Exponentiated MarshallOlkin family of distributions
 Cícero R. B. Dias^{1},
 Gauss M. Cordeiro^{1},
 Morad Alizadeh^{3},
 Pedro Rafael Diniz Marinho^{1}Email author and
 Hemílio Fernandes Campos Coêlho^{2}
https://doi.org/10.1186/s4048801600512
© The Author(s) 2016
Received: 26 February 2016
Accepted: 8 September 2016
Published: 5 November 2016
Abstract
We study general mathematical properties of a new class of continuous distributions with three extra shape parameters called the exponentiated MarshalOlkin family of distributions. Further, we present some special models of the new class and investigate the shapes and derive explicit expressions for the ordinary and incomplete moments, quantile and generating functions and probability weighted moments. We discuss the estimation of the model parameters by maximum likelihood and show empirically the potentiality of the family by means of two applications to real data.
Keywords
Introduction

W[G(x)]∈[a,b];

W[G(x)] is differentiable and monotonically nondecreasing;

W[G(x)]→a as x→−∞ and W[G(x)]→b as x→∞.
Some special models
α  λ  p  G(x)  Reduced distribution 

    0  G(x)  Exponentiated Generalized Class of Distributions (Cordeiro et al. 2013) 
1      G(x)  MarshalOlkin family of distributions (Marshall and Olkin 1997) 
1    0  G(x)  Proportional hazard rate model (Gupta and Gupta 2007) 
  1  0  G(x)  Proportional reversed hazard rate model (Gupta and Gupta 2007) 
1  1  0  G(x)  G(x) 
1      1−e^{−x }  Exponential  Geometric distribution (Adamidis and Loukas 1998) 
      1−e^{−x }  Generalized Exponential  Geometric distribution (Silva et al. 2010) 
1      \(1\mathrm {e}^{\beta x^{\gamma }}\)  WeibullGeometric distribution (BarretoSouza et al. 2011) 
      \(1\mathrm {e}^{\beta x^{\gamma }}\)  Exponentiated WeibullGeometric distribution (Mahmoudi and Shiran 2012) 
 i.
to make the kurtosis more flexible compared to the baseline model;
 ii.
to produce a skewness for symmetrical distributions;
 iii.
to construct heavytailed distributions for modeling real data;
 iv.
to generate distributions with symmetric, leftskewed, rightskewed or reversedJ shape;
 v.
to define special models with all types of the hrf;
 vi.
to provide consistently better fits than other generated models under the same baseline distribution.
This paper is organized as follows. In Section 2, we define the new family of distributions and provide a physical interpretation. Five of its special distributions are discussed in this section. In Section 3, some properties of the EMOG family are presented. The shape of the density and hazard rate functions are described analytically, two useful linear mixtures are provided. We derive a power series for the quantile function (qf) and we provide two general formulae for the moments. The incomplete moments are investigated and we derive the moment generating function (mgf) and determine the mean deviations. Estimation of the model parameters by maximum likelihood is performed in Section 4. Applications to two real data sets illustrate the performance of the EMO family in Section 5. The paper is concluded in Section 6.
The new family
where g(x;ξ) is the baseline pdf. This density function will be most tractable when the functions G(x) and g(x) have simple analytic expressions. Hereafter, a random variable X with density function (3) is denoted by X∼EMOG(p,α,λ,ξ). Henceforth, we can omit sometimes the dependence on the baseline vector ξ of parameters and write simply G(x)=G(x;ξ), f(x)=f(x;p,α,λ,ξ), etc.
follows the density function (3).
2.1 Special EMO distributions
Special EMO distributions
Distribution  q(·)  G(x)  g(x) 

