## The Annals of Statistics

- Ann. Statist.
- Volume 4, Number 2 (1976), 237-251.

### Properties of Generalized Sequential Probability Ratio Tests

Bennett Eisenberg, B. K. Ghosh, and Gordon Simons

#### Abstract

We consider generalized sequential probability ratio tests (GSPRT's), which are not necessarily based on independent or identically distributed observations, to distinguish between probability measures $P$ and $Q$. It is shown that if $T$ is any test in a wide class of GSPRT's, including all SPRT's, and $T'$ is any rival test possessing error probabilities and sample sizes no greater than those of $T$, then $T'$ must be equivalent to $T$. This notion of optimality of $T$ is weaker than that of Kiefer and Weiss but the results are stronger than theirs. It is also shown that, if an SPRT $T'$ has at least one error probability strictly less than that of another SPRT $T$ with the other error probability no larger, $T'$ requires strictly more observations than $T$ some of the time, under both $P$ and $Q$, and never fewer observations. This assertion generalizes Wijsman's conclusions. The methods used in this paper are quite general, and are different from those of the earlier authors.

#### Article information

**Source**

Ann. Statist., Volume 4, Number 2 (1976), 237-251.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176343404

**Digital Object Identifier**

doi:10.1214/aos/1176343404

**Mathematical Reviews number (MathSciNet)**

MR395099

**Zentralblatt MATH identifier**

0323.62053

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62L10: Sequential analysis

**Keywords**

Generalized sequential probability ratio test admissibility optimality sequential probability ratio test likelihood ratio

#### Citation

Eisenberg, Bennett; Ghosh, B. K.; Simons, Gordon. Properties of Generalized Sequential Probability Ratio Tests. Ann. Statist. 4 (1976), no. 2, 237--251. doi:10.1214/aos/1176343404. https://projecteuclid.org/euclid.aos/1176343404