Properties and Tests for Some Classes of Life Distributions

A life distribution and its survival function F = 1 F with finite mean y = /q F(x)dx are said to be HNBUE (HNWUE) if F(x)dx < (>) U exp(-t/y) for t > 0. The major part of this thesis deals with the class of HNBUE (HNWUE) life distributions. We give different characterizations of the HNBUE (HNWUE) property and present bounds on the moments and on the survival function F when this is HNBUE (HNWUE). We examine whether the HNBUE (HNWUE) property is preserved under some reliability operations and study some test statistics for testing exponentiality against the HNBUE (HNWUE) property. The HNBUE (HNWUE) property is studied in connection with shock models. Suppose that a device is subjected to shocks governed by a counting process N = {N(t): t > 0}. The probability that the device survives beyond t is then 00 H(t) = S P(N(t)=k)P, , k=0 where P^ is the probability of surviving k shocks. We prove that H is HNBUE (HNWUE) under different conditions on N and * ^or instance we study the situation when the interarrivai times between shocks are independent and HNBUE (HNWUE). We also study the Pure Birth Shock Model, introduced by A-Hameed and Proschan (1975), and prove that H is IFRA and DMRL under conditions which differ from those used by A-Hameed and Proschan. Further we discuss relationships between the total time on test transform Hp^(t) = /q ^F(s)ds , where F \t) = inf { x: F(x) > t}, and different classes of life distributions based on notions of aging. Guided by properties of we suggest test statistics for testing exponentiality agains t IFR, IFRA, NBUE, DMRL and heavy-tailedness. Different properties of these statistics are studied. Finally, we discuss some bivariate extensions of the univariate proper­ ties NBU, NBUE, DMRL and HNBUE and study some of these in connection with bivariate shock models.