A NECESSARY AND SUFFICIENT CONDITION FOR CONVERGENCE OF ERROR PROBABILITY ESTIMATES IN K-NN DISCRIMINATION

Let（X,θ）be Rd×{1,…,s}valued random vector,（Xj,θj）,j=1,…,n,be its observedvalues, be the K-nearest neighbor estimate of θj,R（K） be the limit of error probabilityand be the error probability estimate.In this paper it is shown thatAε>0, constants α>0,c<∞ such thatif add only if there is no unregular atom of（X,θ）defined below and the various conver-gences Rnk→Rk are equivalent.