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k Invariant Decision Rules
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a-field a-priori G average risk function Bayes optimal risk Cartesian product characteristic function classical standard component scheme compound decision problem compound decision rule compound decision scheme convergence in probability defined dishonest decision rule distributed according Doctor of Philosophy du(x equation exists an integer exp(-iy finite Hannan and Robbins honest compound decision inequality invariant compound decision Lemma 7a loss function m=j+l m=l m=l m=l n=i+l Markov's inequality matrix measurable function measurable space mN mN multinomial distribution Observe positive definite probability measures proof of Lemma proof of Theorem prove Lemma prove Theorem r x s compound r x s compound decision random variable respect to a-priori right hand side simple compound decision simple statistical decision simple symmetric compound sin2 smallest eigenvalue strictly bounded sufficiently large symmetric compound decision symmetric sets Theorem 2a zero uniformly