Preservation of Some Ageing Properties by Order Statistics

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.