Confidence, Likelihood, Probability: Statistical Inference with Confidence Distributions

521 Pages·2016·14.01 MB·English

by  Tore Schweder

Confidence, Likelihood, Probability
This livelybook laysoutamethodology of confidence distributionsand putsthem
through their paces. Among other merits, they lead to optimal combinations of
confidence from different sources of information, and they can make complex
models amenable to objective and indeed prior-free analysis for less subjectively
inclined statisticians. The generous mixture of theory, illustrations, applications
and exercises is suitable for statisticians at all levels of experience, as well as for
data-orientedscientists.
Some confidence distributions are less dispersed than their competitors. This
concept leads to a theory of risk functions and comparisons for distributions of
confidence. Neyman–Pearson type theorems leading to optimal confidence are
developedandrichlyillustrated.Exactandoptimalconfidencedistributionsarethe
goldstandardforinferredepistemicdistributionsinempiricalsciences.
Confidencedistributionsandlikelihoodfunctionsareintertwined, allowingprior
distributionstobemadepartofthelikelihood. Meta-analysisinlikelihoodtermsis
developedandtakenbeyondtraditionalmethods,suitingitinparticulartocombining
informationacrossdiversedatasources.
TORE SCHWEDER isaprofessorofstatisticsintheDepartmentofEconomicsand
attheCentreforEcologyandEvolutionarySynthesisattheUniversityofOslo.
NILS LID HJORT is a professor of mathematical statistics in the Department of
MathematicsattheUniversityofOslo.
CAMBRIDGE SERIES IN STATISTICAL AND
PROBABILISTIC MATHEMATICS
EditorialBoard
Z.Ghahramani(DepartmentofEngineering,UniversityofCambridge)
R.Gill(MathematicalInstitute,LeidenUniversity)
F.P.Kelly(DepartmentofPureMathematicsandMathematicalStatistics,UniversityofCambridge)
B.D.Ripley(DepartmentofStatistics,UniversityofOxford)
S.Ross(DepartmentofIndustrialandSystemsEngineering,UniversityofSouthernCalifornia)
M.Stein(DepartmentofStatistics,UniversityofChicago)
This series of high-quality upper-division textbooks and expository monographs covers all aspects of
stochasticapplicablemathematics.Thetopicsrangefrompureandappliedstatisticstoprobabilitytheory,
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Acompletelistofbooksintheseriescanbefoundatwww.cambridge.org/knowledge.
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Confidence, Likelihood,
Probability
Statistical Inference with
Confidence Distributions
Tore Schweder
UniversityofOslo
Nils Lid Hjort
UniversityofOslo
32AvenueoftheAmericas,NewYork,NY10013-2473,USA
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Firstpublished2016
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Schweder,Tore.
Confidence,likelihood,probability:statisticalinferencewithconfidence
distributions/ToreSchweder,UniversityofOslo,NilsLidHjort,UniversityofOslo.
pages cm.–(Cambridgeseriesinstatisticalandprobabilisticmathematics)
Includesbibliographicalreferencesandindex.
ISBN978-0-521-86160-1(hardback)
1. Mathematicalstatistics. 2. Probability. 3. Appliedstatistics.
I. Hjort,NilsLid. II. Title.
QA277.5.S39 2016
519.2–dc23 2015016878
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Contents
Preface pagexiii
1 Confidence,likelihood,probability:Aninvitation 1
1.1 Introduction 1
1.2 Probability 4
1.3 Inverseprobability 6
1.4 Likelihood 7
1.5 Frequentism 8
1.6 Confidenceandconfidencecurves 10
