## The Annals of Statistics

### Nearly Optimal Sequential Tests of Composite Hypotheses

Tze Leung Lai

#### Abstract

A simple class of sequential tests is proposed for testing the one-sided composite hypotheses $H_0: \theta \leq \theta_0$ versus $H_1: \theta \geq \theta_1$ for the natural parameter $\theta$ of an exponential family of distributions under the 0-1 loss and cost $c$ per observation. Setting $\theta_1 = \theta_0$ in these tests also leads to simple sequential tests for the hypotheses $H: \theta < \theta_0$ versus $K: \theta > \theta_0$ without assuming an indifference zone. Our analytic and numerical results show that these tests have nearly optimal frequentist properties and also provide approximate Bayes solutions with respect to a large class of priors. In addition, our method gives a unified approach to the testing problems of $H$ versus $K$ and also of $H_0$ versus $H_1$ and unifies the different asymptotic theories of Chernoff and Schwarz for these two problems.

#### Article information

Source
Ann. Statist. Volume 16, Number 2 (1988), 856-886.

Dates
First available in Project Euclid: 12 April 2007

http://projecteuclid.org/euclid.aos/1176350840

Digital Object Identifier
doi:10.1214/aos/1176350840

Mathematical Reviews number (MathSciNet)
MR947582

Zentralblatt MATH identifier
0657.62088

JSTOR