A Family of Lifetime Distributions

A four-parameter family of Weibull distributions is introduced, as an example of a more general class created along the lines of Marshall and Olkin, 1997. Various properties of the distribution are explored and its usefulness in modelling real data is demonstrated using maximum likelihood estimates. 1. Introduction Probability distributions are often used in survival analysis for modeling data, because they offer insight into the nature of various parameters and functions, particularly the failure rate (or hazard) function. Throughout the last decades, a considerable amount of research was devoted to the creation of lifetime models with more than the classical increasing and decreasing hazard rates; apparently, the motivation for this trend was to provide with more freedom of choice in the description of complex practical situations (see e.g., [1–9], and the references therein). In this paper a general class of models is introduced, by adding an extra parameter to a distribution in the sense of Marshall and Olkin [10], and subsequently used in developing a four-parameter modified Weibull extension distribution, with various failure rate curves that compete well with other alternatives in fitting real data. Specifically, Xie et al. [11] generalized the Chen [12] distribution by adding the lacking scale parameter, thus creating a three-parameter Weibull distribution; although the variety of shapes of the reliability curves was not enriched, the resulting model provided better fit to real data. The proposed distribution extends the Xie et al. [11] distribution by adding a shape parameter; it will be seen that compared to the previous and other models, the cost of the addition is balanced by the improvement in fitting real data. The paper is organized as follows. Section 2 includes the general class of models and some properties. The proposed four-parameter Weibull model is introduced in Section 3 and some properties and reliability aspects are studied. The parameters are estimated by the method of maximum likelihood and the observed information matrix is obtained; the fit of the proposed distribution to two sets of real data is examined against three and two parameter competitors. 2. The Class of Distributions It is possible to generalize a distribution by adding a shape parameter, in the sense of Marshall and Olkin [10]. Thus, starting with a distribution with survival function , the survival function of the proposed family with the additional parameter is given by and when , then . The probability density and hazard functions are readily found to be