Estimation of Jumps, Reliability and Hazard Rate

Abstract

Let $F(x)$ be a probability distribution function. Assuming the singular part to be identically zero, it is well known (see e.g. Cramer [1] pp. 52, 53) that $F(x)$ can be decomposed into $F(x) = F_1(x) + F_2(x)$ where $F_1(x)$ is an everywhere continuous function and $F_2(x)$ is a pure step function with steps of magnitude, say, $S_\nu$ at the points $x = x_\nu, \nu = 1, 2, \cdots, \infty$ and that finally both $F_1(x)$ and $F_2(x)$ are non-decreasing and are uniquely determined. In this paper the problem of estimating the jump $S_i$ corresponding to the saltus $x = x_i$ is considered. Also considered are the problems of estimation of reliability and hazard rate. Based on a random sample $X_1, X_2, \cdots X_n$ of size $n$ from the distribution $F(x)$, consistent and asymptotically normal classes of estimators are obtained for estimating the jump $S_i$ corresponding to the saltus $x = x_i$. Based on the earlier work of the author [2] on estimation of probability density, consistent and asymptotically normal estimates are obtained for the reliability and hazard rate.