The Marshall-Olkin-Kumarswamy-G family of distributions

A new family of continuous distribution is proposed by using Kumaraswamy-G (Cordeiro and de Castro, 2011) distribution as the base line distribution in the Marshal-Olkin (Marshall and Olkin, 1997) 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, Renyi entropy, quantile function and random sample generation have been investigated. Asymptotes, shapes and stochastic ordering are also investigated. The parameter estimation by methods of maximum likelihood, their large sample standard errors and confidence intervals and method of moment are also presented. Two members of the proposed family are compared with corresponding members of Kumaraswamy-Marshal-Olkin-G family (Alizadeh et al., 2015) by fitting of two real life data sets.


Introduction
One of the preferred area of research in the filed of the probability distribution is that of generating new distributions starting with a base line distribution by adding one or more additional parameters notable among them are Azzalini's skewed family (Azzalini, 1985), Marshall-Olkin extended ) (MOE family (Marshall and Olkin, 1997), exponentiated family ) (EF of distributions (Gupta et al., 1998), or by composite methods of combining two or more known competing distribution through transformations like beta-generated ) ( G beta  family (Eugene et al., 2002

Formulas and notations
Here 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):

Marshall-Olkin Extended ) (MOE family of distributions
Starting with a given baseline distribution with probability density function (pdf) ) (t f , cumulative distribution function (cdf) ) (t F , Marshall and Olkin (1997) proposed a new flexible semi parametric family of distributions and defined a new survival function (sf) ) (t F MO by introducing an additional The sf ) (t F MO of the MOE family of distributions is defined by . The parameter ' '  is known as the tilt parameter as since the hrf of the new family is shifted below (above) for the hrf of the base line distribution (Nanda and Das, 2012). That is for all MO and ) (t h are the hrf's of the MOE and baseline distributions respectively. Now (2) and ) ( . Other reliability measures like the hrf, rhrf and chrf associated with (1) are ) ( Where ) (t h and ) (t r is the hrf and rhrf of the baseline distribution. It is obvious that many new families can be derived from Marshall-Olkin set up by considering different base line distribution in the equation (1). These new families are usually termed as Marshall-Olkin extended F distribution. For details see Tahir et al. (2015). Jayakumar and Mathew (2008) generalized the Marshall-Olkin set up by exponentiating the Marshall-Olkin survival function This method is called method of Lehmann alternative 1 due to Lehmann (1953) Tahir et al. (2015) propose another generalization through Lehmann alternative 2 due to Lehmann (1953) by exponentiating the Marshall-Olkin cdf as [for more on Marshall -Olkin family check Barakat and 1 1 ] and 0 , 0   b a are shape parameters in addition to those in the baseline distribution which partly govern skewness and variation in tail weights. For a lifetime random variable ' 't , the sf, hrf, rhrf and chrf for distribution in (4) are given respectively by b a KwG KwG Recently, Alizadeh et al., (2015) proposed the Kumaraswamy Marshal-Olkin family of distributions by using the Marshal-Olkin (Marshall and Olkin 1997) cdf in that Kumaraswamy-G (Cordeiro and de Castro 2011) family and studied its many properties.
The main motivation behind the present article is to propose another family of continuous probability distribution that generalizes the Kumaraswamy-G (Cordeiro and de Castro 2011) family and the Marshall-Olkin Extended family (Marshall and Olkin 1997) by integrating the former as the base line distribution in the later. We call this new family the Marshall-Olkin Kumaraswamy-G family of distribution which encompasses many known families of distributions and study some its general properties, parameter estimation and real life application in the present article.
The rest of this article is organized in seven sections. In section 2 the new family is defined along with its physical basis. Next section is devoted to presenting some important special cases of the family along with their shape and main reliability characteristics. In section 4 we discuss some important general results of the proposed family, while different methods of estimation of parameters are presented in section 5. In section 6 we present two examples of comparative data fitting. The article ends with a concluding discussion and remark in the final section followed by an appendix to derive asymptotic confidence bounds.

New Marshall-Olkin
, Similarly using equation (4) in (2) we get the cdf and hence the sf of G MOKw  respectively as , the pdf in (6) reduces to that of G Kw  (Cordeiro and de Castro, 2011) , the pdf in (6) reduces to that of MO (Marshall and Olkin, 1997).

Genesis of the distribution
let N be a positive integer random variable independent of the i T 's given by the pgf of a geometric distribution with parameter  , say .We can verify that equation . For both cases, equation (8)

Shape of the density and hazard functions
Here we have plotted the pdf and hrf of the G MOKw  for some choices of the distribution G and parameter values 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 for hazard.

Special models
In this section we provide some special cases of the G MOKw  family of distributions and list their main distributional characteristics.

