## The Annals of Statistics

### Extended Optimality of Sequential Probability Ratio Tests

Albrecht Irle

#### Abstract

The problem of sequentially testing two simple hypotheses for a stochastic process is considered. It is shown that, for arbitrary distributions $P_0$ and $P_1$, the following optimality holds for an SPRT which stops on its boundaries: If $\alpha$ and $\beta$ represent the error probabilities of the SPRT and a competing test has error probabilities $\alpha' \leq \alpha$ and $\beta' \leq \beta$ then $E_0g(D_{\tau'}) \geq E_0g(D_\tau)$ for any convex function $g$ satisfying some minor requirement, provided $P_1(\tau' < \infty) = 1$ for the competing test. Here $D_\tau$ and $D_{\tau'}$ denote the terminal likelihood ratios under the SPRT and the competitor. An analogous statement holds for expectation under $P_1$, and several applications of this optimality result are given.

#### Article information

Source
Ann. Statist., Volume 12, Number 1 (1984), 380-386.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176346416

Digital Object Identifier
doi:10.1214/aos/1176346416

Mathematical Reviews number (MathSciNet)
MR733523

Zentralblatt MATH identifier
0551.62056

JSTOR