JOINT ASYMPTOTIC NORMALITY OF NONPARAMETRIC RECURSIVE DENSITY ESTIMATORS AT A FINITE NUMBER OF DISTINCT POINTS

Abstract

Let x₁, …, x_n be independent and identically distributed R^P-valued random vectors with a common unknown density ƒ. We consider a recursive estimator ƒ_n consisting of kernel functions based on X₁, …, X_n. Let x₁, …, x_q be any distinct points on R^P. Under certain conditions, the joint asymptotic normality of (nh_??_)^1/2(ƒ_n(x₁)-ƒ(x₁), …, ƒ_n(x_q)-ƒ(x_q)) is shown.