Table Of ContentDeGruyterGraduate
Höpfner(cid:2)AsymptoticStatistics
Reinhard Höpfner
Asymptotic Statistics
With a View to Stochastic Processes
De Gruyter
MathematicsSubjectClassification2010:62F12,62M05,60J60,60F17,60J25
Author
Prof.Dr.ReinhardHöpfner
JohannesGutenbergUniversityMainz
Faculty8:Physics,MathematicsandComputerScience
InstituteofMathematics
Staudingerweg9
55099Mainz
Germany
[email protected]
ISBN978-3-11-025024-4
e-ISBN978-3-11-025028-2
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Preface
At Freiburg in 1991 I gave my very first lectures on local asymptotic normality; the
first time I explained likelihood ratio processes or minimum distance estimators to
studentsinacoursewas1995atBonn,duringavisitingprofessorship.Then,during
a short time at Paderborn and now almost 15 years at Mainz, I lectured from time
to time on related topics, or on parts of these. In cooperation with colleagues in the
fieldofstatisticsofstochasticprocesses(wheremostofthetimeIwaslearning)and
indiscussions with good students (whoexposed metoveryprecisequestions onthe
underlying mathematical notions), the scope of subjects entering the topic increases
steadily;initially,mypersonalpreferencewasonnullrecurrentMarkovprocessesand
onlocalasymptoticmixednormality.Later,lecturingonsuchtopics,afirsttentative
versionofascriptcameintolifeatsomepoint.Itwentonaugmentingandcompleting
itself in successive steps of approximation. It is now my hope that the combination
oftopicsmayservestudentsandinterestedreaderstogetacquaintedwiththepurely
statisticaltheoryononehand,developedcarefullyina‘probabilistic’style,andonthe
otherhandwithsome(thereareothers)typicalapplicationstostatisticsofstochastic
processes.
Thepresentbookcanbereadindifferentways,accordingtopossiblydifferentmath-
ematical preferences of a reader. In the author’s view, the core of the book are the
Chapters5(Gaussianshiftmodels),6(mixednormalandquadraticmodels),7(local
asymptoticswherethelimitmodelisaGaussianshiftoramixednormaloraquadratic
experiment, often abbreviated as LAN, LAMN or LAQ), and finally 8 (examples of
statisticalmodelsinacontextofdiffusionprocesseswherelocalasymptoticsoftype
LAN,LAMNorLAQappear).
Areaderwhowantstoconcentrateonthestatisticaltheoryaloneshouldskipchap-
(cid:2)
tersorsubsectionsmarkedbyanasterisk :heorshewouldreadonlytheSections5.1
and6.1,andthenallsubsectionsofChapter7.Thisrouteincludesanumberofexam-
ples formulated in the classical i.i.d. framework, and allows to follow the statistical
theorywithoutgaps.
(cid:2)
Incontrast,chaptersorsubsectionsmarkedbyanasterisk aredesignedforreaders
withaninterestinbothstatisticsandstochasticprocesses.Thisreaderisassumedtobe
acquaintedwithbasicknowledgeoncontinuous-time martingales,semi-martingales,
Ito formula and Girsanov theorem, and may go through the entire Chapters 5 to 8
consecutively. In view of the stochastic process examples in Chapter 8, he or she
may consult from time to time the Appendix Section 9 for further background and
forreferences(onsubjectssuchasHarrisrecurrence,positiveornull,convergenceof
martingales,andconvergenceofadditivefunctionalsofaHarrisprocess).
viii Preface
Inbothcases,areadermaypreviouslyhaveconsultedorhavereadtheSections1.1
and1.2aswellasChapters3and4forstatisticalnotionssuchasscoreandinforma-
tion in classical definition, contiguity or L2-differentiability, to be prepared for the
coreofthebook.GivenSections1.1and1.2,Chapters3and4canbereadindepen-
dentlyofeachother.Onlyfewbasicnotionsofclassicalmathematicalstatistics(such
as sufficiency, Rao–Blackwell theorem, exponential families, ...) are assumed to be
known.
