Statistical analysis of old age behavior

A random life is characterized by a nonnegative random variable X having survival function (sf) F¯(x)=P(X>x),x⩾0. Associated with any life, two notions are important in life testing. These are the random remaining life at age t, Xt, a random variable with sf F¯t(x)=F¯(x+t)/F¯(t),x,t⩾0, and the corresponding stationary renewal life or the equilibrium life denoted by X˜, whose sf isW¯F(α)=1μ∫x∞F¯(u)du,x⩾0,where μ=E(X) assumed finite. Thus X˜ may be used to identify “old age.” Note that X˜ is unobservable but can be studied through X itself. In the current investigation, inequalities of the moments of X are derived from the ageing behavior of X˜. We then show that if X˜ is harmonic new is better than used in expectation and if E(X2) exists, then the moment generating function of X exists and its upper bound is obtained. We also use moments inequalities derived from the ageing behavior of X˜ to test that X˜ is exponential against that it belongs to one of several ageing classes.