E M O N(p,α,λ,μ,σ ^{2})  \(\frac {\alpha \lambda (1p)}{\sigma }\)  \(\phi \left (\frac {x\mu }{\sigma }\right)\)  \(\phi \left (\frac {x\mu }{\sigma }\right)\) 
E M O F r(p,α,λ,β,σ)  α λ(1−p)β  \(\exp \left \{\left (\frac {\sigma }{x}\right)^{\beta }\right \}\)  \(\sigma ^{\beta } x^{\beta 1}\exp \left \{\left (\frac {\sigma }{x}\right)^{\beta }\right \}\) 
E M O G a(p,α,λ,a,b)  \(\frac {\alpha \lambda (1p)b^{a}}{\Gamma (a)}\)  \(\frac {\gamma (a,bx)}{\Gamma (a)}\)  \(\frac {b^{a}}{\Gamma (a)}x^{a1}e^{bx}\) 
E M O B(p,α,λ,a,b)  \(\frac {\alpha \lambda (1p)}{B(a,b)}\)  \(\frac {{\int \limits _{0}^{x}}w^{a1}(1w)^{b1}dw}{B(a,b)}\)  \(\frac {1}{B(a,b)}x^{a1}(1x)^{b1}\) 
E M O G u(p,α,λ,μ,σ)  \(\frac {\alpha \lambda (1p)}{\sigma }\)  \(\text {exp}\left \{{\text {exp}\left [\left (\frac {x\mu }{\sigma }\right)\right ]}\right \}\)  \(\text {exp}\left \{\text {exp}\left [\frac {(x\mu)}{\sigma }\right ]\frac {(x\mu)}{\sigma }\right \}\) 
Some properties of the EMOG family
We investigate some properties of the EMOG in this section.
3.1 Asymptotic and shapes
Proposition 1
Proposition 2
If x=x _{0} is a root of (6) then it corresponds to a local maximum if λ(x)>0 for all x<x _{0} and λ(x)<0 for all x>x _{0}. It corresponds to a local minimum if λ(x)<0 for all x<x _{0} and λ(x)>0 for all x>x _{0}. It gives a point of inflexion if either λ(x)>0 for all x≠x _{0} or λ(x)<0 for all x≠x _{0}.
There may be more than one root to (8). Let τ(x)=d ^{2} log[h(x)]/d x ^{2}. If x=x _{0} is a root of (8) then it refers to a local maximum if τ(x)>0 for all x<x _{0} and τ(x)<0 for all x>x _{0}. It corresponds to a local minimum if τ(x)<0 for all x<x _{0} and τ(x)>0 for all x>x _{0}. It gives an inflexion point if either τ(x)>0 for all x≠x _{0} or τ(x)<0 for all x≠x _{0}.
3.2 Linear mixtures
where (for k≥0) h _{ k+1}(x;ξ)=(k+1) g(x;ξ) G(x;ξ)^{ k } denotes the density function of the random variable Y _{ k+1}∼expG(k+1). Equation (10) reveals that the EMOG density function is a linear mixture of expG density functions. Thus, some of its mathematical properties can be derived directly from those properties of the expG distribution. For example, the ordinary and incomplete moments and (mgf) of X can be obtained from those quantities of the expG distribution. Some structural properties of the expG distributions are welldefined by Mudholkar and Hutson (1996), Gupta and Kundu (2001) and Nadarajah and Kotz (2006), among others.
The formulae derived throughout the paper can be easily handled in most symbolic computation software platforms such as Maple, Mathematica and Matlab. These platforms have currently the ability to deal with analytic expressions of formidable size and complexity. Established explicit expressions to calculate statistical measures can be more efficient than computing them directly by numerical integration. The infinity limit in these sums can be substituted by a large positive integer such as 20 or 30 for most practical purposes.
3.3 Quantile power series
where the coefficients a _{ i }’s are suitably chosen real numbers depending on the parameters of the parent distribution. For several important distributions such as the normal, Student t, gamma and beta distributions, Q _{ G }(u) does not have explicit expressions but it can be expanded as in Eq. (11).
Clearly, c _{ n,i } can be easily evaluated numerically from c _{ n,0},…,c _{ n,i−1} and then from the quantities a _{0},…,a _{ i }.
where f _{ r,m } is obtained from the e _{ m }’s using (13).
3.4 Moments
Hereafter, we shall assume that G is the cdf of a random variable Z and that F is the cdf of a random variable X having density function (3). The rth ordinary moment of X can be obtained from the (r,k)th Probability Weighted Moment (PWM) of Z defined by
The PWMs for some wellknown distributions will be determined in the following sections using alternatively Eqs. (17) and (19).
3.5 Incomplete moments
The integral in (21) can be evaluated at least numerically for most baseline distributions.
Equations (21) and (22) are the main results of this section.
3.6 Generating function
where M _{ k+1}(s) is the generating function of the expG(k+1) distribution. Hence, M(s) can be determined from an infinite linear combination of the expG generating functions.
respectively.
3.7 Mean deviations
Equation (28) is the basic quantity to compute the mean deviations for the expG distributions. A simple application of (27) and (28) can be conducted to the exponentiated MarshallOlkin Weibull (EMOW) distribution. The exponentiated Weibull density function (for x>0) with power parameter k+1, shape parameter c and scale parameter β is given by
where \(\gamma (a,x)= {\int _{0}^{x}} w^{a1}\,\mathrm {e}^{w}\mathrm {d}w\).
Applications of these equations are straightforward to obtain Bonferroni and Lorenz curves. These curves are defined (for a given probability π) by \(B(\pi)=m_{1}(q)/(\pi \,\mu ^{\prime }_{1})\) and \(L(\pi)=m_{1}(q)/\mu ^{\prime }_{1}\), respectively, where q=F ^{−1}(π)=Q(π) comes from the qf of X for a given probability π.
Estimation
Several approaches for parameter estimation were proposed in the literature but the maximum likelihood method is the most commonly employed. The maximum likelihood estimators (MLEs) enjoy desirable properties and can be used when constructing confidence intervals for the model parameters. The normal approximation for these estimators in large samples can be easily handled either analytically or numerically. So, we consider the estimation of the unknown parameters for this family from complete samples only by maximum likelihood. Here, we determine the MLEs of the parameters of the new family of distributions from complete samples only.
where h ^{(ξ)}(·) means the derivative of the function h with respect to ξ. For interval estimation of the model parameters, we can derive the observed information matrix J _{ n }(θ), whose elements can be obtained from the authors upon request. Let \(\widehat {\theta }\) be the MLE of θ. Under standard regularity conditions (Cox and Hinkley 1974), we can approximate the distribution of \(\sqrt {n}(\widehat {\theta }  \theta)\) by the multivariate normal N _{ r }(0,K(θ)^{−1}), where \(K(\theta) = {\lim }_{n \rightarrow \infty }n^{1}J_{n}(\theta)\) is the unit information matrix and r is the number of parameters of the new distribution.
Applications
Descriptive statistics
Statistics  Real data sets  

