A General Theorem With Applications on Exponentially Bounded Stopping Time, Without Moment Conditions

Abstract

The stopping time $N$ of a sequential test based on an i.i.d. sequence $X, X_1, X_2, \cdots$ with common distribution $P$ is defined as the first integer $n \geqq 1$ such that $l_1 < L_n < l_2$ is violated, where $L_n$ is a statistic depending only on $X_1, \cdots, X_n$. If for all $l_1, l_2$ there exist constants $c > 0, \rho < 1$ such that $P(N > n) < c\rho^n, n = 1, 2, \cdots,$ then $N$ is called exponentially bounded under $P$. In the contrary case $P$ is termed obstructive. A general theorem is proved whose conclusion is of the from that $N$ is exponentially bounded under $P$ unless $P\{f(X) = 0\} = 1$, with $f$ a function that depends on the particular testing problem. Among the applications presented there are two new results. The first is for the sequential linear hypotheses $F$-test. The function $f$ is found, and the distributions for which $P\{f(X) = 0\} = 1$ are shown to be supported on spheres. All these $P$'s, and only these, are obstructive. The second result concerns the sequential two-sample Wilcoxon test for the equality of the distributions of random variables $X$ and $Y$. Here the result is simple: $P$ is obstructive if and only if $P(X = Y) = 1$. Two auxiliary results of independent interest are presented. The first is a generalization of the exponential convergence of empirical to theoretical distribution function. The second one shows that if random variables $X$ and $Y$ have joint distribution $P$ and marginal distribution function $F, G$, respectively, then $P\{F(Y) = G(X)\} = 1$ if and only if $P(X = Y) = 1$.