## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 41, Number 4 (1970), 1322-1333.

### Stopping Time of a Rank-Order Sequential Probability Ratio Test Based on Lehmann Alternatives II

#### Abstract

We are motivated by Stein's proof (Stein (1946), Wald (1947), pages 157-158) of the termination of a sequential probability ratio test in the case of independent and identically distributed random variables. Extending his ideas to take certain "dependencies" into account we examine the rank-order sequential probability ratio test based on a Lehmann alternative studied in a paper with the above title by I. R. Savage and the author (1966) (referred to as SS I in the rest of this paper). We prove that this test terminates with probability one and that the stopping time has a finite moment generating function under a very mild condition on the bivariate random variables which resembles the Stein-condition, namely that a certain random variable $V(X_1, Y_1)$, defined in (32), is not identically equal to 0. Finally the asymptotic normality of the logarithm of the likelihood ratio of the rank order is established using the well-known Chernoff-Savage Theorem.

#### Article information

**Source**

Ann. Math. Statist., Volume 41, Number 4 (1970), 1322-1333.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177696906

**Digital Object Identifier**

doi:10.1214/aoms/1177696906

**Mathematical Reviews number (MathSciNet)**

MR267697

**Zentralblatt MATH identifier**

0229.62042

**JSTOR**

links.jstor.org

#### Citation

Sethuraman, J. Stopping Time of a Rank-Order Sequential Probability Ratio Test Based on Lehmann Alternatives II. Ann. Math. Statist. 41 (1970), no. 4, 1322--1333. doi:10.1214/aoms/1177696906. https://projecteuclid.org/euclid.aoms/1177696906