Sections1.3and1.4areofcomplementarycharacterandmaybeskipped;theydis-
cuss naive belief in maximum likelihood and provide some background in order to
appreciatethetheoremsofChapter7.
Chapter2standsisolatedandcanbereadseparatelyfromallotherchapters.Ini.i.d.
frameworkwe study in detail one particular class of estimators for the unknown pa-
rameterwhich‘worksreasonablywell’inalargevarietyofstatisticalproblemsunder
weakassumptions.Fromatheoreticalpointofview,thisallowstoexplicitlyconstruct
estimatorsequenceswhichconvergeatacertainrate.Fromapracticalpointofview,
wefinditinterestingtostart–priortoalloptimalityconsiderationsinlaterchapters–
withestimatorsthattoleratesmalldeviationsfromtheoreticalmodelassumptions.
Fruitfulexchangesandcooperationsoveralongperiodoftimehavecontributedto
thescopeoftopicstreatedinthisbook,andIwouldliketothankmycolleagues,coau-
thorsandfriendsforthosemanylongandstimulatingdiscussionsaroundsuccessive
projects relatedtoour joint papers. Their influence, well visible inthe relevantparts
of the book, is acknowledged with deep gratitude. In a similar way, I would like to
thankmycoauthorsandpartnersuptonowinother(formalorquiteinformal)coop-
erations.Therearesometeachersandcolleaguesinprobabilityandstatisticstowhom
I owe much, either for encouragement and help at decisive moments of my mathe-
matical life, or for mathematical discussions on specific topics, and I would like to
takethisopportunitytoexpressmygratitude.Furthermore,Ihavetothankthosewho
fromthebeginningallowedmetolearnandtostarttodomathematics,and–beyond
mathematics,sharingeverydaylife–myfamily.
Concerning a more recent time period, I would like to thank my colleague Eva
Löcherbach,myPhDstudentMichaelDietheraswellasTobiasBergandSimonHol-
bach:theyagreedtoreadlongerorshorterpartsofthistextinclose-to-finalversions
andmadecriticalandhelpfulcomments;remainingerrorsaremyown.
Mainz,June2013 ReinhardHöpfner
Contents
Preface vii
1 ScoreandInformation 1
1.1 Score,Information,InformationBounds ........................ 2
1.2 EstimatorSequences,AsymptoticsofInformationBounds ......... 15
1.3 HeuristicsonMaximumLikelihoodEstimatorSequences .......... 23
1.4 ConsistencyofMLEstimatorsviaHellingerDistances ............ 30
2 MinimumDistanceEstimators 42
2.1 StochasticProcesseswithPathsinLp.T,T ,(cid:2)/ .................. 43
2.2 MinimumDistanceEstimatorSequences ....................... 55
2.3 SomeCommentsonGaussianProcesses........................ 68
2.4 AsymptoticNormalityforMinimumDistanceEstimatorSequences .. 75
3 Contiguity 85
3.1 LeCam’sFirstandThirdLemma ............................. 86
3.2 ProofsforSection3.1andsomeVariants ....................... 92
4 L2-differentiableStatisticalModels 108
4.1 Lr-differentiableStatisticalModels ........................... 109
4.2 LeCam’sSecondLemmafori.i.d.Observations ................. 119
5 GaussianShiftModels 127
5.1 GaussianShiftExperiments.................................. 127
(cid:2)
5.2 BrownianMotionwithUnknownDriftasaGaussianShift
Experiment .............................................. 141
6 QuadraticExperimentsandMixedNormalExperiments 148
6.1 QuadraticandMixedNormalExperiments ..................... 148
(cid:2)
6.2 LikelihoodRatioProcessesinDiffusionModels ................ 160
(cid:2)
6.3 TimeChangesforBrownianMotionwithUnknownDrift ......... 168
