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 greater than or equal to 0. Associated with any life, two notions are important in life testing. These are the random remaining life at age t, X-t, a random variable with sf F-t (x) = F(x + t)/F (t), x, t greater than or equal to 0, and the corresponding stationary renewal life or the equilibrium life denoted by X, whose sf is W-F(alpha) = 1/mu integral(x)(infinity) F(u) du, x greater than or equal to 0, where mu = E(X) assumed finite. Thus may be used to identify "old age." Note that, 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 is harmonic new is better than used in expectation and if E (X-2) 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 : to test that is exponential against that it belongs to one of several ageing classes. (C) 2004 Elsevier B.V. All rights reserved.