On the comparison in hazard rate ordering of fail-safe systems

Let X-1,..., X-n be a set of independent exponential random variables with Xi having hazard rate lambda(i), i = 1,..., n. Let Y-1,..., Y-n be another set of independent exponential random variables with common hazard rate lambda. In this note, we characterize the comparison, according to the hazard rate ordering, between the second order statistic from X-i's and that of Y-i's. Thus, we show that X-2:n is larger in the hazard rate ordering than Y-2:n if and only if lambda >= root Sigma 1 <= i <= j <= n(lambda i lambda j)/(n 2) and X-2:n is smaller in the same ordering than Y-2:n if and only if lambda <= Sigma(n)(i=1) lambda(i) - max(1 <= i <= n lambda i)/n - 1 These results are related to the stochastic comparison of fail-safe systems in reliability. (C) 2007 Elsevier B.V. All rights reserved.