Asymptotic Properties of Kernel Estimators of the Radon-nikodym Derivative With Applications To Discriminant-analysis

Let F and G be cumulative distribution functions and denote by h the Radon-Nikodym derivative of G with respect to F. Two i.i.d. samples of sizes n and m = m(n) pertaining respectively to F and G are given. The uniform rate of convergence of the grade estimate fin,m of the Radon-Nikodym derivative is shown to be O ((log m/(mb(m)))(1/2) + b(m)(2)) a.s., where {b(m)} denotes the bandwidth parameter: The proof uses the exponential inequality for the oscillation modulus of continuity for empirical processes given by Mason, Shorack and Wellner (1983). The result is applied to study asymptotic properties of a discriminant rule pertaining to ($) over cap h(n,m). It is established that its risk converges exponentially fast to Bayes risk. Finally, an estimator for Gini separation measure is introduced and its rate of strong consistency is obtained.