The Annals of Statistics

Minimax Estimation of Location Parameters for Spherically Symmetric Unimodal Distributions Under Quadratic Loss

Ann R. Cohen Brandwein and William E. Strawderman

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Abstract

Families of minimax estimators are found for the location parameter of a $p$-variate $(p \geqq 3)$ spherically symmetric unimodal distribution with respect to general quadratic loss. The estimators of James and Stein, Baranchik, Bock and Strawderman are all considered for this general problem. Specifically, when the loss is general quadratic loss given by $L(\delta, \theta) = (\delta - \theta)'D(\delta - \theta)$ where $D$ is a known $p \times p$ positive definite matrix, one main result, for one observation, $X$, on a multivariate s.s.u. distribution about $\theta$, presents a class of minimax estimators whose risk dominate the risk of $X$, provided $p \geqq 3$ and trace $D \geqq 2d_L$ where $d_L$ is the maximum eigenvalue of $D$. This class is given by $\delta_{a,r}(X) = (1 - a(r(\|X\|^2)/\|X\|^2)) X$ where $0 \leqq r(\bullet) \leqq 1, r(\|X\|^2)$ is nondecreasing, $r(\|X\|^2)/\|X\|^2$ is nonincreasing, and $0 \leqq a \leqq (c_0/E_0(\|X\|^{-2}))((\operatorname{trace} D/d_L) - 2)$, where $c_0 = 2p/((p + 2)(p - 2))$ when $p \geqq 4$ and $c_0 \approx .96$ when $p = 3$.

Article information

Source
Ann. Statist., Volume 6, Number 2 (1978), 377-416.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176344131

Digital Object Identifier
doi:10.1214/aos/1176344131

Mathematical Reviews number (MathSciNet)
MR467992

Zentralblatt MATH identifier
0402.62019

JSTOR
links.jstor.org

Subjects
Primary: 62C99: None of the above, but in this section
Secondary: 62F10: Point estimation 62H99: None of the above, but in this section

Keywords
Minimax estimation spherically symmetric unimodal multivariate location parameter

Citation

Brandwein, Ann R. Cohen; Strawderman, William E. Minimax Estimation of Location Parameters for Spherically Symmetric Unimodal Distributions Under Quadratic Loss. Ann. Statist. 6 (1978), no. 2, 377--416. doi:10.1214/aos/1176344131. https://projecteuclid.org/euclid.aos/1176344131


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