Huber Behavior of ML Estimates

Huber Behavior of - THE BEHAVIOR OF MAXIMUM LIKELIHOOD ESTIMATES UNDER NONSTANDARD CONDITIONS PETER J HUBER SWISS FEDERAL INSTITUTE OF TECHNOLOGY

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Unformatted text preview: THE BEHAVIOR OF MAXIMUM LIKELIHOOD ESTIMATES UNDER NONSTANDARD CONDITIONS PETER J. HUBER SWISS FEDERAL INSTITUTE OF TECHNOLOGY 1.Introductionandsummary This paperproves consistencyand asymptoticnormalityof maximum like- lihood (ML) estimatorsunderweakerconditionsthanusual. In particular, (i)itisnot assumedthatthetruedistributionunderlyingthe observations belongs to theparametricfamily defining theML estimator,and (ii)theregularityconditionsdo not involvethesecondandhigherderivatives ofthelikelihoodfunction. The needfortheoremsonasymptoticnormalityofML estimatorssubjectto (i)and (ii)becomes apparentinconnectionwithrobust estimation problems; forinstance,ifone triestoextendtheauthor'sresultson robustestimationof a location parameter [4]to multivariate and other more general estimation problems. Wald's classical consistency proof [6] satisfies (ii) and can easily bemodified to showthattheML estimatorisconsistentalsoincase (i), thatis,itconverges to the 0o characterizedbytheproperty E(logf(x, 0)- log f(x,O0)) < 0for0. Oo, wheretheexpectationistakenwithrespecttothetrueunderlyingdistribution. Asymptotic normality ismore troublesome. Daniels [1]proved asymptotic normalitysubjectto (ii),butunfortunatelyheoverlookedthatacrucialstepin hisproof (theuse ofthe centrallimittheorem in (4.4)) isincorrect without condition (2.2)of Linnik [5]; thisconditionseemstobetoorestrictiveformany purposes. In section 4 we shallprove asymptotic normality, assuming that the ML estimatorisconsistent.Forthesakeofcompleteness,sections2and 3contain, therefore, two differentsetsofsufficientconditionsforconsistency. Otherwise, thesesectionsareindependentofeachother.Section5 presentstwo examples. 2. Consistency:caseA Throughoutthissection,whichrephrasesWald'sresultsonconsistencyofthe ML estimatorin aslightlymoregeneralsetup,theparameterset0 isalocally compactspacewithacountablebase, (X, 2I, P)isaprobabilityspace,andp(x,0) issomereal-valuedfunctiononX X 0. 221 222 FIFTH BERKELEY SYMPOSIUM: HUBER Assume that xI, X2,- **are independent random variables with values in I having thecommon probabilitydistributionP. Let Tn(xI, *- , xn)beany se- quence offunctions Tn:X'ne* 0, measurableor not, suchthat in I n (1)- E p(xi,T.)- inf- E p(xi,0)-°0 almost surely (orin probability-more precisely,outerprobability). We want to givesufficientconditionsensuringthateverysuchsequenceconvergesalmost surely(orinprobability)towardsomeconstant0o. IfdP = f(x,O0)d,and p(x,0) =-logf(x,0)forsome measure uion (X,9) and some family ofprobabilitydensitiesf(x,0),then the ML estimatorof 0o evidentlysatisfiescondition(1). Convergence of Tn shallbe proved under thefollowingsetofassumptions. ASSUMPTIONS. (A-1).Foreachfixed0 E 0, p(x, 0)is91-measurable,andp(x,0)isseparable in thesense ofDoob: thereisa P-null setN and a countable subset 0' C 0 such thatforeveryopensetU C 0 and every closedintervalA, thesets (2) {xlp(x, 0) E A, VO E U}, {xlp(x, 0) E A,VO E U n et} differ by atmosta subsetofN....
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This note was uploaded on 12/26/2011 for the course ECON 245a taught by Professor Staff during the Fall '08 term at UCSB.

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Huber Behavior of - THE BEHAVIOR OF MAXIMUM LIKELIHOOD ESTIMATES UNDER NONSTANDARD CONDITIONS PETER J HUBER SWISS FEDERAL INSTITUTE OF TECHNOLOGY

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