–JarrettEstimateoftheStandardErrorofxˆq..... 69
Contents v
3.5.4 RandS-PLUSFunctionmjse.............................. 70
3.5.5 TheHarrell–DavisEstimator............................... 71
3.5.6 RandS-PLUSFunctionhd................................. 72
ˆ
3.5.7 ABootstrapEstimateoftheStandardErrorofθ ........... 72
q
3.5.8 RandS-PLUSFunctionhdseb............................. 72
3.6 AnM-EstimatorofLocation........................................ 73
3.6.1 ComputinganM-EstimatorofLocation.................... 79
3.6.2 RandS-PLUSFunctionmest.............................. 81
3.6.3 EstimatingtheStandardErroroftheM-Estimator.......... 81
3.6.4 RandS-PLUSFunctionmestse............................ 84
3.6.5 ABootstrapEstimateoftheStandardErrorofµˆm.......... 84
3.6.6 RandS-PLUSFunctionmestseb........................... 85
3.7 One-StepM-Estimator.............................................. 85
3.7.1 RandS-PLUSFunctiononestep........................... 86
3.8 W-Estimators....................................................... 87
3.9 TheHodges–LehmannEstimator................................... 88
3.10 SkippedEstimators................................................. 88
3.10.1 RandS-PLUSFunctionsmomandbmean.................... 89
3.11 SomeComparisonsoftheLocationEstimators...................... 90
3.12 MoreMeasuresofScale............................................. 92
3.12.1 TheBiweightMidvariance................................. 93
3.12.2 RandS-PLUSFunctionbivar............................. 96
3.12.3 ThePercentageBendMidvariance......................... 96
3.12.4 RandS-PLUSFunctionpbvar............................. 98
3.12.5 TheInterquartileRange.................................... 98
3.12.6 RandS-PLUSFunctionidealf............................ 99
3.13 SomeOutlierDetectionMethods.................................... 99
3.13.1 RulesBasedonMeansandVariances...................... 99
3.13.2 AMethodBasedontheInterquartileRange................ 100
3.13.3 Carling’sModification..................................... 100
3.13.4 AMAD-MedianRule...................................... 101
3.13.5 RandS-PLUSFunctionsoutboxandout................... 101
3.14 Exercises........................................................... 102
vi Contents
Chapter4 ConfidenceIntervalsintheOne-SampleCase ........... 105
4.1 ProblemsWhenWorkingwithMeans............................... 105
4.2 Theg-and-hDistribution........................................... 110
4.3 InferencesAbouttheTrimmedMean............................... 113
4.3.1 RandS-PLUSFunctiontrimci............................ 117
4.4 BasicBootstrapMethods............................................ 117
4.4.1 ThePercentileBootstrapMethod.......................... 118
4.4.2 RandS-PLUSFunctiononesampb.......................... 119
4.4.3 Bootstrap-tMethod........................................ 119
4.4.4 BootstrapMethodsWhenUsingaTrimmedMean......... 121
4.4.5 Singh’sModification....................................... 125
4.4.6 RandS-PLUSFunctionstrimpbandtrimcibt............. 126
4.5 InferencesAboutM-Estimators..................................... 127
4.5.1 RandS-PLUSFunctionsmestciandmomci................ 129
4.6 ConfidenceIntervalsforQuantiles.................................. 130
4.6.1 AlteranativeMethodfortheMedian....................... 132
4.6.2 RandS-PLUSFunctionsqmjci,hdci,sint,
qci,andqint.............................................. 133
4.7 ConcludingRemarks............................................... 134
4.8 Exercises........................................................... 135
Chapter5 ComparingTwoGroups.................................. 137
5.1 TheShiftFunction.................................................. 139
5.1.1 TheKolmogorov–SmirnovTest............................ 142
5.1.2 RandS-PLUSFunctionsks,kssig,kswsig,
andkstiesig.............................................. 145
5.1.3 TheSBandandWBandfortheShiftFunction............. 147
5.1.4 RandS-PLUSFunctionssbandandwband................. 148
5.1.5 ConfidenceBandfortheDecilesOnly...................... 151
5.1.6 RandS-PLUSFunctionshifthd........................... 153
5.1.7 RandS-PLUSFunctionsg2plotandsplotg2.............. 155
5.2 Student’stTest..................................................... 155
Contents vii
5.3 TheYuen–WelchTest............................................... 159
5.3.1 RandS-PLUSFunctionyuen.............................. 161
5.3.2 ABootstrap-tMethodforComparingTrimmedMeans..... 162
5.3.3 S-PLUSFunctionyuenbt................................... 165
5.4 InferencesBasedonaPercentileBootstrapMethod.................. 167
5.4.1 ComparingM-Estimators.................................. 168
5.4.2 ComparingTrimmedMeans............................... 169
5.4.3 RandS-PLUSFunctionstrimpb2,pb2gen,andm2ci....... 169
5.5 ComparingMeasuresofScale....................................... 170
5.5.1 ComparingVariances...................................... 170
5.5.2 RandS-PLUSFunctioncomvar2........................... 171
5.5.3 ComparingBiweightMidvariances........................ 171
5.5.4 RandS-PLUSFunctionb2ci.............................. 171
5.6 PermutationTests.................................................. 172
5.6.1 RandS-PLUSFunctionpermg............................. 173
5.7 SomeHeteroscedastic,Rank-BasedMethods........................ 173
