CHARACTERIZATION THROUGH MOMENTS OF THE RESIDUAL LIFE AND CONDITIONAL SPACINGS

SUMMARY. In this work, we give a general method to obtain a distribution function F(x) through the moment of the residual life defined by hk(x) = E((X x) k | X x), for k = 1,2,3,..., both in continuous and discrete cases. We also characterize F(x) through moments of conditional spacings of order statistics, which have applications in the context of the k-out-of-n systems. Moreover, we study characterizations based on relations between failure rate function and left censored moment functions, mk(x) = E(X k | X x). Let X be a random variable (r.v.), usually representing the life length for a certain unit (where this unit can have multiple interpretations), then r.v. (X x | X x), represents the residual life of a unit with age x. Several functions are defined related to the residual life. The failure rate function, defined by: r(x) = f(x) 1 F(x ) ...(1.1) represents the failure rate of X (or F) at age x, for x 2 D = {t 2 R : F(t ) < 1}, where F(x) = P(X x), F(x ) = limz!x F(z) and f(x) is the density function when X is absolutely continuous, or f(x) = P(X = x) when X is discrete. Another interesting function is the mean residual life function, defined by h1(x) = E(X x | X x), for x 2 D, and it represents the expected additional