Minimax Estimation of Location Parameters for Spherically Symmetric Distributions with Concave Loss

Brandwein, Ann Cohen; Strawderman, William E.

doi:10.1214/aos/1176344953

Cited by 60 publications

(33 citation statements)

References 7 publications

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“…Brown [7] proved that the best invariant estimator of a location vector is inadmissible for a wide class of distributions and loss functions if the dimension is at least three. James and Stein's [13] result remains true if the distribution of X is spherically symmetric and p 4 as shown by Brandwein [2], Brandwein and Strawderman [3,4,6], Fan and Fang [11] and others; see the review article by Brandwein and Strawderman [5]. Under the assumption that the components of X are independent, identically and symmetrically (iis) distributed about their respective means, Shinozaki [15] investigated the bounds of a and b in (1) which involve the second and the fourth moments of the component distributions.…”

Section: Introductionmentioning

confidence: 92%

Estimation of location parameters for spherically symmetric distributions

Izmirlian

2006

Journal of Multivariate Analysis

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Estimation of the location parameters of a p × 1 random vector X with a spherically symmetric distribution is considered under quadratic loss. The conditions of Brandwein and Strawderman [Ann. Statist. 19(1991Statist. 19( ) 1639Statist. 19( -1650 under which estimators of the formwhere V has a uniform distribution in the sphere centered at with a radius R, and (iii) 0 < a 1/[pE 0 ( X −2 )]. In this paper, we not only drop their condition (ii) to show the dominance of X + ag(X) over X, but also obtain a new bound for a which is sometimes better than that obtained by Brandwein and Strawderman. Specifically, the new bound of a is 0 < a < [ 1 /(p 2 −1 )][1 − (p − 1) 1 /(p −1 2 )] −1 with i =E 0 ( X i ) for i =−1, 1, 2. The generalization to concave loss functions is also considered. Additionally, we investigate estimators of the location parameters when the scale is unknown and the observation contains a residual vector.

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Section: Introductionmentioning

confidence: 92%

Estimation of location parameters for spherically symmetric distributions

Izmirlian

2006

Journal of Multivariate Analysis

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“…c 2 1 dx (2) . From (7), one sees that d c (i)ÂV(c) is positive, continuous for c # (0, 1] and Without loss of generality, we assume q 2 =q 3 .…”

Section: Main Results and The Proofsmentioning

confidence: 99%

“…From (7), one sees that d c (i)ÂV(c) is positive, continuous for c # (0, 1] and Without loss of generality, we assume q 2 =q 3 . Define $ 2 (X (2) )= (1&1Âb(a+|X (2) | 2 )) X (2) . By Lemma 4 in Brown and Hwang (1989), there exists a constant K>0 independent of c such that 2#E(W c (X (2) &% (2) )&W c ($ 2 (X (2) )&% (2) To prove Theorem 2, we partition % and X as universally dominates X (2) under [W n c ( } ), c>0].…”

Section: Main Results and The Proofsmentioning

confidence: 99%

Universal Inadmissibility of Least Squares Estimator

Shi

2000

Journal of Multivariate Analysis

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For a p-dimensional normal distribution with mean vector % and covariance matrix I p , it is known that the maximum likelihood estimator % of % with p 3 is inadmissible under the squared loss. The present paper considers possible extensions of the result to the case where the loss is a member of a general class of losses of the form L( |$&%| Q ), where L is nondecreasing and |$&%| Q denotes the1Â2 with respect to a given positive definite matrix Q, which, without loss of generality, may be assumed to be diagonal, i.e., Q=diag(q 1 , ..., q p ), q 1 >q 2 q 3 } } } q p >0. For the case where q 1 >q 2 =q 3 = } } } =q p >0, L. D. Brown and J. T. Hwang (1989, Ann. Statist. 17, 252 267) showed that there exists an estimate of % universally dominates % if and only if p 4. This paper further extends Brown and Hwang's result to the case in which q 1 >q 2 and at least there are two equal elements among q 2 , } } } , q p&1 ; namely, we show that, for this case, there exists an estimate of % which universally dominates % if and only if p 4. For a general Q, we gives a lower bound on p that implies the least squares estimators is universally inadmissible. Academic PressAMS 1991 subject classifications: 62C05, 62H12, 62J07.

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“…See Brandwein and Strawderman [2] and the papers in their references. Shinozaki [6] obtained similar results in the case where XL, X~,..., Xp are independent, identically and symmetrically distributed p random variables, by applying integration by parts to three typical distributions; uniform, double exponential and t. Since Stein [8] used integration by parts for estimating the location parameter of the normal distribution, it has been shown to apply to simultaneous estimation problems in general continuous exponential family by many authors.…”

Section: Introductionmentioning

confidence: 99%

“…Berger [1] showed some results for losses which are polynomial in the coordinates of (8-#) for the normal case, and Brandwein and Strawderman [2] for the spherically symmetric distribution when the loss is a nondecreasing concave function of quadratic loss. Here we also study a special form of Berger's loss function for the uniform distribution.…”

Section: Introductionmentioning

confidence: 99%

Simultaneous estimation of location parameters of the distribution with finite support

Akai

1986

Ann Inst Stat Math

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SummaryLet X~, i=l,...,p be the ith component of the px1 vector X= (X,, X~,..., X~,)'. Suppose that XL, X~,..-, X~ are independent and that X, has a probability density which is positive on a finite interval, is symmetric about G and has the same variance. In estimation of the location vector ~=(8~, 82,..., G)' under the squared error loss function explicit estimators which dominate X are obtained by using integration by parts to evaluate the risk function. Further, explicit dominating estimators are given when the distributions of X,'s are mixture of two uniform distributions. For the loss function L(d, o)=ll~-stl 4 such an estimator is also given when the distributions of X,'s are uniform distributions.

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Minimax Estimation of Location Parameters for Spherically Symmetric Distributions with Concave Loss

Cited by 60 publications

References 7 publications

Estimation of location parameters for spherically symmetric distributions

Estimation of location parameters for spherically symmetric distributions

Universal Inadmissibility of Least Squares Estimator

Simultaneous estimation of location parameters of the distribution with finite support

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