5.7.1 RandS-PLUSFunctionmee................................ 174
5.7.2 HandlingTiedValues..................................... 176
5.7.3 RandS-PLUSFunctionscidandbmp...................... 180
5.8 ComparingTwoIndependentBinomials............................ 181
5.8.1 Storer–KimMethod....................................... 182
5.8.2 Beal’sMethod............................................. 183
5.8.3 RandS-PLUSFunctionstwobinomandtwobici........... 184
5.9 ComparingDependentGroups..................................... 184
5.9.1 ComparingDeciles........................................ 185
5.9.2 RandS-PLUSFunctionshiftdhd.......................... 186
5.9.3 ComparingTrimmedMeans............................... 188
5.9.4 RandS-PLUSFunctionyuend............................. 190
5.9.5 ABootstrap-tMethodforMarginalTrimmedMeans....... 191
5.9.6 RandS-PLUSFunctionydbt.............................. 192
5.9.7 PercentileBootstrap:ComparingM-EstimatorsandOther
MeasuresofLocationandScale............................ 192
5.9.8 RandS-PLUSFunctionbootdpci.......................... 194
5.9.9 ComparingVariances...................................... 195
viii Contents
5.9.10 TheSignTestandInferencesAboutthe
BinomialDistribution...................................... 196
5.9.11 RandS-PLUSFunctionbinomci........................... 198
5.10 Exercises........................................................... 199
Chapter6 SomeMultivariateMethods.............................. 203
6.1 GeneralizedVariance............................................... 203
6.2 Depth.............................................................. 204
6.2.1 MahalanobisDepth........................................ 204
6.2.2 HalfspaceDepth........................................... 205
6.2.3 ComputingHalfspaceDepth.............................. 207
6.2.4 RandS-PLUSFunctionsdepth2,depth,fdepth,fdepthv2,
unidepth,depth2.for,depth3.for,fdepth.for,
fdepthv2.for,andufdepth.for........................... 210
6.2.5 ProjectionDepth........................................... 211
6.2.6 RandS-PLUSFunctionspdisandpdis.for............... 212
6.2.7 OtherMeasuresofDepth.................................. 213
6.2.8 RandS-PLUSFunctionzdepth............................ 214
6.3 SomeAffine-EquivariantEstimators................................ 214
6.3.1 Minimum-VolumeEllipsoidEstimator..................... 215
6.3.2 TheMinimum-CovarianceDeterminantEstimator......... 216
6.3.3 S-EstimatorsandConstrainedM-Estimators............... 216
6.3.4 Donoho–GaskoGeneralizationofaTrimmedMean........ 218
6.3.5 RandS-PLUSFunctionsdmeananddmean.for............. 219
6.3.6 TheStahel–DonohoW-Estimator.......................... 220
6.4 MultivariateOutlierDetectionMethods............................. 221
6.4.1 ARelplot.................................................. 222
6.4.2 RandS-PLUSFunctionrelplot........................... 223
6.4.3 TheMVEMethod......................................... 226
6.4.4 TheMCDMethod......................................... 226
6.4.5 RandS-PLUSFunctionscov.mveandcov.mcd............. 226
6.4.6 RandS-PLUSFunctionsoutmveandout................... 227
6.4.7 TheMGVMethod......................................... 228
6.4.8 RandS-PLUSFunctionoutmgv............................ 231
Contents ix
6.4.9 AProjectionMethod....................................... 231
6.4.10 RandS-PLUSFunctionsoutproandoutpro.for.......... 233
6.4.11 CommentsonChoosingaMethod......................... 234
6.5 ASkippedEstimatorofLocationandScatter........................ 236
6.5.1 RandS-PLUSFunctionssmean,smean.for,wmcd,wmve,
mgvmean,andspat......................................... 238
6.6 ConfidenceRegionandInferenceBasedontheOPEstimatorof
Location............................................................ 240
6.6.1 RandS-PLUSFunctionssmeancrandsmeancr.for........ 241
6.6.2 InferencesBasedontheMGVEstimator................... 243
6.6.3 RandS-PLUSFunctionsmgvcr............................ 244
6.7 Two-SampleCase.................................................. 244
6.7.1 RandS-PLUSFunctionssmean2andsmean2.for.......... 245
6.8 MultivariateDensityEstimators.................................... 245
6.9 ATwo-Sample,Projection–TypeExtensionofthe
Wilcoxon–Mann–WhitneyTest..................................... 247
6.9.1 RandS-PLUSFunctionsmulwmwandmulwmw.for.......... 249
6.10 ARelativeDepthAnalogoftheWilcoxon–Mann–WhitneyTest..... 250
6.10.1 RandS-PLUSFunctionsmwmwandmwmw.for............... 252
6.11 ComparisonsBasedonDepth....................................... 253
6.11.1 RandS-PLUSFunctionslsqs3,lsqs3.for,anddepthg2... 256
6.12 ComparingDependentGroupsBasedonAllPairwiseDifferences... 259
6.12.1 RandS-PLUSFunctiondfried............................ 261
6.13 Exercises........................................................... 262
Chapter7 One-WayandHigherDesignsfor
IndependentGroups...................................... 265
7.1 TrimmedMeansandaOne-WayDesign............................ 266
7.1.1 AWelch-TypeAdjustedDegreesofFreedomProcedure... 266
7.1.2 RandS-PLUSFunctiont1way............................. 268
7.1.3 AGeneralizationofBox’sMethod......................... 270
7.1.4 RandS-PLUSFunctionbox1way........................... 270
7.1.5 ComparingMedians....................................... 271