ASYMPTOTIC NORMALITY OF THE KERNEL ESTIMATE UNDER DEPENDENCE CONDITIONS - APPLICATION TO HAZARD RATE

Let X1,X2,...,Xn be p-dimensional random vectors coming from a strictly stationary sequence satisfying any one of the four standard modes of mixing, and let f be the probability density function of the X's with respect to Lebesque measure. The hazard rate at x is defined to be r(x) = f{hook}(x) F ̄(x), where F ̄(x) = P(X>x), and the inequality is to be understood co-ordinate-wise; r(x) is defined for x in Rp for which F ̄(x)>0. In a previous paper, the quantity r(x) was estimated by r ̂n(x), constructed in terms of the usual kernel estimate of f{hook}(x), f{hook} ̂n(x), and the natural estimate of F ̄(x), F ̄n(x); also, strong consistency was established, both pointwise and uniform over certain sets. One of the purposes of the present paper is to establish asymptotic normality of a suitably normalized version of r ̂n(x) as n tends to infinity. The proof of this result hinges on the asymptotic normality of suitable normalized versions of f{hook} ̂n(x) and F ̄n(x). The former is established herein, under any one of the four standard kinds of mixing; the latter is presented elsewhere. The asymptotic behavior of the variance and of the covariance of f{hook} ̂n(x) is also studied. © 1990.