Lower bounds for Bayes error estimation

We give a short proof of the following result. Let (X,Y) be any distribution on N×{0,1}, and let (X₁,Y₁),...,(X_n,Y_n) be an i.i.d. sample drawn from this distribution. In discrimination, the Bayes error L*=inf_gP{g(X)≠Y} is of crucial importance. Here we show that without further conditions on the distribution of (X,Y), no rate-of-convergence results can be obtained. Let φ_n(X₁,Y₁,...,X_n,Y_n ) be an estimate of the Bayes error, and let {φ_n(.)} be a sequence of such estimates. For any sequence {a_n} of positive numbers converging to zero, a distribution of (X,Y) may be found such that E{|L*-φ_n(X₁,Y ₁,...,X_n,Y_n)|}⩾a_n often converges infinitely