## The Annals of Mathematical Statistics

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

J. Sethuraman

#### 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

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