## The Annals of Statistics

### Optimal Stopping and Sequential Tests which Minimize the Maximum Expected Sample Size

Tze Leung Lai

#### Abstract

Among all sequential tests with prescribed error probabilities of the null hypothesis $H_0: \theta = -\theta_1$ versus the simple alternative $H_1: \theta = \theta_1$, where $\theta$ is the unknown mean of a normal population, we want to find the test which minimizes the maximum expected sample size. In this paper, we formulate the problem as an optimal stopping problem and find an optimal stopping rule. The analogous problem in continuous time is also studied, where we want to test whether the drift coefficient of a Wiener process is $-\theta_1$ or $\theta_1$. By reducing the corresponding optimal stopping problem to a free boundary problem, we obtain upper and lower bounds as well as the asymptotic behavior of the stopping boundaries.

#### Article information

Source
Ann. Statist. Volume 1, Number 4 (1973), 659-673.

Dates
First available in Project Euclid: 12 April 2007

http://projecteuclid.org/euclid.aos/1176342461

Digital Object Identifier
doi:10.1214/aos/1176342461

Mathematical Reviews number (MathSciNet)
MR426317

Zentralblatt MATH identifier
0261.62062

JSTOR