J. Appl. Prob. 50, 848–860 (2013) Printed in England  Applied Probability Trust 2013 STOCHASTIC COMPARISONS OF RESIDUAL LIFETIMES AND INACTIVITY TIMES OF COHERENT SYSTEMS NITIN GUPTA, ∗ Jaypee University of Information Technology and Indian Institute of Technology Kharagpur Abstract Under the assumption of independent and identically distributed (i.i.d.) components, the problem of the stochastic comparison of a coherent system having used components and a used coherent system has been considered. Necessary and sufficient conditions on structure functions have been provided for the stochastic comparison of a coherent system having used/inactive i.i.d. components and a used/inactive coherent system. As a consequence, for r -out-of-n systems, it has been shown that systems having used i.i.d. components stochastically dominate used systems in the likelihood ratio ordering. Keywords: Likelihood ratio order; coherent system; residual life; inactivity time 2010 Mathematics Subject Classification: Primary 90B25 Secondary 60E15 1. Introduction In some practical situations, one has to make a choice between a used system of n components and a system made up of n used components. The used system or the system made up of used components has a lifetime in terms of the residual life. Let X be a random variable with probability density function f(·), distribution function F(·), and survival function ¯ F(·) = 1 - F(·). The residual lifetime and the inactivity time of X with age/time t ≥ 0 is defined as X t = (X - t | X>t) and X (t) = (t - X | X ≤ t), respectively. For comprehensive details on the residual lifetime and the inactivity time, we refer the reader to [2], [3], and [12]. The stochastic comparisons and reliability properties of the residual lifetime and the inactivity time have been discussed by [6], [7], [10], [11], and [14]. Let us denote by η(·) =-f ′ (·)/f (·), the eta function of the random variable X. The eta function plays a vital role in the study of the reliability characteristics. We refer the reader to [4] for an overview of the eta function. Throughout this paper, terms such as ‘increasing’ and ‘decreasing’ will be used to denote ‘nondecreasing’ and ‘nonincreasing’, respectively. To make the paper self-contained, we include below some definitions which are standard in the literature (see [13]). Definition 1.1. Let Z i ,i = 1, 2, be two random variables with probability density functions g i (·), distribution functions G i (·), and survival functions ¯ G i (·) = 1 -G i (·), i = 1, 2. Then the Received 15 March 2012; revision received 12 December 2012. ∗ Postal address: Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-721302, India. Email address: nitin.gupta@maths.iitkgp.ernet.in 848 https://www.cambridge.org/core/terms. https://doi.org/10.1239/jap/1378401240 Downloaded from https://www.cambridge.org/core. IP address: 54.161.69.107, on 14 Jun 2020 at 23:26:21, subject to the Cambridge Core terms of use, available at