On the convergence rates of kernel estimator and hazard estimator for widely dependent samples
In this paper, we establish a Bernstein-type inequality for widely orthant dependent random variables, and obtain the rates of strong convergence for kernel estimators of density and hazard functions, under some suitable conditions.
Strong Convergence Rate of Recursive Probability Density Estimator for NA Sequences
Under suitable conditions, we discuss the strong convergence rate of recursive probability density estimator. At the same time, the strong convergence rate of the hazard rate estimator is investigated.
Asymptotic normality of the kernel estimate under dependence conditions: application to hazard rate
Abstract 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) = 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.