On a Nonparametric Estimation of the Failure Rate Function,

Based on a sequence of independent and identically distributed random variables from an absolutely continuous distribution function F(x) with the probability density function f(x), and estimate (r bar)(x) of the failure rate function r(x) = f(x)/(1-F(x)) was given by Watson and Leadbetter (1964). The asymptotic normality of r(x) was shown by the same authors. In the present paper some further asymptotic results are obtained. It is shown that (r bar)(x) converges to r(x) strongly at the continuity point of f(x). Necessary and sufficient conditions for the strong uniform convergence are obtained. Finally, the asymptotic joint normality of the estimate evaluated at a finite set of distinct points of f(x) is established where f(x) is twice differentiable with bounded derivates.