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Domination Over the Best Invariant Estimator Is, When Properly DOCUMENT RESUME ED 394 999 TM 024 995 AUTHOR Brandwein, Ann Cohen; Strawderman, William E. TITLE James-Stein Estimation. Program Statistics Research, Technical Report No. 89-86. INSTITUTION Educational Testing Service, Princeton, N.J. REPORT NO ETS-RR-89-20 PUB DATE Apr 89 NOTE 46p. PUB TYPE Reports Evaluative/Feasibility (142) Statistical Data (110) EDRS PRICE MF01/PCO2 Plus Postage. DESCRIPTORS Equations (Mathematics); *Estimation (Mathematics); *Maximum Likelihood Statistics; *Statistical Distributions IDENTIFIERS *James Stein Estimation; Nonnormal Distributions; *Shrinkage ABSTRACT This paper presents an expository development of James-Stein estimation with substantial emphasis on exact results for nonnormal location models. The themes of the paper are: (1) the improvement possible over the best invariant estimator via shrinkage estimation is not surprising but expected from a variety of perspectives; (2) the amount of shrinkage allowable to preserve domination over the best invariant estimator is, when properly interpreted, relatively free from the assumption of normality; and (3) the potential savings in risk are substantial when accompanied by good quality prior information. Relatively, much less emphasis is placed on choosing a particular shrinkage estimator than on demonstrating that shrinkage should produce worthwhile gains in problems where the error distribution is spherically symmetric. In addition, such gains are relatively robust with respect to assumptions concerning distribution and loss. (Contains 1 figure and 53 references.) (Author/SLD) *********************************************************************** * Reproductions supplied by EDRS are the best that can be made * * from the original document. * *********************************************************************** US. OlPAOTOPINT Of MOICATION Moo al Elea Monet Romorch and improvement "PERMISSION TO REPRODUCE THIS EDUCATION& RESOURCES INFORMATION MATERIAL HAS BEEN GRANTED BY F--"---it.,1/4:44%f4s1,c CENTER (ERIC) ne clocurnal has hailm reproduced as rectsvod from th person or orpsnisstion feiehtU/V onoinahng 0 Miner Chance tetan made 10 employs roroductiOn Quality Pointsol via./ or opinions statudinthisdocu mnt do nol ri.cseanly mpriestrit official TO THE EDUCATIONAL RESOURCES OEM Position or policy INFORMATION CENTER (ERIC)." James-Stein Estimation Ann Cohen Brandwein and William E. Strawderman PROGRAM STATISTICS RESEARCH TECHNICAL REPORT NO. 89-86 BEST COPY AVAILABLE EDUCATIONAL TESTING SERVICE PRINCETON, NEW JERSEY 08541 2 THE JAMES-STEIN ESTIMATION Ann Cohen Brandwein and William E. Strawderman Program Statistics Research Technical Report No. 89-86 Research Report No. 89-20 Educational Testing Service Princeton, New Jersey 08541-0001 April 1989 Copyright c 1989 by Educational Testing Service. All rights reserved. The Program Statistics Research TechnicalReport Series is designed to make the working papers of the ResearchStatistics Group at Educational Testing Servicegenerally available. The series con- sists of reports by the members of theResearch Statistics Group as well as their external and visiting statisticalconsultants. Reproduction of any portion of a ProgramStatistics Research Technical - Report requires the written consent of theauthor(s). 4 TABLE OF CONTENTS' Page 1. Introduction 1 2. A Geometric Hint 3 3. An Empirical Bayes Account 4 4. Some Spherical Normal Theory 5 5. More Normal Theory 8 6. Scale Mixtures of Normals. .. 18 7. Other Spherically Symmetric Distributions 22 8. Multiple Observations 27 9. Other Loss Functions 28 10. Comments 33 References 36 bst r a ct This paper presents an expository development of JamesStein estimation with substantial emphasis on exact results for nonnormal location models. The themes of the paper area) that the improvement possible over the best invariant estimator via shrinkage estimation is not surprisingbut expected from a variety uf perspectives; b) that the amount of shrinkage allowable to preservedomination over the best invariant estimator, is, when properly interpreted, relativelyfree from the assumption of normality; and c) the potential savings in risk are substantialwhen accompanied by good quality prior information. 