Open Access
Translator Disclaimer
September, 1986 On the Asymptotic Formula for the Probability of a Type I Error of Mixture Type Power One Tests
Moshe Pollak
Ann. Statist. 14(3): 1012-1029 (September, 1986). DOI: 10.1214/aos/1176350047

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 $y\in\omega$.

Citation

Download Citation

Moshe Pollak. "On the Asymptotic Formula for the Probability of a Type I Error of Mixture Type Power One Tests." Ann. Statist. 14 (3) 1012 - 1029, September, 1986. https://doi.org/10.1214/aos/1176350047

Information

Published: September, 1986
First available in Project Euclid: 12 April 2007

zbMATH: 0606.62085
MathSciNet: MR856803
Digital Object Identifier: 10.1214/aos/1176350047

Subjects:
Primary: 62L10
Secondary: 62F05

Rights: Copyright © 1986 Institute of Mathematical Statistics

JOURNAL ARTICLE
18 PAGES


SHARE
Vol.14 • No. 3 • September, 1986
Back to Top