## The Annals of Statistics

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

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

#### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/aos/1176342461

**Mathematical Reviews number (MathSciNet)**

MR426317

**Zentralblatt MATH identifier**

0261.62062

**JSTOR**

links.jstor.org

**Keywords**

6225 6245 6062 Generalized sequential probability ratio test symmetric test optimal stopping rule continuation region Brownian motion space-time process least excessive majorant harmonic function free boundary problem

#### Citation

Lai, Tze Leung. Optimal Stopping and Sequential Tests which Minimize the Maximum Expected Sample Size. Ann. Statist. 1 (1973), no. 4, 659--673. doi:10.1214/aos/1176342461. https://projecteuclid.org/euclid.aos/1176342461