Statistica Sinica 5(1995), 261-278

ASYMPTOTIC PROPERTIES OF KERNEL ESTIMATORS OF

THE RADON-NIKODYM DERIVATIVE WITH

APPLICATIONS TO DISCRIMINANT ANALYSIS

Irène Gijbels and Jan Mielniczuk

Université Catholique de Louvain and Polish Academy of Sciences

Abstract: 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 of the Radon-Nikodym derivative is shown to be a.s., where {bm} 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 . 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.

Key words and phrases: Bayes rule, density ratio, discriminant rule, Gini separation measure, grade density, misclassification errors, modified kernel estimate, nonparametric kernel estimation, Radon-Nikodym derivative, strong uniform consistency.