The Kumaraswamy Generalized Marshall-Olkin-G family of distributions

Another new family of continuous probability distribution is proposed by using Generalized Marshal-Olkin distribution as the base line distribution in the Kumaraswamy-G distribution. This family includes (Cordeiro and de Castro, 2011) and (Jayakumar and Mathew, 2008) families special case besides a under of other distributions. The probability density function (pdf) and the survival function (sf) are expressed as series to observe as a mixture of the Generalized Marshal-Olkin distribution. Series expansions pdf of order statistics are also obtained. Moments, moment generating function, R\'enyi entropies, quantile function, random sample generation and asymptotes are also investigated. Parameter estimation by method of maximum likelihood and method of moment are also presented. Finally the proposed model is compared to the Generalized Marshall-Olkin Kumaraswamy extended family (Handique and Chakraborty, 2015) by considering four examples of real life data modeling.


Introduction
Recently, some efforts have been made to define new families of continuous distributions to 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 with four comparative data modelling applications 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.
If T is a continuous random variable with pdf, , then its Survival function (sf): Hazard rate function (hrf): Reverse hazard rate function (rhrf): and 1 1 ] 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 Extended ( GMOE ) family of distribution
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 ) 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 G and ) (t h are respectively the pdf, cdf, sf and hrf of the baseline distribution. We denote the family of distribution with pdf (5) as ) , , ,

Kumaraswamy Generalized Marshall-Olkin-G ( G KwGMO  ) family of distribution
We now propose a new extension of the G Kw  (Cordeiro and de Castro, 2011) family by considering the cdf and pdf of GMO (Jayakumar and Mathew, 2008) distribution in (4) and (5) as the ) ( and ) ( t F t f respectively in the G Kw  formulation in (2) and call it G KwGMO  distribution. The resulting expression for the pdf of G KwGMO  is given by The cdf, sf, hrf, rhrf and chrf of G KwGMO  distribution are respectively given by : G KwGMO  reduces to some known families of distributions as:

Shape of the density and hazard function
Here we have plotted the pdf and hrf of the G KwGMO  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 of density and hazard rate function including the bath tub shaped free hazard.

Some special G KwGMO  distributions
In this section we provide some special cases of the G KwGMO  family of distributions and list their main distributional characteristics.
Let the base line distribution be exponential with parameter , 0 then for the E KwGMO  model we get the pdf and cdf respectively as: Considering the Lomax distribution (Ghitany et al. 2007) with pdf and cdf given by the pdf and cdf of the L KwGMO  distribution are given by

The
Next by taking the Gompertz distribution (Gieser et al. 1998) with pdf and cdf The pdf and the cdf of the extended Weibull (EW) distributions of Gurvich et al. (1997) is is a non-negative monotonically increasing function which depends on the parameter vector  . and ) : By considering EW as the base line distribution we derive pdf and cdf of the EW KwGMO  as ( l Important models can be seen as particular cases with different choices of ) : (1) ) : (2) ) : : Gompertz distribution.

The
The modified Weibull (MW) distribution (Sarhan and Zaindin 2013) with cdf and pdf is given The corresponding pdf and cdf of EMW KwGMO  are given by The cdf, sf, hrf, rhrf and chrf of EMW KwGMO  distribution are respectively given by The pdf and cdf of the exponentiated Pareto distribution, of Nadarajah (2005), are given respectively by . Thus the pdf and the cdf of EEP KwGMO  distribution are given by

family of distributions
In this section we derive some general results for the proposed G KwGMO  family.

Expansions of pdf and sf
By using binomial expansion in (6), we obtain Alternatively, we can expand the pdf as Similarly an expansion for the survival function of Where,

Order statistics
Where,

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) and (12) the th s moment of T can be written either as Therefore the moments of the ) , , , can be expresses in terms of the

Moment generating function
The moment generating function of G KwGMO  family can be easily expressed in terms of those of the exponentiated GMO (Jayakumar and Mathew, 2008) distribution using the results of section 5.1. For example using equation (13) it can be seen that ) (s M X is the mgf of a GMO (Jayakumar and Mathew, 2008) 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). Using binomial expansion in (6) we can write Thus the Rényi entropy of T can be obtained as

Quantile function and random sample generation
We shall now present a formula for generating G KwGMO  random variable by using inversion method by inverting the cdf or the survival function.
The p th Quantile p t for G KwGMO  can be easily obtained from (19) as For example, let the base line distribution be exponential with parameter , 0   having pdf and cdf as 0 , respectively. Therefore the p th Quantile

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

Maximum likelihood method
The model parameters of the G KwGMO  distribution can be estimated by maximum likelihood. Let 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 (20).
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

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 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 As an illustration on the MLE method its large sample standard errors, confidence interval in the case of ) , , , , is discussed in an appendix.

Real life applications
In this subsection, we consider fitting of four real data sets to compare the proposed . Where k the number of parameters is, n the sample size and l is the maximized value of the log-likelihood function under the considered model. In these applications method of maximum likelihood will be used to obtain the estimate of parameters.

Example I:
Here we consider the following data set of 346 nicotine measurements made from several brands of cigarettes in 1998. The data have been collected by the Federal Trade Commission which is an independent agency of the US government, whose main mission is the promotion of consumer protection.

Example II:
This data set consists of 100 observations of breaking stress of carbon fibres (in Gba) given by Nichols and Padgett (2006

Example III:
The Carbon Fibres data set of the 66 observations on the breaking stress of carbon fibres (in Gba) as reported in Nichols and Padgett (2006)

Example IV:
Here, we use the following real data sets which gives the time to failure ) 10 ( 3 h of turbocharger of one type of engine given in Xu et al. (2003).
A visual comparison of the closeness of the fitted densities with the observed histogram of the data and fitted cdfs with empirical cdfs for example I, II, III and IV are presented in the figures 3, 4, 5 and 6 respectively. These plots indicate that the proposed distributions provide a closer fit to these data.

Conclusion
Kumaraswamy

E KwGMO 
The pdf of the E KwGMO  distribution is given by For a random sample of size n from this distribution, the log-likelihood function for the parameter vector The components of the score vector The asymptotic variance covariance matrix for mles of the parameters of Where the elements of the information matrix can be derived using the following second partial derivatives:             e  t  e  e  t  e  e  e  a   0   2   2  2   is the derivative of the digamma function.