$\Gamma$-Minimax Selection Procedures in Treatments Versus Control Problems

Abstract

Let $X$ be a random variable with distribution function $F(x\mid\theta)$ over a sample space $\mathscr{H}$ for each $\theta$ in $\Theta$, the parameter space. Consider the decision problem with loss function $L(\theta, a)$ for each $a$ in $\mathscr{A}$, the action space. For a decision rule $\delta(x)$ mapping $\mathscr{H}$ into $\mathscr{A}$ the risk function is $R(\theta, \delta) = \int L(\theta, \delta(x)) dF(x \mid \theta)$. If $\tau(\theta)$ is a distribution over $\Theta$, the expected risk of using rule $\delta(x)$ is then $r(\tau, \delta) = \int_\Theta R(\theta, \delta)d\tau(\theta)$. In many problems a priori information will be incomplete. Suppose that our prior information consists of a class $\Upsilon$ of distributions over $\Theta$. One method of utilizing such partial prior information to obtain a decision rule is given by Blum and Rosenblatt [1]. DEFINITION 1.1. The rule $\delta_0(x)$ is a $\Upsilon$-minimax decision rule if $\sup_{\tau \epsilon \Upsilon} r(\tau, \delta_0) = \inf_\delta\sup_{\tau \in\Upsilon} r(\tau, \delta)$. The use of partial prior information by Menges [5] and Hodges and Lehmann [2] may also be considered as satisfying the $\Upsilon$-minimax criterion for suitable choices of $\Upsilon$. In this paper the $\Upsilon$-minimax principle is applied to the problem of selecting treatment populations which have larger translation parameters than that of a control population. Let $S_0, S_1, \cdots, S_k$ be $k + 1$ independent random variables with respective probability density functions $f_0(s - \theta_0), f_1(s - \theta_1), \cdots, f_k(s - \theta_k)$. The random variables $S_0, S_1,\cdots, S_k$ may represent sufficient statistics from the control and $k$ treatment populations, respectively. We assume that each $f_i(s), i = 0, 1, \cdots, k$, is a Polya frequency function of order two $(PF_2)$, that is, if $x_1 < x_2$ and $y_1 < y_2$ then \begin{equation*}\begin{vmatrix}f_i(x_1 - y_1) & f_i(x_1 - y_2) \\ f_i(x_2 - y_1) & f_i(x_2 - y_2)\end{vmatrix} \geqq 0.\end{equation*} Hence $f_i(s - \theta_i)$ has a monotone likelihood ratio in its translation parameter. In Section 2, necessary notation, the loss function, and the incomplete prior information are introduced. In Section 3, a $\Upsilon$-minimax decision rule is found, for the case in which the control population parameter $\theta_0$ is known, by finding a rule which is Bayes with respect to a least favorable prior in the class of prior distributions $\Upsilon$. The case in which the control population parameter is unknown is treated in Section 4. Rules are derived which are $\Upsilon$-minimax among procedures for which the decision to select or reject the $i$th population depends only on $S_i$ and $S_0$. When specialized to normal populations with common known variance $\sigma^2$, a $\Upsilon$-minimax rule selects (rejects) the $i$th population as $\bar{X}_i - \bar{X}_0 \geqq (<)$ a constant, where the constant depends on $\Delta$ (a known constant used to define "positive" and "negative" populations), $\sigma^2$, the sample sizes, the ratio of the losses for the two kinds of incorrect decisions, and the ratio of the prior probabilities of negative and positive populations. (An analogous result is found in Section 3 for the known control case.) Section 5 gives comparisons of a $\Upsilon$-minimax rule with a Bayes competitor based on independent normal priors for the case of normal populations with common known variance. Some comments on a theorem by Y. L. Tong [7] are given in Section 6. Whenever a new criterion is being considered by the statistical community, it is important to see if the criterion is fruitful in a variety of situations. In the treatment versus control problems discussed here, the $\Upsilon$-minimax criterion leads to simple explicit rules which compare favorably with Bayes rules that require stronger assumptions on the prior distribution. We hope that this work will encourage others to obtain and apply $\Upsilon$-minimax rules in different settings.