A New Generalized Family of Lifetime Distributions Motivated by Parallel and Series Structures

Abstract

Any given system can be represented as a parallel arrangement of series structures. Motivated by this fact, a general family of distributions is introduced, by adding two extra parameters to a distribution (called baseline distribution), twice compounding with power series distribution. The new family can allow various hazard rate curves that compete well with other alternatives in fitting real data. We derive formal expressions for its moments, generating function, mean residual lifetime and other reliability functions. Certain characterizations of the new family are presented in terms of the ratio of two truncated moments as well as based on the hazard rate function. The maximum likelihood estimation technique is used to estimate the model parameters and a simulation study is conducted to investigate the performance of the maximum likelihood estimates. Finally, two applications of the model with real data sets are presented to illustrate the usefulness of the proposed distribution.The purpose of this paper is to introduce a new family of lifetime distributions by compounding a lifetime distribution and twice the power series distribution, which is referred to as the lifetime PS 2 family of distributions. The compounding procedure follows the ideas of Marshall and Olkin [21]. The proposed family is motivated by a system consisting of parallel components with each component consisting of a series of components, i.e, a system made of parallel and series structures. According to Ross [28], any system can be represented either as a series arrangement of parallel structures or as a parallel arrangement of series structures. This new family is the first family of distributions motivated by a system made of parallel and series components. Applications of the parallel and series systems can be found in the areas of nuclear power systems [25] and modeling crystal deformation [13]. The lifetime PS 2 family of distributions contains as special cases all the compounded lifetime distributions constructed by Marshall and Olkin method such as the generalized exponential geometric distribution [4], exponential power series [9], Weibull power series [22], exponentiated extended Weibull power series [32] and the inverse Weibull geometric distribution [20].The rest of the paper is organized as follows. In Section 2, the new family of distributions are introduced. Section 3 derives some of its mathematical properties; the density, survival, hazard rate and moment generating functions are given in this section. Section 4 deals with various characterizations of the new family of distributions. Section 5 presents some special cases of the new distribution. Estimation of the parameters of the new distribution via the maximum likelihood method and some related inferences and finite sample behaviors of the maximum likelihood estimates are investigated in Section 6. In Section 7, a simulation study is performed to illustrate the behavior of asymptotic variances of maximum likelihood estimations (MLEs). Illustrative examples to two real da...

The Type II Exponentiated Half Logistic-Gompertz-G Power Series Class of Distributions: Properties and Applications

“…The t th moment of the distribution of the k th order statistic from TIIEHL-Gom-GPS class of distributions can be readily obtained from equation (13). Visit the appendix for derivations of the pdf of the k th order statistic.…”

“…Various distributions have been proposed in existing literature to model lifetime data using compounding method. Examples include the type II Exponentiated Half-Logistic-Topp-Leone-G power series class of distributions by Moakofi et al [17], a new generalized family of lifetime distributions by Goldoust et al [13], a new two-sided class of lifetime distributions by Kharazmi et al [15], and the Odd Log-Logistic Transmuted-G family of distributions by Alizadeh et al [2].…”