The Kumaraswamy Marshal-Olkin family of distributions

Abstract

A new family of continuous probability distributions is proposed by using Kumaraswamy-G distribution as the base line distribution in the Marshall-Olkin construction. A number of known distributions are derived as particular cases. Various properties of the proposed family like formulation of the pdf as different mixture of exponentiated baseline distributions, order statistics, moments, moment generating function, Rényi entropy, quantile function and random sample generation have been investigated. Asymptotes, shapes and stochastic ordering are also investigated. Characterizations of the proposed family based on truncated moments, hazard function and reverse hazard function are also presented. The parameter estimation by method of maximum likelihood, their large sample standard errors and confidence intervals and method of moment are also discussed. Two members of the proposed family are compared with different sub models and also with the corresponding members of Kumaraswamy-Marshall-Olkin-G family by fitting of two real life data sets.

The Marshall-Olkin-Kumaraswamy-G family of distributions

“…Recently, Alizadeh et al, (2015a) proposed the Kumaraswamy Marshall-Olkin family of distributions by using the Marshall-Olkin cdf in that Kw-G family and studied its many properties. The main motivation behind the present article is to propose a new family of continuous probability distributions that generalizes the Kw-G family as well as the Marshall-Olkin extended family by integrating the former as the base line distribution in the latter.…”

“…In this subsection, we consider fitting two real data sets to show that the distributions from the proposed MOKw-G family can provide better model than the corresponding distributions from KwMO-G (Alizadeh et al, 2015a) by considering the Frechet and the Exponential distribution as our G. Further we compare it with its sub models namely MO-G, Kw-G to show its superiority. Here the parameters are estimated by numerical maximization of log-likelihood function and their asymptotic standard errors are computed using large sample approach.…”

“…Notable among them are the Azzalini's skewed family (Azzalini, 1985), Marshall-Olkin extended (MOE) family (Marshall and Olkin, 1997), exponentiated family of distributions (Gupta et al, 1998), or the composite methods of combining two or more known competing distributions through transformations like betagenerated family (Eugene et al, 2002;Jones 2004), gamma-generated familiy (Zografos andBalakrishnan 2009, Ristic andBalakrishnan, 2012), Kumaraswamy-G (Kw-G) family (Cordeiro and de Castro, 2011), McDonald-G family (Alexander et al, 2012), beta extended-G family , Kumaraswamybeta generalized family (Pescim et al, 2012), exponentiated transformed transformer family (Alzaghal et al 2013), exponentiated generalized family , geometric exponential-Poisson family (Nadarajah et al, 2013a), truncated-exponential skew symmetric family (Nadarajah et al, 2013b), logisticgenerated family (Torabi and Montazari, 2014), Kumaraswamy Marshall-Olkin-G family (Alizadeh et al, 2015a), generalized odd log-logistic family (Cordeiro et al, 2016), generalized gamma-Weibull distribution (Meshkat et al 2016) and generalized transmuted-G family (Nofal et al, 2017). While the additional parameter(s) bring in more flexibility at the same time they also complicates the mathematical form of the resulting distribution, often considerably enough to render it not amenable to further analytical and numerical manipulations.…”

The Burr X Generator of Distributions for Lifetime Data

“…Many generalized distribution functions are constructed in a similar manner, for example the exponentiated gamma, exponentiated Fréchet and exponentiated Gumbel distributions [23], although the way they defined the cdfs of the last two distributions is slightly different. Several other classes can be mentioned such as the Marshall-Olkin-G (MO-G) family by Marshall and Olkin [22], beta generalized-G (BG-G) family by Eugene et al [14], Kumaraswamy-G (Kw-G) family by Cordeiro and de Castro [9] and exponentiated generalized-G (EG-G) family by Cordeiro et al [8], the Lomax generator of distributions by Cordeiro et al [12], beta odd log-logistic generalized (BOLL-G) by Cordeiro et al [11], beta Marshall-Olkin (BMO-G) by Alizadeh et al [4], Kumaraswamy odd log-logistic (KwOLL-G) by Alizadeh et al [6], Kumaraswamy Marshall-Olkin (KwMO-G) by Alizadeh et al [5], generalized transmuted-G (GT-G) by Nofal et al [24], transmuted exponentiated generalized-G (TExG-G) by Yousof et al [26], Kumaraswamy transmuted-G by Afify et al [2] and transmuted geometric-G by Afify et al [1].…”

Generalized Transmuted Family of Distributions: Properties and Applications

“…So, several generators based on one or more parameters have been proposed to generate new distributions. Some well-known generators are Marshal-Olkin generated family (MO-G) [33], the beta-G by Eugene et al [20] , Jones [31], Kumaraswamy-G (Kw-G for short) by Cordeiro and de Castro [16], McDonald-G (Mc-G) by Alexander et al [1], gamma-G (type 1) by Zografos and Balakrishanan [57], gamma-G (type 2) by Ristić and Balakrishanan [47] , , gamma-G (type 3) by Torabi and Montazari [55], log-gamma-G by Amini et al [7], logistic-G by Torabi and Montazari [56], exponentiated generalized-G by Cordeiro et al [18], Transformed-Transformer (T-X) by Alzaatreh et al [5], exponentiated (T-X) by Alzaghal et al [6], Weibull-G by Bourguignon et al [12], Exponentiated half logistic generated family by Cordeiro et al [15], Kumaraswamy Odd log-logistic by Alizadeh et al [3], Lomax Generator by Cordeiro et al [19], a new Weibull-G by Tahir et al [51], Logistic-X by Tahir et al [52], Kumaraswamy Marshal-Olkin family by Alizadeh et al [4], Beta Marshal-OLkin family by Alizadeh et al [2], type I half-logistic family by Cordeiro et al [14] and Odd Generalized Exponential family by Tahir et al [53].…”