On the Sample Size and Simplification of a Class of Sequential Probability Ratio Tests

Abstract

$T$ is the sequential probability ratio test (SPRT) based on the sequence $\{X_n\}$ whose family of distributions $\{P_\theta, \theta \varepsilon \Theta\}$ satisfies certain sufficiency, monotone likelihood ratio, and consistency assumptions. Sufficiency reduces the criterion of $T$ to $q\theta_2^n/q\theta_1^n$, where $q_{\theta n}$ is the density of $X_n, \theta_1$ and $\theta_2$ are the values of $\theta$ specified by the hypothesis and alternative respectively, and $\theta_1 < \theta_2$. It is assumed that $q_{\theta n}(x)$ has the asymptotic form: $q_{\theta n}(x) \sim f_{\theta n}(x) \equiv K(n)C(\theta, x)e^{nh(\theta, x)}$ as $n \rightarrow \infty$. The Simplified SPRT $T^\ast$ is proposed where $T^\ast$ uses the criterion $e^{nh(\theta_2,\cdot)}/e^{nh(\theta_1,\cdot)}$. The following conditions are relevant: Condition C states that $h(\theta,x)$ has a unique maximum at $x = \theta$ and $h(\theta,\theta)$ is free of $\theta$; Condition D(i) requires $\Delta_{\theta n}(x) \equiv q_{\theta n}(x)/f_{\theta n}(x)$ to be bounded for all $\theta, x$, and $n$; and Condition D(ii) states that for each $\theta,\Delta_{\theta n}(x) \rightarrow 1$ uniformly in $x$ for $x$ in a neighborhood of $x = \theta$. Let $N$ and $N^\ast$ be the sample sizes of $T$ and $T^\ast$ respectively, and let $\theta_0$ be the solution of $h(\theta_2, x) = h(\theta_1, x)$. It is shown in Section 3 that Conditions C and D imply: $P_\theta(N^\ast > n) < \gamma\delta^n/n^{\frac{1}{2}}$, where $0 < \delta < 1, \gamma < \infty$ and $\theta \neq \theta_0$. The same is true for $N$. Thus, the moment generating functions of $N$ and $N^\ast$ exist, and inequalities for the expected values of $N$ and $N^\ast$ are readily obtained with respect to any $P_\theta, \theta \neq \theta_0$. The following monotonicity properties of $E_\theta N$ and $E_\theta N^\ast$ are established under an additional condition in Section 4: the expected values increase for $\theta$ less than a certain interval containing $\theta_0$ and decrease for $\theta$ greater than this interval. Several examples are discussed in Section 5, and the conditions are checked in Section 6.