Open Access
June, 1965 Estimation of Jumps, Reliability and Hazard Rate
V. K. Murthy
Ann. Math. Statist. 36(3): 1032-1040 (June, 1965). DOI: 10.1214/aoms/1177700075


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.


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V. K. Murthy. "Estimation of Jumps, Reliability and Hazard Rate." Ann. Math. Statist. 36 (3) 1032 - 1040, June, 1965.


Published: June, 1965
First available in Project Euclid: 27 April 2007

zbMATH: 0134.36103
MathSciNet: MR177474
Digital Object Identifier: 10.1214/aoms/1177700075

Rights: Copyright © 1965 Institute of Mathematical Statistics

Vol.36 • No. 3 • June, 1965
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