## The Annals of Statistics

### Asymptotic Optimality of Invariant Sequential Probability Ratio Tests

Tze Leung Lai

#### Abstract

It is well known that Wald's SPRT for testing simple hypotheses based on i.i.d. observations minimizes the expected sample size both under the null and under the alternative hypotheses among all tests with the same or smaller error probabilities and with finite expected sample sizes under the two hypotheses. In this paper it is shown that this optimum property can be extended, at least asymptotically as the error probabilities tend to 0, to invariant SPRTs like the sequential $t$-test, the Savage-Sethuraman sequential rank-order test, etc. In fact, not only do these invariant SPRTs asymptotically minimize the expected sample size, but they also asymptotically minimize all the moments of the sample size distribution among all invariant tests with the same or smaller error probabilities. Modifications of these invariant SPRTs to asymptotically minimize the moments of the sample size at an intermediate parameter are also considered.

#### Article information

Source
Ann. Statist., Volume 9, Number 2 (1981), 318-333.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176345398

Digital Object Identifier
doi:10.1214/aos/1176345398

Mathematical Reviews number (MathSciNet)
MR606616

Zentralblatt MATH identifier
0459.62069

JSTOR