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Robust Tests of Mean Vector in Symmetrical Multivariate Distributions

N. C. Giri and B. K. Sinha
Sankhyā: The Indian Journal of Statistics, Series A (1961-2002)
Vol. 49, No. 2 (Jun., 1987), pp. 254-263
Published by: Springer on behalf of the Indian Statistical Institute
Stable URL: http://www.jstor.org/stable/25050647
Page Count: 10
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Robust Tests of Mean Vector in Symmetrical Multivariate Distributions
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Abstract

Let $X=(X_{ij})=(\mathbf{\mathit{X}}_{1},\ldots,\mathbf{\mathit{X}}_{n})\text{'}$, $\mathbf{\mathit{X}}_{i}^{\prime}=({\rm X}_{i1},\ldots ,X_{ip})$ be a n×p random matrix with probability density function $f_{X}(x)=|\Sigma|^{-n/2}q({\rm tr}\ \Sigma ^{-1}(x-\mathbf{\mathit{eu}}^{\prime})^{\prime}(x-\mathbf{\mathit{eu}}^{\prime}))$ where x ε X = {x:n×p matrix| rank of x = p}, $\boldsymbol{\mu}=(\mu _{1},..,\mu _{p})^{\prime}\epsilon R^{p},\mathbf{\mathit{e}}=(1,..,1)^{\prime}$ n-vector and Σ > 0 (positive definite). Set $Q_{1}=[q\colon [0,\infty)\rightarrow [0,\infty)$, $\underset R^{np}\to{\int }q({\rm tr}\ u^{\prime }u)du=1$, q thrice continuously differentiable, $\underset R^{np}\to{\int }q^{(2)}({\rm tr}\ u^{\prime }u)\ du<\infty $, $\underset R^{np}\to{\int }q^{(3)}({\rm tr}\ u^{\prime }u+\varepsilon )du<\infty $ for some ε > 0}, $Q_{2}=[q\colon [0,\infty)\rightarrow [0,\infty)$, $\underset R^{np}\to{\int }q({\rm tr}\ u^{\prime }u)du=1$, q convex}. Assume that n > p so that $S\equiv X^{\prime}(I_{n}-\mathbf{\mathit{ee}}!n)X>0$ with probility one. It is proved that for testing $H_{10}\colon \boldsymbol{\mu}={\bf 0}$ versus the alternative $H_{{\bf 11}}\colon \boldsymbol{\mu}\neq {\bf 0}$, the Hotelling's $T^{{\bf 2}}\text{-}\text{-test}$ is locally minimax for $q\epsilon Q_{1}$, and for testing $H_{2{\bf 0}}\colon \boldsymbol{\mu}_{(1)}={\bf 0}$ versus the alternative $H_{2{\bf 1}}\colon \boldsymbol{\mu}_{(1)}\neq {\bf 0}$, the appropriate Hotelling's $T^{2}\text{-test}$ is UMPI for $q\epsilon Q_{2}$ and locally minimax for $q\epsilon Q_{1}$. In the second case $\boldsymbol{\mu}_{(1)}=(\mu _{1},..,\mu _{p_{1}})^{\prime},p_{1}

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