## The Annals of Mathematical Statistics

### Optimum Character of the Sequential Probability Ratio Test

#### 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

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

Digital Object Identifier
doi:10.1214/aoms/1177730197

Mathematical Reviews number (MathSciNet)
MR26779

Zentralblatt MATH identifier
0032.17302

JSTOR