Statistica Sinica

Irène Gijbels and Jan Mielniczuk

Abstract:LetFandGbe cumulative distribution functions and denote byhthe Radon-Nikodym derivative ofGwith respect toF. Two i.i.d. samples of sizesnandm=m(n) pertaining respectively toFandGare given. The uniform rate of convergence of the grade estimate of the Radon-Nikodym derivative is shown to be a.s., where {b} 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._{m}

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.