Applied Mathematical Sciences
Given that a unit is of age t, the remaining life after time t is random. The expected value of this random residual life is called the mean residual life at time t. Specifically, if T is the life of a component with distribution function F, then δF (t) = E(T −t|T > t) is called the mean residual life function (MRLF). It is well known that the class of distributions with decreasing mean residual life (DMR) contains the class of distributions with increasing hazard rate (IHR). In this note, exponential length-biased approximations, bounds and stability results on the distance between residual life reliability functions with monotone weight functions and the exponential counterpart in the class of distribution functions with increasing or decreasing hazard rate and mean residual life functions are established. Some examples are presented.
Oluyede, Broderick O., Marvis Pararai.
"Inequalities and Exponential Approximations for Residual Life Reliability Functions."
Applied Mathematical Sciences, 2 (39): 1919-1928.