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On the Inequality of Cover and Hart in Nearest Neighbor Discrimination

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1 Author(s)
Devroye, L. ; 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.

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Pattern Analysis and Machine Intelligence, IEEE Transactions on  (Volume:PAMI-3 ,  Issue: 1 )