## The Annals of Statistics

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

### Nearly Optimal Sequential Tests of Composite Hypotheses

#### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/aos/1176350840

**Mathematical Reviews number (MathSciNet)**

MR947582

**Zentralblatt MATH identifier**

0657.62088

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62L10: Sequential analysis

Secondary: 62L05: Sequential design 62L15: Optimal stopping [See also 60G40, 91A60]

**Keywords**

Bayes sequential tests diffusion approximations exponential family Kullback-Leibler information generalized sequential likelihood ratio tests boundary crossing probabilities

#### Citation

Lai, Tze Leung. Nearly Optimal Sequential Tests of Composite Hypotheses. Ann. Statist. 16 (1988), no. 2, 856--886. doi:10.1214/aos/1176350840. https://projecteuclid.org/euclid.aos/1176350840