Unbiasedness of Tests for Homogeneity

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.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.

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Unbiasedness of Tests for Homogeneity

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