Equivariant Procedures in the Compound Decision Problem with Finite State Component Problem

Abstract

Let $(\mathscr{X}, \mathscr{B}, P)$ be a probability measure space for each $P \in \mathscr{P} = \{F_0, \cdots, F_m\}, \mathscr{A}$ be an action space and $L$ be a loss function defined on $\mathscr{X} \times \mathscr{P} \times \mathscr{A}$ such that for each $i$, $c_i = \int_{\mathbf{V}_a} L(x, F_i, a) dF_i(x) < \infty$. In the compound problem, consisting of $N$ components each with the above structure, we consider procedures equivariant under the permutation group. With $\rho_{ij} = \mathbf{V}_{B\in\mathscr{B}}|F_i(B) - F_j(B)| \text{and} K(\rho) = .5012\ldots\rho(1 - \rho)^{-\frac{3}{2}},$ we show that the difference between the simple and the equivariant envelopes is bounded by \begin{equation*}\tag{T1} \{2K(\rho) \sum_i c_i^2\}^{\frac{1}{2}}N^{-\frac{1}{2}}\quad \text{where} \rho = \mathbf{V}_{i,j}\rho_{ij},\end{equation*} and by \begin{equation*}\tag{T2} 2^m\{2K(\rho') \sum_i c_i^2\}^{\frac{1}{2}}N^{-\frac{1}{2}} \text{where} \rho' = \mathbf{V}\{\rho_{ij}\mid\rho_{ij} < 1\}\end{equation*} The bound (T1) is finite iff the $F_i$ are pairwise non-orthogonal and (T2) is designed to replace it otherwise.