Minimax Sequential Tests of Some Composite Hypotheses

Abstract

Let $X(t), t \geqq 0$, be a Wiener process with unknown mean $\mu$ per unit time and unit variance per unit time. Thus, $X(0) = 0$ and for any $t_2 > t_1 \geqq 0, X(t_2) - X(t_1)$ is normally distributed with mean $(t_2 - t_1)\mu$ and variance $t_2 - t_1$. Furthermore, for any sequence $0 \leqq t_{11} < t_{12} \leqq t_{21} < t_{22} \leqq \cdots \leqq t_{k1} < t_{k2},$ the random variables $X(t_j2) - X(t)j1), j = 1, \cdots, k,$ are independent. The process may be observed continuously beginning at $t = 0$ and the problem is to decide between the hypotheses that $\mu \leqq \mu_0$ and $\mu > \mu_0$, where $\mu_0$ is a given number, which without loss of generality is taken as 0. Thus the hypotheses are \begin{align*}\tag{1.1}H_0 &: \mu \leqq 0 \\ H_1 &: \mu > 0.\end{align*} It is assumed that the cost of observing the process for a time $t$ is $bt$, where $b > 0$, and that $W_i(\mu)$, the cost of accepting $H_i(i = 0, 1)$ when $\mu$ is the true mean, is of the form \begin{align*}W_0(\mu) &= \begin{cases}0 & \text{for} \mu\leqq 0\\c\mu^r & \text{for} \mu > 0\\ \tag{1.2}W_1(\mu) = \begin{cases}c |\mu |^r & \text{for} \mu \leqq 0\\0 \text{for} \mu > 0\end{cases}\end{cases}\end{align*} where $c > 0$ and $0 < r \leqq 2$. The main result of this paper is that under these conditions the minimax decision procedure is a certain sequential probability ratio test (SPRT). The reason for restricting $r$ to the interval $0 < r \leqq 2$ will be brought out in the derivation given in Section 3. In Section 6, the analogous problem of testing the hypotheses (1.1) about the mean of a normal distribution is considered. The minimax procedure found for the Wiener process provides, in an obvious fashion, an approximation to the minimax procedure for this problem. Approximations of this type have been discussed in the literature. For $r = 1$, Moriguti [10] and Maurice [9] have found the approximate minimax procedure in a certain class of symmetric SPRT's. The same procedure is mentioned by Johnson in the discussion following [8]. Breakwell [2], [3], [4], has treated similar problems for the binomial and Poisson distributions. The work to be presented here not only puts all of this on a rigorous basis for the Wiener process but shows that, for the Wiener process, the minimax SPRT is in fact minimax among all decision procedures. Finally, it is shown in Section 6 that if the cost per observation is large, the true minimax procedure for the normal decision problem is to take exactly one observation and then accept one of the hypotheses.