1.7 Fiducialprobabilityandconfidence 14
1.8 WhynotgoBayesian? 16
1.9 Notesontheliterature 19
2 Inferenceinparametricmodels 23
2.1 Introduction 23
2.2 Likelihoodmethodsandfirst-orderlarge-sampletheory 24
2.3 Sufficiencyandthelikelihoodprinciple 30
2.4 Focusparameters,pivotsandprofilelikelihoods 32
2.5 Bayesianinference 40
2.6 Relatedthemesandissues 42
2.7 Notesontheliterature 48
Exercises 50
3 Confidencedistributions 55
3.1 Introduction 55
3.2 Confidencedistributionsandstatisticalinference 56
3.3 Graphicalfocussummaries 65
3.4 Generallikelihood-basedrecipes 69
3.5 Confidencedistributionsforthelinearregressionmodel 72
3.6 Contingencytables 78
3.7 Testinghypothesesviaconfidenceforalternatives 80
3.8 Confidencefordiscreteparameters 83
vii
viii Contents
3.9 Notesontheliterature 91
Exercises 92
4 Furtherdevelopmentsforconfidencedistribution 100
4.1 Introduction 100
4.2 Boundedparametersandboundedconfidence 100
4.3 Randomandmixedeffectsmodels 107
4.4 TheNeyman–Scottproblem 111
4.5 Multimodality 115
4.6 Ratiooftwonormalmeans 117
4.7 Hazardratemodels 122
4.8 ConfidenceinferenceforMarkovchains 128
4.9 Timeseriesandmodelswithdependence 133
4.10 Bivariatedistributionsandtheaverageconfidencedensity 138
4.11 Devianceintervalsversusminimumlengthintervals 140
4.12 Notesontheliterature 142
Exercises 144
5 Invariance,sufficiencyandoptimalityforconfidence
distributions 154
5.1 Confidencepower 154
5.2 Invarianceforconfidencedistributions 157
5.3 Lossandriskfunctionsforconfidencedistributions 161
5.4 Sufficiencyandriskforconfidencedistributions 165
5.5 Uniformlyoptimalconfidenceforexponentialfamilies 173
5.6 Optimalityofcomponentconfidencedistributions 177
5.7 Notesontheliterature 179
Exercises 180
6 Thefiducialargument 185
6.1 Theinitialargument 185
6.2 Thecontroversy 188
6.3 Paradoxes 191
6.4 FiducialdistributionsandBayesianposteriors 193
6.5 Coherencebyrestrictingtherange:Invarianceorirrelevance? 194
6.6 Generalisedfiducialinference 197
6.7 Furtherremarks 200
6.8 Notesontheliterature 201
Exercises 202
7 Improvedapproximationsforconfidencedistributions 204
7.1 Introduction 204
7.2 Fromfirst-ordertosecond-orderapproximations 205
Contents ix
7.3 Pivottuning 208
7.4 Bartlettcorrectionsforthedeviance 210
7.5 Median-biascorrection 214
7.6 Thet-bootstrapandabc-bootstrapmethod 217
7.7 Saddlepointapproximationsandthemagicformula 219
7.8 Approximationstothegoldstandardintwotestcases 222
7.9 Furtherremarks 227
7.10 Notesontheliterature 228
Exercises 229
8 Exponentialfamiliesandgeneralisedlinearmodels 233
8.1 Theexponentialfamily 233
8.2 Applications 235
8.3 AbivariatePoissonmodel 241
8.4 Generalisedlinearmodels 246
8.5 Gammaregressionmodels 249
8.6 Flexibleexponentialandgeneralisedlinearmodels 252
8.7 Strauss,Ising,Potts,Gibbs 256
8.8 Generalisedlinear-linearmodels 260
8.9 Notesontheliterature 264
Exercises 266
9 Confidencedistributionsinhigherdimensions 274
9.1 Introduction 274
9.2 Normallydistributeddata 275
9.3 Confidencecurvesfromdeviancefunctions 278
9.4 Potentialbiasandthemarginalisationparadox 279
9.5 Productconfidencecurves 280
9.6 Confidencebandsforcurves 284
9.7 Dependenciesbetweenconfidencecurves 291
9.8 Notesontheliterature 292
Exercises 292
10 Likelihoodsandconfidencelikelihoods 295
10.1 Introduction 295
10.2 Thenormalconversion 298
10.3 Exactconversion 301
10.4 Likelihoodsfrompriordistributions 302
10.5 Likelihoodsfromconfidenceintervals 305
10.6 Discussion 311
10.7 Notesontheliterature 312
Exercises 313