Exponentially Bounded Stopping Time of Sequential Probability Ratio Tests for Composite Hypotheses

Abstract

Let $N$ be the stopping variable of a SPRT for testing one composite hypothesis against another, based on i.i.d. observations $Z_1, Z_2, \cdots$ with common distribution $P. P$ need not belong to the model. $N$ is termed exponentially bounded if for every choice of stopping bounds there exists $c < \infty$ and $\rho < 1$ such that $P\{N > n\} < c\rho^n$; if this does not hold $P$ is called obstructive. The main theorem presents sufficient conditions, both on the model and on $P$, for $N$ to be exponentially bounded. Under weaker conditions the theorem proves $P\{N < \infty\} = 1$. Two applications of the theorem are given: 1. In the problem of testing $\sigma = \sigma_1$ against $\sigma = \sigma_0$ in a normal population with unknown mean it is proved that $N$ is exponentially bounded for every $P$ except if $P\{Z_1 = \zeta \pm a\} = \frac{1}{2} (\zeta$ arbitrary and $a^2$ a given function of $\sigma_1$ and $\sigma_2)$ in which case $P$ is obstructive. 2. In the sequential $t$-test it is proved that $N$ is exponentially bounded for every $P$ for which $Z_1^2$ has finite $\operatorname{mgf}$ and is not a member of a certain family of two-point distributions.