The Marshall-Olkin extended Weibull family of distributions

Abstract

We introduce a new class of models called the Marshall-Olkin extended Weibull family of distributions based on the work by Marshall and Olkin (Biometrika 84:641-652, 1997). The proposed family includes as special cases several models studied in the literature such as the Marshall-Olkin Weibull, Marshall-Olkin Lomax, Marshal-Olkin Fréchet and Marshall-Olkin Burr XII distributions, among others. It defines at least twenty-one special models and thirteen of them are new ones. We study some of its structural properties including moments, generating function, mean deviations and entropy. We obtain the density function of the order statistics and their moments. Special distributions are investigated in some details. We derive two classes of entropy and one class of divergence measures which can be interpreted as new goodness-of-fit quantities. The method of maximum likelihood for estimating the model parameters is discussed for uncensored and multi-censored data. We perform a simulation study using Markov Chain Monte Carlo method in order to establish the accuracy of these estimators. The usefulness of the new family is illustrated by means of two real data sets.Mathematics Subject Classification (2010): 60E05; 62F03; 62F10; 62P10

A new class of defective models based on the Marshall–Olkin family of distributions for cure rate modeling

“…Table 1 lists ten extended Weibull distributions which will be used to illustrate Theorem 2.1. They were selected from [36].…”

Marshall-Olkin extended Burr III distribution

“…where ( , , ) is the joint prior distribution of the parameters, ( ; ) and ( ; , , ) are defined in (16), and B is the normalizing constant. Bayesian and credible interval estimates of the parameters are obtained numerically using Markov chain Monte Carlo (MCMC) techniques.…”

“…These includes the Marshall-Olkin extended (MOE) Weibull proposed by Ghitany et al [9], MOE Pareto proposed by Ghitany [7], MOE gamma proposed by Ristic et al [15], MOE Lomax using censored data proposed by Ghitany et al [8], MOE exponential distribution proposed by Srivastava et al [19], MOE Uniform distribution proposed by Jose and Krishna [12], both MOE power log-normal and MOE log-logistic distributions proposed by Gui [10], [11], and MOE Burr type XII distribution proposed by Al-Saiari et al [1]. In addition, Santos-Neto et al [16] introduced the MOE Weibull family of distributions and studied some of its mathematical properties. Barreto-Souza et al [2] and Cordeiro and Lemonte [5] provided general results of the Marshall and Olkin's family of distributions.…”

On Bivariate Exponentiated Extended Weibull Family of Distributions

“…Further, we refer the reader to Nadarajah and Kotz (2005) and Pham and Lai (2007). In recent years, many authors worked on this class of distributions such as the beta extended Weibull family by Cordeiro et al (2012), the extended Weibull power series distributions by Silva et al (2013), the complementary extended Weibull power series class of distributions by Cordeiro and Silva (2014), the Marshall-Olkin extended Weibull family of distributions by Santos-Neto et al (2014) and the exponentiated extended Weibull power series class of distributions by Tahmasebi and Jafari (2015).…”