## The Annals of Statistics

- Ann. Statist.
- Volume 3, Number 3 (1975), 581-598.

### Termination, Moments and Exponential Boundedness of the Stopping Rule for Certain Invariant Sequential Probability Ratio Tests

#### 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.

#### Article information

**Source**

Ann. Statist., Volume 3, Number 3 (1975), 581-598.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176343124

**Digital Object Identifier**

doi:10.1214/aos/1176343124

**Mathematical Reviews number (MathSciNet)**

MR370969

**Zentralblatt MATH identifier**

0328.62053

**JSTOR**

links.jstor.org

**Subjects**

Primary: 6245

Secondary: 6225 6030

**Keywords**

Invariant sequential probability ratio tests termination with probability one exponentially bounded stopping rule obstructive distributions invariantly sufficient sequence likelihood ratio asymptotic behavior of moments

#### Citation

Lai, Tze Leung. Termination, Moments and Exponential Boundedness of the Stopping Rule for Certain Invariant Sequential Probability Ratio Tests. Ann. Statist. 3 (1975), no. 3, 581--598. doi:10.1214/aos/1176343124. https://projecteuclid.org/euclid.aos/1176343124