On the Asymptotic Formula for the Probability of a Type I Error of Mixture Type Power One Tests

Abstract

Let $X_1,X_2,\cdots$ be iid with density $f_y$ with respect to a sigma finite measure $\mu$ where ${f_y}_(y\in\omega$, $\omega\subseteqR$ is an exponential family. Let F be a probability measure on $\omega$ and let $\theta_0\in\omega$. Define $T(B,F)=\min \big\{n \left| \int_omega \frac{f_y(X_1) \1dots f_y(X_n)} {f_{\theta_0}(X_1)\dots f_{\theta_0 (X_1) \1dots f_{\theta_0} (X_n)} dF(y)\geq B \big\}$, $T (B,F) =\infty$ if no such n exists. Previous studies have found that if F has a positive and continuous density with respect to Lebesgue measure on $\omega$, then $BP_\theta\0(t(B,F)<\infty)\rigtharrow_{B\rigtharrow\infty}\int_\omega\ int^\infty_0\exp\{-x\}dH_\theta(x)dF(\theta)$, where $H_\theta$ are certain measures arising in a renewal-theoretic context. Here we show that in a nonlattice context, this convergence holds for general probability measures F. We also show that the convergence is uniform for all probability measures F whose support is contained in an arbitrary interval [a,b] interior to $\omega$, if the distribution of $X_1$ is strongly nonlattice for all $y\in\omega$.