## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 19, Number 3 (1948), 326-339.

### Optimum Character of the Sequential Probability Ratio Test

A. Wald and J. Wolfowitz

#### Abstract

Let $S_0$ be any sequential probability ratio test for deciding between two simple alternatives $H_0$ and $H_1$, and $S_1$ another test for the same purpose. We define $(i, j = 0, 1):$ $\alpha_i(S_j) =$ probability, under $S_j$, of rejecting $H_i$ when it is true; $E_i^j (n) =$ expected number of observations to reach a decision under test $S_j$ when the hypothesis $H_i$ is true. (It is assumed that $E^1_i (n)$ exists.) In this paper it is proved that, if $\alpha_i(S_1) \leq \alpha_i(S_0)\quad(i = 0,1)$, it follows that $E_i^0 (n) \leq E_i^1 (n)\quad(i = 0, 1)$. This means that of all tests with the same power the sequential probability ratio test requires on the average fewest observations. This result had been conjectured earlier ([1], [2]).

#### Article information

**Source**

Ann. Math. Statist. Volume 19, Number 3 (1948), 326-339.

**Dates**

First available in Project Euclid: 28 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aoms/1177730197

**Digital Object Identifier**

doi:10.1214/aoms/1177730197

**Mathematical Reviews number (MathSciNet)**

MR26779

**Zentralblatt MATH identifier**

0032.17302

**JSTOR**

links.jstor.org

#### Citation

Wald, A.; Wolfowitz, J. Optimum Character of the Sequential Probability Ratio Test. Ann. Math. Statist. 19 (1948), no. 3, 326--339. doi:10.1214/aoms/1177730197. http://projecteuclid.org/euclid.aoms/1177730197.