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March, 1958 On the Distribution of a Statistic Based on Ordered Uniform Chance Variables
Shanti S. Gupta, Milton Sobel
Ann. Math. Statist. 29(1): 274-281 (March, 1958). DOI: 10.1214/aoms/1177706726


The exact distribution of a statistic based on the $r$ smallest of $n$ independent observations from a unit uniform distribution is derived. In life-testing terminology, this statistic includes as special cases (i) the sum of the $r$ earliest failure times, (ii) the total observed life up to the $r$th failure, and (iii) the sum of all $n$ failure times. The density, cumulative distribution function (c.d.f.) and first four moments of the general statistic are summarized in Sec. 2. Section 3 gives the derivation of the density and c.d.f. The moments are obtained from the moment generating function in Sec. 4. Asymptotic normality under certain conditions is proved in Sec. 5 and illustrations of the rapidity of approach to normality are given in Sec. 6.


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Shanti S. Gupta. Milton Sobel. "On the Distribution of a Statistic Based on Ordered Uniform Chance Variables." Ann. Math. Statist. 29 (1) 274 - 281, March, 1958.


Published: March, 1958
First available in Project Euclid: 27 April 2007

zbMATH: 0086.35104
MathSciNet: MR95558
Digital Object Identifier: 10.1214/aoms/1177706726

Rights: Copyright © 1958 Institute of Mathematical Statistics

Vol.29 • No. 1 • March, 1958
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