In the discrimination problem the random variable θ, known to take values in \1,⋯, M\, is estimated from the random vector X. All that is known about the joint distribution of (X, θ) is that which can be inferred from a sample (X₁, θ₁),⋯, (Xₙ, θₙ) of size n drawn from that distribution. A discrimination rule is any procedure which determines a decision θ̂ for θ from X and (X₁, θ₁),⋯, (Xₙ, θₙ). A rule is called k-local if the decision θ̂ depends only on X and the pairs (Xᵢ, θᵢ) for which Xᵢ is one of the k-closest to X from X₁,⋯, Xₙ. It is shown that for any k-local discrimination rule, the mean-square difference between the probability of error for the rule and its deleted estimate is bounded by $A/n$ where A is an explicitly given small constant which depends only on M and k. Thus distribution-free confidence intervals can be placed about probability of error estimates for k-local discrimination rules.
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Rogers et al. (1978) studied this question.