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December, 1972 Maximum Likelihood Estimation of Life-Distributions from Renewal Testing
L. H. Crow, I. N. Shimi
Ann. Math. Statist. 43(6): 1827-1838 (December, 1972). DOI: 10.1214/aoms/1177690854


In 1965 Marshall and Proschan (1965) (see also Grenander (1956)), considered the maximum likelihood estimation for life-distributions with monotone failure rate over the support of the distribution functions. They considered data arising from a testing plan which does not allow censoring, time-truncation or replacements. In the present paper we consider the maximum likelihood estimation of life-distributions with monotone failure rates over the interval $\lbrack 0, T)$, where $T$ is a fixed positive real number, and no other assumptions about the distribution or its failure rate are given outside that interval. The following renewal type testing plan is used, which allows for time-truncation and replacement. At time zero, the beginning of the testing, $n$ new items from the population to be tested are put on test. When an item fails it is instantaneously replaced with another new item from the same population and at time $T$ all testing is stopped. The maximum likelihood estimates of the distribution function and its failure rate over $\lbrack 0, T)$ are given and shown to be uniformly strongly consistent as $n$ tends to infinity.


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L. H. Crow. I. N. Shimi. "Maximum Likelihood Estimation of Life-Distributions from Renewal Testing." Ann. Math. Statist. 43 (6) 1827 - 1838, December, 1972.


Published: December, 1972
First available in Project Euclid: 27 April 2007

zbMATH: 0254.62066
MathSciNet: MR356420
Digital Object Identifier: 10.1214/aoms/1177690854

Rights: Copyright © 1972 Institute of Mathematical Statistics

Vol.43 • No. 6 • December, 1972
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