On a Multiple Decision Rule

Abstract

Let $X = (X_1, \cdots, X_k)$ be a random vector whose distribution depends on a parameter vector $\theta = (\theta_1, \cdot, \theta_k)$. A standard procedure $\phi^\ast$ is considered for selecting a set of $m < k$ coordinate values corresponding to the $m$ largest components of $\theta$. $\phi^\ast$ is given as follows: Select the $m$ coordinates corresponding to the $m$ largest components of $x$, the observed value of $X$. Break ties, if any, with randomization. Some optimal properties of $\phi^\ast$ are known, given that the loss function and the distribution of $X$ have certain invariance and monotonicity properties. It is shown in this paper that $\phi^\ast$ is a Bayes decision rule if $X$ is "stochastically increasing" in $\theta$.