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In the discrimination problem the random variable, known to take values in 1, , M, is estimated from the random vector X. All that is known about the joint distribution of (X, ) is that which can be inferred from a sample (X₁, ₁), , (X₍, ₍) of size n drawn from that distribution. A discrimination nde is any procedure which determines a decision for from X and (X₁, ₁), , (X₍, ₍). 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 by A/ n where A is an explicitly given constant depending only on M and the potential function. The O (n ^-1/2) behavior is shown to be the best possible for one of the most commonly encountered rules of this type.
Devroye et al. (Sat,) studied this question.