The
Let the base line distribution be exponential with parameter , 0 Considering the Lomax distribution (Ghitany et al., 2007) with pdf and cdf given by Next by taking the Gompertz distribution (Gieser et al., 1998) The extended Weibull (EW) distributions of Gurvich et al. (1997) has the cdf is a non-negative monotonically increasing function which depends on the parameter vector  . The corresponding pdf is Some important particular cases of EW can be seen as follows: Here we derive EW MOKw  by considering EW as the base line distribution as pdf: The cdf and pdf of the modified Weibull (MW) distribution (Sarhan and Zaindin, 2013) is given respectively by The corresponding pdf of EMW MOKw  is given by Power Log-normal distribution introduced by Nelson and Dognanksoy (1992) by extending the Lognormal distribution and its density and cdf are respectively given by, , it reduces to the usual Log-normal distribution.
With this as the base line distribution we get the pdf of , it reduces to the MOKw -Log-normal distribution.

The
The pdf and cdf of the exponentiated Pareto distribution, of Nadarajah (2005) are given respectively Thus the pdf of EEP MOKw  distribution is given by The cdf and pdf of the extended power distribution are . The corresponding pdf of EP MOKw  distribution is then given by pdf:

Series expansions
Consider the series representation This is valid for using (10) in (6), we Where is the cdf of G Kw  (Cordeiro and de Castro 2011) distribution in equation (4). Similarly an expansion for the survival function of ] can be derives as The density function (6) can also be expressed as Hence for 1   using (10) we get , the survival function of G MOKw  can be expressed as

Order statistics
Now using the general expansion of the G MOKw  distribution pdf and sf we get the pdf of the th i order statistics for of the Again using the general expansion of the pdf and sf of G MOKw  distribution we get the pdf of the th i order statistics for of the The density function of the th i order statistics of G MOKw  distribution can be expressed as These results play important role and may be used to obtain explicit expressions for the moments and moment generating function (mgf) of the G MOKw  distribution and of its order statistics in a general framework and also for special models using the corresponding results of exponentiated -G distributions.

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 From equations (11), (12) and (14) The PWM can generally be used for estimating parameters quantiles of generalized distributions. These moments have low variance and don't possess severe biases, and they compare favourably with estimators obtained by maximum likelihood (Alizadeh et al., 2015).

Moment generating function
The moment generating function of G MOKw  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 4.1. For example using equation (15) Where X follows exponentiated G Kw  (Cordeiro and de Castro, 2011) distribution.

Renyi Entropy
The entropy of a random variable is a measure of uncertainty. The Rényi entropy is defined as For furthers details, see Song (2001). For ) 1 , 0 (   using expansion (10) in (6) we can write the Rényi entropy of T can be obtained as The density function (6) can be expressed as using expansion (10) we get, the Rényi entropy of T also can be obtained as

Quantile function and random sample generation
We shall now present a formula for generating G MOKw  random variable by using inversion method by inverting the cdf or the survival function.
Now combining (19) and (20) we get for . Therefore, the th p quantile p t , of E MOKw  is given by

Asymptotes and shapes
Here we investigate the asymptotic shapes of the proposed family following the method followed in Alizadeh et al., (2015). Proposition 1. The asymptotes of equations (6), (7) and (9) as 0  t are given by The asymptotes of equations (6), (7) and (9) as   t are given by The shapes of the density and hazard rate functions can be described analytically. The critical points of the G MOKw  density function are the roots of the equation: The number of roots of (22) may be more than one. In particular, if 0 t t  is a root of (22) 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 The number of roots of (23) may be more than one. In particular, ff 0 t t  is a root of (23) 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 MOKw  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 ratio . These four stochastic orders are related to each other, . Which is always less than 0.
The remaining statements follow from the implications (24).

Maximum likelihood estimation for G MOKw 
The model parameters of the G MOKw  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 softwares like R,

Mathematica.
By taking the partial derivatives of the log-likelihood function with respect to b a, ,  and β components of the score vector can be obtained as follows:

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 loglikelihood 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   n the As an illustration on the MLE method its large sample standard errors, confidence interval in 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 following (Barreto-Souzai et al., 2013) we get One can use (28)

Real life applications
In this subsection, we consider fitting of two real data sets to show that the distributions from the proposed G MOKw  distribution can be a better model than the corresponding distributions from It may be noted that and l n k , where k is the number of parameters in the statistical model, n the sample size and l is the maximized value of the log-likelihood function under the considered model.
We have used the maximum likelihood method for estimating the model parameters.

Example I:
Here

Example II:
This data set consists of 100 observations of breaking stress of carbon fibres (in Gba) given by Nichols     The pdf of the E MOKw  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 parameters of Where the elements of the information matrix can be derived using the following second partial derivatives:            is the derivative of the digamma function.