Reliability Properties of Reversed Residual Lifetime

Abstract

If the random variable X denotes the lifetime (X ≥ 0, with probability one) of a unit, then the random variable X _t  = (t − X|X ≤ t), for a fixed t > 0, is known as `time since failure', which is analogous to the residual lifetime random variable used in reliability and survival analysis. The reversed hazard rate function, which is related to the random variable X _t , has received the attention of many researchers in the recent past [(cf. Shaked, M., Shanthikumar, J. G., 1994). Stochastic Orders and Their Applications. New York: Academic Press]. In this paper, we define some new classes of distributions based on the random variable X _t and study their interrelations. We also define a new ordering based on the mean of the random variable X_t and establish its relationship with the reversed hazard rate ordering.

Keywords:

• Reversed coefficient of variation
• Reversed hazard rate
• Reversed hazard rate in average
• Reversed lifetime
• Reversed mean residual
• Reversed residual life
• Reversed variance residual

Acknowledgments

The authors are thankful to the anonymous referee for valuable comments. The financial assistance from CSIR (# 25(0125)/02/EMR-II) is gratefully acknowledged by the first author.