Existence, Uniqueness and Monotonicity of Sequential Probability Ratio Tests

Abstract

Existence and uniqueness questions concerning SPRT's for testing one distribution against another in the case of independent and identically distributed random variables are considered under general conditions. Tests that need not take an observation $(n \geqq 0)$, as well as tests that are required to take at least one observation $(n \geqq 1)$ are being considered. SPRT's are allowed to have arbitrary stopping rule at the stopping bounds. The error point $\alpha$ of a test is the vector of error probabilities, and $A(u, v)$ is the set of all error points of SPRT's with stopping bounds $u, v(u \leqq v)$. For given $u < v$ there are four nonrandomized SPRT's (three if $u = v$), where $u$ may be a stopping point or a continuation point, and similarly $v$. It is shown in Section 4, Theorem 6, that the error points of these four tests are the extreme points of the convex set $A(u, v)$. A special mixture of these four tests is denoted $R(s, t)$, its error point $\alpha(s, t)$, where $s = (u, \lambda)$ and $t = (v, \mu)$ are the randomized stopping bounds, and $s, t$ may be considered points in an ordered topological space $Z$. Let $D = \{\alpha: \alpha_i \leqq \alpha_i(s, s)$ for some $s \varepsilon Z$ and $i = 1, 2\}$. Using the characterization of SPRT's as Bayes tests, it is shown in Theorem 2 that there is a SPRT with given error point $\alpha^\ast$ if and only if $\alpha^\ast_1 + \alpha^\ast_2 \leqq 1$ in the case $n \geqq 0$, and $\alpha^\ast \varepsilon D$ in the case $n \geqq 1$. In Theorem 1 a somewhat stronger result is claimed, namely that the SPRT in Theorem 2 can be taken to be a test of the form $R(s, t)$. The main tool in dealing with the uniqueness question is the monotonicity Theorem 3, stating that if of a test $R(s, t) s$ is decreased or $t$ increased, and the new $s, t$ have positions $u, v$, then the change $\Delta\alpha$ in the error point satisfies $u\Delta\alpha_1 + \Delta\alpha_2 \leqq 0$ and $\Delta\alpha_1 + \Delta\alpha_2/v \leqq 0$, with at least one strict inequality unless the new test is equivalent to the old one. Theorem 4 says that a test of the form $R(s, t)$ with given error point is unique up to an equivalence. On the other hand, a SPRT is in general not unique. Theorem 5 claims only uniqueness of the stopping bounds $u, v$, up to an equivalence, in the case of a SPRT with given error point.