The New Kumaraswamy Kumaraswamy Family of Generalized Distributions with Application

Finding the best fitted distribution for data set becomes practically an important problem in world of data sets so that it is useful to use new families of distributions to fit more cases or get better fits than before. In this paper, a new generating family of generalized distributions so called the Kumaraswamy Kumaraswamy (KW-KW) family is presented. Four important common families of distributions are illustrated as special cases from the KW KW family. Moments, probability weighted moments, moment generating function, quantile function, median, mean deviation, order statistics and moments of order statistics are obtained. Parameters estimation and variance covariance matrix are computed using maximum likelihood method. A real data set is used to illustrate the potentiality of the KW KW weibull distribution (which derived from the kw kw family) compared with other distributions.


Introduction
The main idea of this paper is based on generating new families of generalized distributions, see Wahed (2006), to derive more generalized distributions from the following integration   1 ; 2 0 ( ; , ) (t ;T) ;0 1; (1) Where 1 ( ; ) G x W and 1 ( ; ) g x W are the cdf and pdf of the baseline distribution, 2 ( ; ) G t T and 2 ( ; ) g t T are the cdf and the pdf of the generator distribution, T is the parameters vector of the generator distribution and W is the parameters vector of the baseline distribution.
The contributions of this paper are four parts. First, we present the pdf and cdf of the new KW KW family of generalized distributions (section 2). Second, we calculate some properties of KW KW family like Moments, probability weighted moments, moment generating function, quantile function, median, mean deviation, order statistics and moments of order statistics (section 3). Third We will consider ( ; , , , ) p t a b  as the pdf of the generator distribution 2 ( ; ) g t T in equation (1), then the cdf of the KW-KW family can be given by substituting equation (2) into equation (1) as follows: is the parameters vector of the generator distribution, W is the parameters vector of the baseline distribution and 1 ( ; ) G x W is the cdf of the baseline distribution.
where a, b, α and β are five shape parameters

Some properties of the Kumaraswamy Kumaraswamy Family
In this section some properties of the kw-kw family for any KW KW generalized distribution will be obtained as follows:

Moments
If X has the pdf (6), then its r th moments can be given from Where,  is the probability weighted moments (PWM) of the baseline distribution.
Second, when  is a real non integer, substituting equation (6)

The Probability Weighted Moments
The probability weighted moments (PWM) can be given from sr sr

First when α is an integer
Substituting equation (5) & (7) into equation (9), Replacing h with r, leads to: Second, when α is a real non integer, Substituting equation (6) & (7) into equation (9), Replacing h with r, leads to: Another form Based on the parent quantile function: First, when  is integer yields

The Moment Generating Function
Generally, the moment generating function is: and using expansion at the last equation leads to Another form Based on the parent quantile function:

Quantile Function and The Median
The quantile function of the KW KW family can be gotten from the cdf of the KW KW family.
As follows:

The Mean Deviation
Basically, the mean deviation is a measure for the amount of scatter in X.
Generally, the mean deviation is expressed by: where, µ is mean and M is median.
These measures can be expressed easily as:

Which is the first incomplete moment
Another form Based on the parent quantile function:

Order Statistics
The pdf of the u th order statistics can be given from, see Arnold et al. (1992)  Substituting equation (7) into last equation, Replacing u+w-1 with r, leads to: First: when α is integer Substituting equation (5) and (11)

Maximum Likelihood Estimators
let X 1 , X 2 … Xn be the iid random variables from the KW-KW (a, b, α, β) distribution and  ,  Then.
Let θ is the vector of the unknown parameters (a,b,α,β,w j ), then the element of the 5 x 5 information matrix I (a,b,α,β,w j ) can be approximated by: 1 (a,b,α,β,w j ) is the variance covariance matrix of the unknown parameters (a,b,α,β,w j ) and the asymptotic distributions of the MLE parameters are

Application
In this section we give a real data to illustrate an example for one distribution of the new family of KW-KW distributions so called the the Kumaraswamy -Kumaraswamy -Weibull distribution to see how the new model works practically and we will use the Mathematica package version 10 to do that. In our example, We used different distributions like the kumaraswamy -kumaraswamy -Weibull (  In Table (1) we compute the MLE of distributions parameters, Kolmogorov-Smirnov (K.S) test statistic, AIC and BIC for every distribution. We find from K.S test statistic that at level of significance 0.05 we can not reject that the data fits all earlier distributions but it fits more the KW -KW -W (a, b, α, β, θ, λ) distribution. We see that the KW -KW -W (a, b, α, β, θ, λ) distribution has the smallest AIC and BIC so the KW -KW -W (a, b, α, β, θ, λ) distribution can be the best fitted distribution compared with earlier distributions In Table (2) and based on the likelihood ratio test, where the KW-KW-W (a, b, α, β, θ, λ) distribution generalizes the E-KW-W (a, α, β, θ, λ) distribution, the KW -W (α, β, θ, λ) distribution, the EG -W (a, β, θ, λ) distribution, the E-W (α, θ, λ) distribution, the E-W (α, θ, λ) distribution and the W(θ, λ) distribution, we find from the p-values that we can reject all null hypotheses when the level of significance is 0.1.

Conclusions
The new Kumaraswamy Kumaraswamy family of generalized distributions can be useful in world of data sets because of its flexible properties and its generalization of some important families of distributions like the families of Exponentiated Kumaraswamy, Kumaraswamy, Exponentiated Generalized and the Exponentiated. It is clear from the application that the new model so called the Kumaraswamy Kumaraswamy Weibull distribution which was derived from the Kumaraswamy Kumaraswamy family practically gave the best fit compared with other distributions. We encourage researchers to do more researches and applications on other distributions of the Kumaraswamy Kumaraswamy family in univariate and multivariate cases.

Appendix
The elements of the observed information matrix I(a,b,α,β,w j ) for the parameters are: Let: 1 log 1 log 1 log 1 log 1 ( ; , ) 1 ,