Annals of Mathematical Statistics

Optimum Character of the Sequential Probability Ratio Test

A. Wald and J. Wolfowitz

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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]).

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

First available in Project Euclid: 28 April 2007

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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.

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