The Annals of Statistics

Nearly Optimal Sequential Tests of Composite Hypotheses

Tze Leung Lai

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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

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

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62L10: Sequential analysis
Secondary: 62L05: Sequential design 62L15: Optimal stopping [See also 60G40, 91A60]

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


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

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