Open Access
January, 1977 Robust Tests for Spherical Symmetry
Takeaki Kariya, Morris L. Eaton
Ann. Statist. 5(1): 206-215 (January, 1977). DOI: 10.1214/aos/1176343755


Let $R^n$ be Euclidean $n$-space and let $O(n)$ be the group of $n \times n$ orthogonal matrices. Consider $\mathscr{F}_0 = \{f\mid f$ is a density on $R^n, f(x) = f(gx), x \in R^n, g \in O(n)\}$, and let $Q = \{q\mid q: \lbrack 0, \infty) \rightarrow \lbrack 0, \infty), q$ is nonincreasing, $\int_{R^n} q(\| x \|^2) dx = 1\}$. If $\Sigma$ is an $n \times n$ positive definite matrix, set $\mathscr{F}_1(\Sigma) = \{f|f(x) = \mid \Sigma|^{-\frac{1}{2}}q(x'\Sigma^{-1}x), q \in Q\}$. For $\mu \in R^1$ and $a_0 \in R^n, \| a_0 \| = 1$, let $\mathscr{F}_2(\mu) = \{f\mid f(x) = q(\| x - \mu a_0\|^2), q \in Q\}$ and $\mathscr{F}_3(\mu) = \{f\mid f(x) = q(\| x - \mu a_0\|^2), q \in Q,\text{and} q \text{convex}\}.$ Uniformly most powerful tests are derived for testing $\mathscr{F}_0$ versus $\mathscr{F}_1(\Sigma)$ and for testing $\mathscr{F}_0$ versus $\{\mathscr{F}_2(\mu) \mid \mu > 0\}$. A uniformly most powerful unbiased test is derived for testing $\mathscr{F}_0$ versus $\{\mathscr{F}_3(\mu) \mid \mu \neq 0\}$.


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Takeaki Kariya. Morris L. Eaton. "Robust Tests for Spherical Symmetry." Ann. Statist. 5 (1) 206 - 215, January, 1977.


Published: January, 1977
First available in Project Euclid: 12 April 2007

zbMATH: 0361.62033
MathSciNet: MR428590
Digital Object Identifier: 10.1214/aos/1176343755

Primary: 62G10
Secondary: 62F05 , 62G35

Keywords: robustness , Student's $t$-test , tests for sphericity , UMP nonparametric tests

Rights: Copyright © 1977 Institute of Mathematical Statistics

Vol.5 • No. 1 • January, 1977
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