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Issue No.01 - January (1981 vol.3)
pp: 75-78
Luc Devroye , School of Computer Science, McGill University, Montreal, P.Q., Canada.
When (X1, ¿1),..., (Xn, ¿n) are independent identically distributed random vectors from IRd X {0, 1} distributed as (X, ¿), and when ¿ is estimated by its nearest neighbor estimate ¿(1), then Cover and Hart have shown that P{¿(1) ¿ ¿}n ¿ ¿ ¿ 2E {¿ (X) (1 - ¿(X))} ¿ 2R*(1 - R*) where R* is the Bayes probability of error and ¿(x) = P{¿ = 1 | X = x}. They have conditions on the distribution of (X, ¿). We give two proofs, one due to Stone and a short original one, of the same result for all distributions of (X, ¿). If ties are carefully taken care of, we also show that P{¿(1) ¿ ¿|X1, ¿1, ..., Xn, ¿n} converges in probability to a constant for all distributions of (X, ¿), thereby strengthening results of Wagner and Fritz.
Luc Devroye, "On the Inequality of Cover and Hart in Nearest Neighbor Discrimination", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.3, no. 1, pp. 75-78, January 1981, doi:10.1109/TPAMI.1981.4767052
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