On Weighted Covariance and Vitality Orderings with Applications to the Exponential Family

In this note, some general results on measures of ageing and conditional orderings as well as related functions for weighted and parent or unweighted distributions are given. Results are presented for vitality measures of ageing. Vitality measure of ageing in a time period describes the increase in average lifespan resulting from surviving that time period. Inequalities, vitality and distortion orderings of the weighted and parent reliability measures are presented. Result on vitality order for the exponential family of distributions and some applications are also presented