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ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
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In the discrimination problem the random variabletheta, known to take values in{1, cdots ,M}, is estimated from the random vectorX. All that is known about the joint distribution of(X, theta)is that which can be inferred from a sample(X_{1}, theta_{1}), cdots ,(X_{n}, theta_{n})of sizendrawn from that distribution. A discrimination nde is any procedure which determines a decisionhat{ theta}forthetafromXand(X_{1}, theta_{1}) , cdots , (X_{n}, theta_{n}). For rules which are determined by potential functions it is shown that the mean-square difference between the probability of error for the nde and its deleted estimate is bounded byA/ sqrt{n}whereAis an explicitly given constant depending only onMand the potential function. TheO(n ^{-1/2})behavior is shown to be the best possible for one of the most commonly encountered rules of this type.