Monotonicity in Selection Problems: A Unified Approach

Abstract

Let $\mathbf{X} = (X_1,\cdots, X_n)$ have a density $g(\mathbf{x}, \lambda)$ which is decreasing in transposition, where $\lambda = (\lambda_1,\cdots, \lambda_n)$. Suppose one wishes to select a subset of $\{1,\cdots, n\}$ containing the subscripts associated with the largest values of the $\lambda_i$'s. Let $S$ be a permutation invariant selection rule which is more likely to select a subset associated with the largest values of the $X_i$'s. Let $A = \{i(1),\cdots, i(k)\} \subset \{1,\cdots, n\}$ and $B = \{j(1),\cdots, j(k)\} \subset \{1,\cdots, n\}$ be such that $\lambda_{i(s)} \geq \lambda_{j(s)}, s = 1,\cdots, k$. The following five inequalities are proved for nonrandomized selection rules. $(|D|$ denotes the number of elements in $D$. $D^c$ denotes the complement of $D$.) $P_\lambda(|S \cap A| \geq (>)m) \geq P(|S \cap B| \geq (>)m)$ for every $m \in R^1, P_\lambda(|S \cap A^c| \leq (<)m) \geq P_\lambda(|S \cap B^c| \leq (<)m)$ for every $m \in R^1$, and $P_\lambda(S = A) \geq P_\lambda(S = B)$. Inequalities for randomized selection rules are also obtained. These generalized monotonicity properties are derived using a unified approach. The results apply to selection rules proposed under several formulations of the selection problem.