Inadmissibility

Inadmissibility - INADMISSIBILITY OF THE USUAL ESTI MATOR FOR THE MEAN OF A MULTI VARIATE NORMAL DISTRIBUTION CHARLES STEIN STANFORD UNIVERSITY 1

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Unformatted text preview: INADMISSIBILITY OF THE USUAL ESTI- MATOR FOR THE MEAN OF A MULTI- VARIATE NORMAL DISTRIBUTION CHARLES STEIN STANFORD UNIVERSITY 1. Introduction Ifone observes therealrandom variables Xi, X,,independentlynormallydis- tributedwithunknown means ti, *, {nand variance 1,itiscustomary toestimate (i by Xi. Ifthelossisthesum ofsquaresoftheerrors,thisestimatorisadmissiblefor n < 2, butinadmissible for n _ 3. Sincetheusualestimatorisbestamong thosewhich transform correctlyundertranslation,anyadmissibleestimatorfor n _ 3involvesan arbitrarychoice.Whiletheresultsofthispaperarenotinaformsuitableforimmediate practicalapplication,thepossibleimprovementover theusualestimator seems tobe large enoughtobeofpracticalimportanceifn islarge. Let X bearandom n-vector whoseexpectedvalueisthecompletelyunknownvec- tor tand whose components are independentlynormallydistributedwith variance 1. We considerthe problemofestimating twiththelossfunctionL givenby (1) L(t, d) = ( -d)I = 2(ti-dj 2 whered isthe vector ofestimates.Insection2 we giveashortproofoftheinadmissi- bilityoftheusual estimator (2) d =t(X) = X, for n 2 3.Forn = 2, the admissibilityof 4, is proved insection4.Forn = 1thead- missibilityof t, iswellknown (see,for example, [1],[2], [3]) and alsofollowsfrom the resultforn = 2. Ofcourse,alloftheresultsconcerningthisproblemapplywithobvious modificationsiftheassumptionthatthecomponentsofX areindependentlydistributed with variance1isreplacedbytheconditionthatthecovariancematrix2ofX isknown andnonsingularandthelossfunction(1)isreplacedby (3) L (, d) = (-d)'2-'( -d). We shallgiveimmediatelybelowaheuristicargumentindicatingthattheusualesti- mator t, may bepoorifn islarge.Withsomeadditionalprecision,thiscouldbemadeto yieldadiscussionoftheinfinitedimensionalcaseoraproofthatforsufficientlylargen theusual estimatorisinadmissible.We chooseanarbitrarypointinthesamplespace independentoftheoutcome ofthe experimentand callittheorigin.Ofcourse,inthe way we have expressedtheproblemthischoicehasalreadybeenmade,butinacorrect coordinate-free presentation,itwouldappearas an arbitrarychoiceofonepointin an affinespace.Now (4) X2 = (X-t)2+ t2+ 2 \ Z I97 I98 THOIRD BERKELEY SYMPOSIUM: STEIN where (5) z_ _t_(X_- hasa univariatenormaldistributionwithmean0andvariance1,andforlargen,wehave (X- 2 = n + Op,(un), so that (6) X2 = n+ 2+O ( t + ) uniformlyin0.(Forthestochasticordernotationop, 0, see[41.) Consequently-when weobserveX2we knowthat t2 isnearly x2- n.Theusualestimator 4. wouldhaveus estimatettolieoutsideoftheconvexset{ t;t .5 X2- cn} (withcslightlylessthan 1)al- thoughwearepracticallysurethattliesinthatset.Itcertainlyseemsmorereasonable tocutX downatleastbyafactorof[(X2- n)/X2]1/2to bringtheestimatewithinthat sphere.Actually,becauseofthecurvatureofthespherecombinedwiththeuncertainty ofour knowledgeof t, the bestfactor,towithintheapproximationconsideredhere, turnsouttobe(X2- n)/X2. For,considertheclassofestimators (7) t(X)= I ( )]x where h isa continuousreal-valuedfunctionwith lim I Ih()I < -.< -....
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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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Inadmissibility - INADMISSIBILITY OF THE USUAL ESTI MATOR FOR THE MEAN OF A MULTI VARIATE NORMAL DISTRIBUTION CHARLES STEIN STANFORD UNIVERSITY 1

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