A Theorem on Obstructive Distributions

Abstract

Let $N$ be the stopping time of a sequential probability ratio test of composite hypothesis, based on the i.i.d. sequence $Z_1, Z_2, \cdots$ with common distribution $P$. If for every choice of stopping bounds there exist constants $c > 0, 0 < \rho < 1$ such that $P\{N > n\} < c\rho^n n = 1, 2, \cdots$, we say that $N$ is exponentially bounded under $P$; otherwise $P$ is called obstructive. A theorem is proved giving sufficient conditions for $P$ to be obstructive. By virtue of this theorem it is possible to exhibit families of obstructive distributions in several examples, including the sequential $t$-test.