Reliability properties of reversed residual lifetime

If the random variable X denotes the lifetime (X ≥ 0, with probability one) of a unit, then the random variable Xt = (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 Xt, has received the attention of many researchers in the recent past [(cf. Snaked, 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 Xt and study their interrelations. We also define a new ordering based on the mean of the random variable Xt and establish its relationship with the reversed hazard rate ordering.