1 I.Introduction. This paper presents an exposiiory development of JamesStein estimation with substantial emphasis on exact results for nonnormal location models. The themes of the paper area) that the improvement possible over the best invariant estimator via shrinkage estimation is not surprising but expectedfrom a variety of perspectives; b) that the amount of shrinkage alkable to preserve domination over the best invariant estimator, is, when properly interpre ,relatively free from the assumption of normality; and c) the potential savings in risk are substantial whenaccompanied by good quality prior information. Relatively, much less emphasis is placed on choosing a particularshrinkage estimator than on demonstrating that shrinkage should produceworthwhile gains in problems where the error distribution is spherically symmetric.Additionally such gains are relatively robust with respect to assumptions concerning distributionand loss. The basic problem, of course, is the estimation of the mean vector0 of a pvariate location parameter family. In the normal case (with identitycovariance) for p = I, the usual estimator, the sample mean, is the maximum likelihoodestimator, the UMVUE, the best equivariant and minimax estimator for nearly arbitrarysymmetric loss, and is admissible for essentially arbitrary symmetric loss. Admissibilityfor quadratic loss was first proved by Hodges and Lehmann (1950) and Girschickand Savage (1951) using the CramerRao inequality and by Blyth (1951) using a limit of Bayes typeargument. For p = 2, the above properties also hold in thenormal case. Stein (1956) proved admissibility using an information inequality argument. Inthat same paper however, Stein proved a result that astonished many and which has led to an enormousand rich literature of substantial importance in statistical theory and practice. Stein (1956) showed that estimators of the form (1 a/(b+11X112))X dominate X for 7 a sufficiently small and h sufficientlylarge when p > 3. James and Stein (1961) sharpened the result and gave an explicit class of dominating estim: t.ors, (1 a/11X112)X for 0 < a < 2(p-2). They also indicated that a version ofthe result holds for general location equivariant estimators with finite fourth moment and for lossfunctions which are concave functions of squared error loss. Brown (1966) showed that inadmissibilityof the best equivariant estimator of location holds for virtually allproblems for p > 3, and, in Brown (1965), that admissibility tends to hold for p = 2. Minimaxity for all pfollows from Kiefer (1957). Section 2 gives a geometrical argument du,?: to Stein whichindicates that shrinkage might be expected to work under quite broad distributionalassumptions. Section 3 gives an empirical Bayes argument in thenormal case which results in the usual JamesStein estimator. Section 4 presents Stein's "unbiased estimator of risk"in the normal case and develops the basic theory for the standard JamesSteinestimator in the normal case. Section 5 describes a variety of extensions of thebasic theory to cover shrinkage towards subspaces, Bayes minimax estimation,nonspherical shrinkage, and limited translation rules. Section 6 considers extensions of the resultsof sections 4 and 5 to scale mixtures of normal distributions. 3 Section 7 presents generalizations togeneral spherically symmetric families of distributions, while section 8 indicates theapplicability of earlier results to the multiple observation case. Section 9 is concerned with resultsfor nonspherical quadratic loss and for nonquadratic loss. ., Section 10 presents some additional comments. 2. A Geometric Hint As in much of the development of thesubject, the following rough geometric argument is basically due toStein (1962). Consider an observation vectorX in p dimensions with mean vector 0 andindependent (or uncorrelated) components.Assume n that the components have equalvariance, a`. The situation is depictedin figure 1. Figure 1 about here. Since E(X-0)' 0 = 0 we expectX-0 and 0 to be nearly orthogonal,especially for estimator of 0 might be too large HI. SinceEIIXII2=pa2+1142, itappears that X as an might be a better estimator. long, and that the projectionof 0 on X or something close to it but perhaps we This projection of course depends on0 and therefore isn't a valid estimator, approximate a. can estimate it.If we denote this projectionby (1a)X the problem is to One way to do this is to assumethat the angle between 0and X-0 is exactly a right 9 FIGURE 1 0 4 2 angle and to assume that 11 X11is exactly equal to its expected value 0'0 -I- p9 a- and i.211.x112 similarly 11X-011-2 is equaltopa2. In thiscase we have 11Y112 = pa2i2.'--..2 Ix!'from triangle BCD and 111112 = 11 0112 (1i)211X112 9 9 2 2 = 11)(11- - (1i) 11X11 from triangle ABD. Equating these expressions we obtain 2 2 2 2 pa a-11X11 = 11)(11 P2 g(la)11X11 or (1-2)l1X11= 11X11 2p2. Thisgives 2 9 2 a = po-/11X11and the suggested estimator is (1 a) X = (1Pa 11X11- The above development does not particularly depend on normality of X or even that 0 is a location vector. Unfortunately, it fails to be a proof of the inadmissibility of X, and also fails to distinguish between different values of p.It is however suggestive that the possibility of improving on the unbiased vector X by shrinkage toward the origin may be quite general. 3. An Empirical
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