Robust Tests for Spherical Symmetry

Let Rⁿ be Euclidean n-space and let $O(n)$ be the group of n × n orthogonal matrices. Consider F₀ = f is a density on Rⁿ, f(x) = f(gx), x ∈ Rⁿ, g ∈ O(n)\, and let Q = q: 0, ∞) → 0, ∞), q is nonincreasing, ∫Rⁿ q(\| x \|²) dx = 1\. If Σ is an n × n positive definite matrix, set F₁(Σ) = |f(x) = Σ|-1/2q(x'Σ⁻¹x), q ∈ Q\. For μ ∈ R¹ and a₀ ∈ Rⁿ, \| a₀ \| = 1, let F₂(μ) = f(x) = q(\| x - μ a₀\|²), q ∈ Q\ and F₃(μ) = f(x) = q(\| x - μ a₀\|²), q ∈ Q,and q convex\. Uniformly most powerful tests are derived for testing F₀ versus F₁(Σ) and for testing F₀ versus ₂(μ) μ > 0\. A uniformly most powerful unbiased test is derived for testing F₀ versus ₃(μ) μ ≠ 0\.