On Optimal Solutions to Compound Statistical Decision Problems
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In a compound decision problem, consisting of n statistically independent copies of the same problem to be solved under the sum of the individual losses, any reasonable compound decision rule δ satisfies a natural symmetry property, entailing that δ(σ(y)) = σ(δ(y)) for any permutation σ. We derive the greatest lower bound on the risk of any such decision rule. The classical problem of estimating the mean of a homoscedastic normal vector is used to demonstrate the theory, but important extensions are presented as well in the context of Robbins's original ideas.
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