Robust Tests for Spherical Symmetry

Abstract

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 $\{\mathscr{F}_3(\mu) \mid \mu \neq 0\}$.