MODELING THE LIFETIME OF LONGITUDINAL ELEMENTS

• Published in: Communications in statistics. Simulation and computation Vol. 30; no. 4; pp. 717 - 741
• Main Authors: Sifuentes, Mario Cantú, Villaseñor Alva, José A., Arnold, Barry C.
• Format: Journal Article
• Language: English
• Published: Colchester Taylor & Francis Group 01.01.2001; Taylor & Francis
• Subjects: Bootstrap; Exact sciences and technology; Generalized Gamma regression model; Lifetime data analysis; Mathematics; Maximum likelihood; Model selection; Parametric inference; Probability and statistics; Sciences and techniques of general use; Size effect; Statistics; Weakest-link principle; Censored data; Monotonic function; Experimental data; Generalized function; Gamma distribution; Parametric method; Lifetime; Survival function; Regression model; Distribution function; Failure time; Survival probability; Maximum likelihood; Weibull distribution
• ISSN: 0361-0918; 1532-4141
• DOI: 10.1081/SAC-100107778

In the study of the stochastic behaviour of the lifetime of an element as a function of its length, it is often observed that the failure time (or lifetime) decreases as the length increases. In probabilistic terms, such an idea can be expressed as follows. Let T be the lifetime of a specimen of length x, so the survival function, which denotes the probability that an element of length x survives till time t, will be given by S T (t, x) = P(T > t/α(x), where α(x) is a monotonically decreasing function. In particular, it is often assumed that T has a Weibull distribution. In this paper, we propose a generalization of this Weibull model by assuming that the distribution of T is Generalized gamma (GG). Since the GG model contains the Weibull, Gamma and Lognormal models as special and limiting cases, a GG regression model is an appropriate tool for describing the size effect on the lifetime and for selecting among the embedded models. Maximum likelihood estimates are obtained for the GG regression model with α(x) = cx b . As a special case this provide an alternative to the usual approach to estimation for the GG distribution which involves reparametrization. Related parametric inference issues are addressed and illustrated using two experimental data sets. Some discussion of censored data is also provided.