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May, 1975 Termination, Moments and Exponential Boundedness of the Stopping Rule for Certain Invariant Sequential Probability Ratio Tests
Tze Leung Lai
Ann. Statist. 3(3): 581-598 (May, 1975). DOI: 10.1214/aos/1176343124

Abstract

It is well known that Wald's SPRT terminates with probability one and in fact the stopping time is exponentially bounded for every distribution $P$ except in the trivial case where the $\log$ likelihood ratio vanishes with probability one. The results in the literature for invariant SPRT's, however, have been considerably less complete. In all the parametric problems studied, moment conditions of certain random variables have been assumed to prove termination, and the finiteness of their moment generating function has also been assumed to prove the exponential boundedness of the stopping rule. In this paper, we try to remove or weaken these conditions for certain invariant SPRT's. In particular, we show that like the Wald SPRT, the sequential $t$- and $F$-tests always terminate with probability one for any distribution $P$ except in trivial cases. However, the stopping rules may fail to be exponentially bounded, and obstructive distributions are also exhibited. Sufficient conditions for exponential boundedness and finiteness of moments of the stopping rule are studied, and asymptotic expressions for the moments of the stopping rule are also given.

Citation

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Tze Leung Lai. "Termination, Moments and Exponential Boundedness of the Stopping Rule for Certain Invariant Sequential Probability Ratio Tests." Ann. Statist. 3 (3) 581 - 598, May, 1975. https://doi.org/10.1214/aos/1176343124

Information

Published: May, 1975
First available in Project Euclid: 12 April 2007

zbMATH: 0328.62053
MathSciNet: MR370969
Digital Object Identifier: 10.1214/aos/1176343124

Subjects:
Primary: 6245
Secondary: 6030, 6225

Rights: Copyright © 1975 Institute of Mathematical Statistics

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Vol.3 • No. 3 • May, 1975
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