Pattern Recognition Letters 1 (1983) 291-292 July 1983 North-Holland A note on nearest neighbour error rates S.J. MAYBANK Department of Computer Science, Birbeck College, University of London, U.K. and Marconi Space and Defence Systems, Frimley, U.K. Received 9 December 1982 Abstract: Let Ak, t, Ek,/denote the acceptance and error rates in the (k, l) case of the nearest neighbour decision rule. We show that if q>k and Aq, m=Ak, I then Eq, m<-Ek, l. Key words: Nearest neighbours, error rates. 1. Introduction In Devijver and Kittler (1982, p. 105) the follow- ing conjecture is made: if Ak, l=Aq, m (k< q< oo), then Ek, l+l <_Eq, m<_Ek,t. (1) In (1), Aab denotes the acceptance rate for the nearest neighbour rule where a pattern, re- presented by a point in Euclidean space, is ac- cepted only if b (b > a/2) out of its first a nearest neighbours in a large sample set of correctly classified patterns are in the same pattern class. The pattern is assigned to this class. Eab denotes the corresponding error rate. The notation is fully explained in Devijver and Kittler (1982). In this note we prove the second inequality of (1). 2. Proof of the inequality First we note that in the case of M pattern classes COl, . . . , (..OM, l ,-~=~ (1-rli)Iqmp(x) dx, (2) where rli = - I']i(X ) is the a posteriori probability of a pattern x lying in class o9i, p(x) is the un- conditional probability density function of x and q q Iffm=j~_m(j)rIJ(1--rli)q-J is the incomplete Beta function (written as I,,(m,q-m+l) in Zelen and Severo (1964, pp. 944, 945)). Now if we set =I ,_Igm then, since Ak, l=Aq, m, t~J~iP(x) dx = 0. i=l From (2) we now have Ek't-Eq'm =- i=1 ~ ! rliJ'7ip(x)dx" By differentiating the integral representation of the incomplete Beta function, we can show that ~te [0, 1] such that rli<_t implies J~,>>_O and rli>_t implies J,i_<0. The result is also true if q=oo, except at 11 i = t. 0167-8655/83/$3.00 © 1983, Elsevier Science Publishers B.V. (North-Holland) 291