The Nearest Neighbor and the Bayes Error Rates

The (k, l) nearest neighbor method of pattern classification is compared to the Bayes method. If the two acceptance rates are equal then the asymptotic error rates satisfy the inequalities Ek,l + 1 ≤ E*(λ) ≤ Ek,l dE*(λ), where d is a function of k, l, and the number of pattern classes, and λ is the reject threshold for the Bayes method. An explicit expression for d is given which is optimal in the sense that for some probability distributions Ek,l and dE* (λ) are equal.