The Generalized Marshall-Olkin-Kumaraswamy-G family of distributions

A new family of distribution is proposed by using Kumaraswamy-G (Cordeiro and de Castro, 2011) distribution as the base line distribution in the Generalized Marshal-Olkin (Jayakumar and Mathew, 2008) Construction. A number of special cases are presented. By expanding the probability density function and the survival function as infinite series the proposed family is seen as infinite mixtures of the Kumaraswamy-G (Cordeiro and de Castro, 2011) distribution. Density function and its series expansions for order statistics are also obtained. Order statistics, moments, moment generating function, R\'enyi entropy, quantile function, random sample generation, asymptotes, shapes and stochastic orderings are also investigated. The methods of parameter estimation by method of maximum likelihood and method of moment are presented. Large sample standard error and confidence intervals for the mles are also discussed. One real life application of comparative data fitting with some of the important sub models of the family and some other models is considered.


Introduction
Generating new distributions starting with a base line distribution by adding one or more additional parameters through various mechanisms is an area of research in the filed of the probability distribution which have seen lot of work of late. The basic motivation of these works is to bring in more flexibility in the modelling different type of data generated from real life situation.
Recently, there is renewed activity in this area to propose and investigate new families of distributions. A recent review paper by Tahir  Important special cases of the family along with their shape and main reliability characteristics are presented in the next section. In section 5 we discuss some general results of the proposed family, while different methods of estimation of parameters along comparative data modelling example are presented in section 6. The article ends with a conclusion in section 7 followed by an appendix to derive asymptotic confidence bounds.

Some formulas and notations
Here first we list some formulas to be used in the subsequent sections of this article.  Jayakumar and Mathew (2008) proposed a generalization of the Marshall and Olkin (1997) family of distributions by using the Lehman second alternative (Lehmann 1953) to obtain the sf ) (t F GMO of the GMOE family of distributions by exponentiation the sf of MOE family of distributions as

Generalized Marshall-Olkin Extended ( GMOE ) family of distributions
) and 0   is an additional shape parameter. When , . The cdf and pdf of the GMOE distribution are respectively and Reliability measures like the hrf, rhrf and chrf associated with (1) are ) ( ) (t F and ) (t h are respectively the pdf, cdf, sf and hrf of the baseline distribution. We denote the family of distribution with pdf (1) and 1 1 ] and 0 , 0   b a are shape parameters in addition to those in the baseline distribution. The sf, hrf, rhrf and chrf of this distribution are respectively given by

Generalized Marshall-Olkin Kumaraswamy-G ) ( G GMOKw  family of distributions
We now propose a new extension of the GMO family by considering the cdf and pdf of G Kw  distribution in (4) and (5) as the ) ( and ) ( t F t f respectively in the GMO formulation in (3) and call it G GMOKw  distribution. The resulting expression for the pdf of G GMOKw  is given by The cdf, sf, hrf, rhrf and chrf of G GMOKw  distribution are respectively given by hrf: , we get back the corresponding expressions for G MOKw  distribution of Handique and (Jayakumar and Mathew, 2008) and

Let
ii

Shape of the density and hazard functions
Here we have plotted the pdf and hrf of the G GMOKw  for some choices of the parameters to study the variety of shapes assumed by the family.
From the plots in figure 1 and 2 it can be seen that the family is very flexible and can offer many different types of shapes. It offers IFR, DFR even bath tub shaped hazard rate.

Some special G GMOKw  distribution
Some special cases of the G GMOKw  family of distributions are presented in this section.

The
Let the base line distribution be exponential with parameter , 0 Considering the Lomax distribution (Ghitany et al. 2007) with respective pdf and cdf Considering the Weibull distribution (Ghitany et al. 2005 ) (t f GMOKwW respectively, and then the corresponding pdf of Fr GMOKw  distribution is obtained with pdf: Considering the Gompertz distribution (Gieser et al. 1998) with pdf and cdf The extended Weibull (EW) distributions of Gurvich et al. (1997) is defined by cdf is a non-negative monotonically increasing function which depends on the parameter vector  . The corresponding pdf is ) , . Many important distributions can be obtained by choosing different expressions The EW GMOKw  is derived by considering EW as the base line distribution with pdf: The modified Weibull (MW) distribution of Sarhan and Zaindin (2013) has cdf and pdf is given by The pdf and cdf of the exponentiated Pareto distribution of Nadarajah (2005), are given respectively by The cdf and pdf of the extended power distribution are respectively given by The corresponding EP GMOKw  distribution is then given by pdf:

Expansions
We know that where (.)  is the gamma function.
using (9) in (6), we obtain can be called a non central G Kw  (Cordeiro and de Castro, 2011) distribution. Similarly an expansion for the survival function of ] can be derives as Alternatively, we can expand the pdf as Where Another expansion of the density function in (6) can be obtained by expressing the pdf as , the survival function of G GMOKw  can be expressed as

Order Statistics
Now using the general expansion of the G GMOKw  distribution pdf and sf we get the pdf of the th i order statistics for of the G GMOKw  for ) (16) reduces to pdf of the th i order statistics for of the ) , , Again using the general expansion of the G GMOKw  distribution pdf and sf we get the pdf of the th i order statistics for of the Therefore the density function of the ith order statistics of G GMOKw  distribution can be expressed as

Probability weighted moments
The probability weighted moments (PWMs), first proposed by Greenwood et al. (1979), are expectations of certain functions of a random variable whose mean exists. The (10), (14) and (15) Therefore the moments of the ) , , , can be expresses in terms of the

Moment generating function
The moment generating function of G GMOKw  family can be easily expressed in terms of those of the exponentiated G Kw  (Cordeiro and de Castro, 2011) distribution using the results of section 5.1. For example using equation (11) it can be seen that ) (s M X is the mgf of a G Kw  (Cordeiro and de Castro, 2011) distribution.

