Simultaneous Estimation of Location Parameters Under Quadratic Loss

Shinozaki, Nobuo

doi:10.1214/aos/1176346410

Cited by 37 publications

(16 citation statements)

References 9 publications

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“…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. Shinozaki also used the integration-by-parts to discuss the ranges of a and b by assuming that X has either uniform, or t or double exponential distributions.…”

Section: Introductionmentioning

confidence: 95%

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: 95%

Estimation of location parameters for spherically symmetric distributions

Izmirlian

2006

Journal of Multivariate Analysis

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“…Therefore, we need only to show that CI.g(X) has smaller volume than C°(X) for every X. for the given function f0. We note that the form of 3~ is the same as the Stein-type estimator which first appeared in Stein (1955) and was also considered in Shinozaki (1984). We can easily verify that j~ and go satisfy Conditions (i)-(iv) and (vi) given in the previous section.…”

Section: A Confidence Set With the Same Coverage Probability And Smalmentioning

confidence: 65%

Improved confidence sets for the mean of a multivariate normal distribution

Shinozaki

1989

Ann Inst Stat Math

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Abstract. A new class of confidence sets for the mean of a p-variate normal distribution (p > 3) is introduced. They are neither spheres nor ellipsoids. We show that we can construct our confidence sets so that their coverage probabilities are equal to the specified confidence coefficient. Some of them are shown to dominate the usual confidence set, a sphere centered at the observations. Numerical results are also given which show how small their volumes are.

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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. See Hudson [4].…”

Section: Introductionmentioning

confidence: 81%

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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Simultaneous Estimation of Location Parameters Under Quadratic Loss

Cited by 37 publications

References 9 publications

Estimation of location parameters for spherically symmetric distributions

Estimation of location parameters for spherically symmetric distributions

Improved confidence sets for the mean of a multivariate normal distribution

Simultaneous estimation of location parameters of the distribution with finite support

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