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September, 1948 Optimum Character of the Sequential Probability Ratio Test
A. Wald, J. Wolfowitz
Ann. Math. Statist. 19(3): 326-339 (September, 1948). DOI: 10.1214/aoms/1177730197


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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A. Wald. J. Wolfowitz. "Optimum Character of the Sequential Probability Ratio Test." Ann. Math. Statist. 19 (3) 326 - 339, September, 1948.


Published: September, 1948
First available in Project Euclid: 28 April 2007

zbMATH: 0032.17302
MathSciNet: MR26779
Digital Object Identifier: 10.1214/aoms/1177730197

Rights: Copyright © 1948 Institute of Mathematical Statistics

Vol.19 • No. 3 • September, 1948
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