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Preservation of Some Ageing Properties by Order Statistics

Published online by Cambridge University Press:  27 July 2009

Neeraj Misra ,
M. Manoharan  and
Harshinder Singh
Neeraj Misra
Affiliation:
Department of Statistics, Panjab University, Chandigarh — 160 014, India
M. Manoharan
Affiliation:
Department of Statistics, Panjab University, Chandigarh — 160 014, India
Harshinder Singh
Affiliation:
Department of Statistics, Panjab University, Chandigarh — 160 014, India
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Abstract

Let X be a continuous random variable denoting the lifetime of a unit. Let Xk:n denote the kth order statistic based on n independent random observations on X. It has been shown that if Xk:n has decreasing failure rate (DFR) for some k, 1 ≤ k ≤ n, then X is DFR. For n ≥ 2, if Xk:n has increasing failure rate (IFR), then Xk:n–1 is also IFR, and if Xk:n is DFR, then Xk:n+1 is also DFR. The log concavity of the density, function is shown to be preserved by the kth order statistic. It has been established that if the density function of Xk:n is log convex then the density function of Xk:n+1 is also log convex. Because a k-out-of-n system of i.i.d. components each having a life distribution that of X has lifetime Xn-k+1:n the results have applications in the study of such systems.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 7 , Issue 3 , July 1993 , pp. 437 - 440
Copyright
Copyright © Cambridge University Press 1993

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References

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