Optimum Character of the Sequential Probability Ratio Test

Let S₀ be any sequential probability ratio test for deciding between two simple alternatives H₀ and H₁, and S₁ another test for the same purpose. We define $(i, j = 0, 1):$ αᵢ(Sⱼ) = probability, under Sⱼ, of rejecting Hᵢ when it is true; Eᵢʲ (n) = expected number of observations to reach a decision under test Sⱼ when the hypothesis Hᵢ is true. (It is assumed that E¹ᵢ (n) exists.) In this paper it is proved that, if αᵢ(S₁) ≤ αᵢ(S₀)(i = 0,1), it follows that Eᵢ⁰ (n) ≤ Eᵢ¹ (n)(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]).