Stress  Gasoline  
Mean  0.2642  0.1966 
Median  0.2500  0.1780 
Mode  0.2500  0.1500 
Variance  0.0376  0.0115 
Skewness  0.9712  0.3867 
Kurtosis  0.8272  –0.6561 
Maximum  0.8500  0.4570 
Minimum  0.0100  0.0280 
n  166  32 
where K(·) is the symmetrical kernel function and \(\int \limits _{\infty }^{\infty }K(x)dx=1\). Furthermore, h>0 is known in literature as bandwith, which is a smoothing parameter. It is possible to find in literature numerous kernel function, as the normal standard distribution, for example. Silverman (1986) demonstrated that for the K standard normal, a reasonable bandwith is given by \(h=\sqrt [5]{\left (4\hat {\sigma }^{5}/3n\right)}\approx 1.06\hat {\sigma }/\sqrt [5]{n}\), where \(\hat {\sigma }\) is defined by the standard deviation of the sample.
In this application, we use the package AdequacyModel. This package is intended to provide a computational support to work with probability distributions, mainly distributions aimed to survival analysis. This package was used to calculate some fitness statistics adjustment such as AIC (Akaike Information Criterion), CAIC (Consistent Akaikes Information Criterion), BIC (Bayesian Information Criterion), HQIC (HannanQuinn information criterion), KS (Test of KolmogorovSmirnov), A ^{∗} (statistic of AndersonDarling) and W ^{∗} (statistic of Cramérvon Mises), which are described by Chen and Balakrishnan (1995), based on the results presented by Stephens (1986). When we want to test if one random sample, denoted by x _{1},x _{2},…,x _{ n }, with empirical distribuction function F _{ n }(x) comes from a specific distribution, we use these statistics. The Cramérvon Mises (W ^{∗}) and AndersonDarling (A ^{∗}) statistics are given by the following expressions:
Goodnessoffit statistics for the data: (I) stress among women in Townsville, Queensland, Australia (II) proportion of crude oil converted to gasoline after distillation and fractionation
Data set  Distribution  A ^{∗}  W ^{∗} 

I  EMOB (p,α,λ,a,b)  0.1554  0.0786 
KwWP(a,b,c,λ,β)  0.4974  0.1305  
II  EMOB (p,α,λ,a,b)  0.0348  0.0983 
KwWP(a,b,c,λ,β)  0.0489  0.0993 
In this study, the MLEs in Table 5 were obtained by global search heuristic method called Particle Swarm optmization  PSO proposed by Eberhart and Kennedy (1995). One of the advantages of using the PSO method in addition to being a robust optimization method is that there is no need to provide initial guesses. However, this is a computationally intensive method. The Appendix A shows the function pso implemented in R. At the end of the code there is a small example of how to use the function to minimize an objective function. The standard errors of the MLEs can be obtained by the bootstrap method. The standard errors were not obtained in these examples due to the use of the PSO method, which is computationally intensive.
Conclusions
MLEs for: (I) stress among women in Townsville, Queensland, Australia (II) proportion of crude oil converted to gasoline after distillation and fractionation
Data set  Distribution  Maximum Likelihood Estimates  MLE  

I  EMOB (p,α,λ,a,b)  –19.8703  1.4276  3.4036  0.2454  0.7793 
KwWP(a,b,c,λ,β)  12.3010  20.1431  0.1647  24.6569  2.3195  
II  EMOB (p,α,λ,a,b)  –1.2077  0.3989  1.9915  4.6421  10.7513 
KwWP(a,b,c,λ,β)  15.2365  8.3171  0.2446  24.9571  10.5076 
Appendix
Declarations
Authors’ contributions
The authors, CRBD, GMC, MA, PRDM and HFCC with the consultation of each other carried out this work and drafted the manuscript together. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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