The Annals of Mathematical Statistics
- Ann. Math. Statist.
- Volume 19, Number 3 (1948), 326-339.
Optimum Character of the Sequential Probability Ratio Test
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 (, ).
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. https://projecteuclid.org/euclid.aoms/1177730197.