Rényi Entropy
The entropy of a random variable is a measure of uncertainty variation and has been used in various situations in science and engineering. The Rényi entropy is defined by For furthers details, see Song (2001). For ) 1 , 0 (   using expansion (9), in can be obtained as Again the density function (6) can be expressed as the Rényi entropy of G GMOKw  also can be derived as

Quantile function and random sample generation
We shall now present a formula for generating G GMOKw  random variable by using inversion method by inverting the cdf or the survival function.
The th p Quantile p t for G GMOKw  can be easily obtained from (19)

Asymptotes and shapes
Here we investigate the asymptotic shapes of the proposed family following the methods followed in Alizadeh et al., (2015).

Proposition 2.
The asymptotes of equations (6), (7) and (8) as 0  t are given by The asymptotes of equations (6), (7) and (8) as The shapes of the density and hazard rate functions can be described analytically. The critical points of the G GMOKw  density function are the roots of the equation: There may be more than one root to (20). If 0 t t  is a root of (20) then it corresponds to a local maximum, a local minimum or a point of inflexion depending on whether The critical points of ) (t h are the roots of the equation There may be more than one root to (21). If 0 t t  is a root of (21) then it corresponds to a local maximum, a local minimum or a point of inflexion depending on whether

Stochastic orderings
In this section we study the reliability properties and stochastic ordering of the G GMOKw  distributions Stochastic ordering properties have applications in diverse fields such as economics, reliability, survival analysis, insurance, finance, actuarial and management sciences (Shaked and Shanthikumar, 2007).
Let X and Y be two random variables with cfds F and G, respectively, survival functions , and corresponding pdf's f, g. Then X is said to be smaller than Y in the likelihood . These four stochastic orders are related to each other, as Which is always less than 0.
The remaining statements follow from the implications (22).

Maximum likelihood method
The model parameters of the G GMOKw  distribution can be estimated by maximum likelihood.
be a random sample of size n from G GMOKw  with parameter corresponds to the parameter vector of the baseline distribution G. Then the log-likelihood function for θ is given by This log-likelihood function can not be solved analytically because of its complex form but it can be maximized numerically by employing global optimization methods available with software's like R, SAS, Mathematica or by solving the nonlinear likelihood equations obtained by differentiating (23).
By taking the partial derivatives of the log-likelihood function with respect to b a, , ,  and β we obtain the components of the score vector , and solving them simultaneously yields the maximum likelihood estimate (MLE)

Asymptotic standard error and confidence interval for the mles:
The asymptotic variance-covariance matrix of the MLEs of parameters can obtained by inverting the Fisher information matrix ) ( I θ which can be derived using the second partial derivatives of the log-likelihood function with respect to each parameter. The th j i elements of ) ( I θ n are given by The exact evaluation of the above expectations may be cumbersome. In practice one can estimate ) ( I θ n by the observed Fisher's information matrix ) ( Î θ n is defined as: Using the general theory of MLEs under some regularity conditions on the parameters as . This result can be used to provide large sample standard errors and also construct confidence intervals for the model parameters. Thus an approximate standard error and confidence interval for the mle of j th parameter j  are respectively given by j j v and As an illustration on the MLE method its large sample standard errors, confidence interval in the case of ) , , , , is discussed in an appendix.

Estimation by method of moments
Here an alternative method to estimation of the parameters is discussed. Since the moments are not in closed form, the estimation by the usual method of moments may not be tractable. Therefore (Handique and Chakraborty, 2015).
is the incomplete beta function.
For a random sample n t t t ,... , 2 1 from a population with survival function (7), the model parameters can be estimated using (24) by solving the equations

Real life applications
We consider one real life data to illustrate the suitability of the Likelihood Ratio Test for nested models: (ii) 1 : , that is the sample is from ) , , the likelihood ratio test statistic is given by LR = ) , ( The plots of the fitted densities and fitted cdf's along with the observed ones are displayed in Figures 3 and 4 indicate that the W GMOKw  provides a best fit to the data considered here.

Conclusion
Generalized Marshall-Olkin extended Kumaraswamy generalized family of distributions is introduced and some of its important properties and parameter estimation are studied. Applications with one real life data set has shown that the W GMOKw  distribution is better than its sub models and also some other competitors with respect to the AIC values.

E GMOKw 
The pdf of the E GMOKw  distribution is given by For a random sample of size n from this distribution, the log-likelihood function for the parameter is given by The components of the score vector The asymptotic variance covariance matrix for mles of the unknown parameters  Where the elements of the information matrix can be derived using the following second partial derivatives:         is the derivative of the digamma function.