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.
Citation
James Hannan. J. S. Huang. "Equivariant Procedures in the Compound Decision Problem with Finite State Component Problem." Ann. Math. Statist. 43 (1) 102 - 112, February, 1972. https://doi.org/10.1214/aoms/1177692706
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