Joint asymptotic normality of kernel estimates under dependence conditions, with application to hazard rate

Abstract

et Xtt =…, ‐1, 0, 1…be R d -valued (d≧1) random variables forming a strictly stationary sequence satisfying a weak dependence condition, and let f be the probability density function of the X's. On the basis of n successive random variables, recursive and nonrecursive kernel estimates of f are considered and the joint asymptotic normality of the proposed estimates, evaluated at a finite number of distinct continuity points of f, is established. By means of an example, it is demonstrated that the recursive estimate may be superior to the nonrecursive one, in the sense of reducing the variance of the asymptotic distribution. The results are applied to the problem of estimating the hazard rate and in establishing the joint asymptotic normality of the proposed estimates. The above mentioned variance reduction possibility carries over to the hazard rate estimation problem.