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The nearest neighbor classification rule is a nonparametric classification procedure that assigns a random vector to one of two populations . Samples of equal size are taken from and and are ordered separately with respect to their distance from . The rule assigns to if the distance of the th sample observation from to is less than the distance of the th sample observation from to ; otherwise is assigned to . This rule is equivalent to the Fix and Hodges, "majority rule"  or the nearest neighbor rule of Cover and Hart . This paper studies some asymptotic properties of this rule including an expression for a consistent upper bound on the probability of misclassification.