Unbiasedness of Tests for Homogeneity

Abstract

Let $X_i, i = 1, 2, \ldots, k$, be independent random variables distributed according to a one-parameter exponential family with parameter $\theta_i$. Assume also that the probability density function of $X_i$ is a Polya frequency function of order two $(PF_2)$. Consider the null hypothesis $H_0: \theta_1 = \theta_2 = \cdots = \theta_k$ against the alternative $K$: not $H_0$. We show that any permutation invariant test of size $\alpha$, whose conditional (on $T = \sum^k_{i = 1}X_i)$ acceptance sections are convex, is unbiased. A stronger result is that any size $\alpha$ test function $\varphi$, which is Schur-convex for fixed $t$, is unbiased. Previously, such a result was known only for the normal and Poisson cases.