## The Annals of Statistics

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

### Asymptotic Optimality of Invariant Sequential Probability Ratio Tests

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 62L10: Sequential analysis

Secondary: 62F05: Asymptotic properties of tests 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 60F15: Strong theorems

**Keywords**

Invariant SPRT Wald-Wolfowitz theorem asymptotic optimality $r$-quick convergence Wald's lower bounds for the expected sample size

#### Citation

Lai, Tze Leung. Asymptotic Optimality of Invariant Sequential Probability Ratio Tests. Ann. Statist. 9 (1981), no. 2, 318--333. doi:10.1214/aos/1176345398. https://projecteuclid.org/euclid.aos/1176345398