Asymptotic normality of the kernel estimate under dependence conditions: application to hazard rate

Abstract

Let X 1 , X 2 ,…, X n 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) = ƒ(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 R p 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 ƒ(x), ƒ ̂ 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 ƒ ̂ 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 ƒ ̂ n (x) is also studied.