Exponentially Bounded Stopping Time of the Sequential t-Test

Abstract

Let N be the stopping time of the sequential t-test, based on the i.i.d. sequence $Z_1,Z_2,...$, for testing that the ratio of mean to standard deviation in a normal population equals $\gamma_1$ agaist the alternative that it equals $\gamma_2$. Let P be the actual distribution of the $Z_i$ (not necessarily normal). It is proved that if $\gamma_1^2\neq\gamma_2^2$ and P is an arbitrary unbounded distribution, then there exist constants c > 0 and $\rho<1$ such that $P(N > n) < c\rho^n, n =1,2,\cdots$.