The New Kumaraswamy Kumaraswamy Weibull Distribution with Application

The Weibull distribution has been used over the past decades for modeling data in many fields so finding generalization of the Weibull distribution becomes very useful to fit more cases or get better fits than before. In this paper, the Kumaraswamy Kumaraswamy Weibull (Kw Kw W) distribution is presented for the first time and we show that it generalizes many important distributions. The probability density function (pdf), the cumulative distribution function (cdf), moments, quantiles, the median, the mode, the mean deviation, the entropy, order statistics, L-moments, extreme value and parameters estimation based on maximum likelihood are obtained for the Kw Kw W distribution. An illustration study and a real data set are used to illustrate the potentiality and application of the new Kw Kw W distribution.


Introduction
presented for the first time the family of beta generalized distributions which has the following cdf and pdf Where G(x;W) and g(x;W) are the cdf and the pdf of the baseline distribution, W is parameters vector of the baseline distributions.
Wahed (2006) presented a general method of constructing extended families of distribution from an existing continuous class using the following equation ( ; , ) (t;T) ;0 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 and W are respectively parameters vectors of generator and baseline distributions.
Cordeiroa and Castro (2010) presented the family of Kumaraswamy generalized distributions which has the following cdf and pdf   ( ;a, b, ) 1 1 ( ; ) ; , 0 ; Where G(x;W) and g(x;W) are the cdf and the pdf of the baseline distribution, W is parameters vector of the baseline distributions.
The main idea of this paper is based on deriving the Kw Kw W distribution from the Kumaraswamy Kumaraswamy family of generalized distributions see Mahmoud et al.(2015), where the generator distribution is the Kumaraswamy Kumaraswamy distribution, see El-Sherpieny and Ahmed (2014), which has the following cdf  


Then, substituting (3) and ( 4) into (1) and ( 2) yields the cdf and the pdf of Kw Kw W distribution as follows , the Kw Kw W distribution has five shape parameters a, b, α, β, θ and has one scale parameter λ.Some important sub-models of the Kw Kw W distribution are illustrated in the following table

An Expansion for The pdf
We shall obtain expansion for the pdf of the Kw Kw W distribution as follows: Using the binomial expansion of the pdf of the Kw Kw W distribution, where b is a real non integer, yields Condition for the expansion of the pdf of the Kw Kw W distribution Since,

An Expansion for The cdf
We shall obtain an expansion for the cdf of the Kw Kw W distribution as follows: Using binomial expansion in (5) yields

Some Properties of the Kw Kw W distribution
In this section some properties of the Kw Kw W distribution will be obtained as follows:

The r th Moment
Generally, the r th moment of a random variable X can be given from , ,  , , , 0 0

The Moment Generating Function
Generally, the moment generating function of a random variable X can be given from ( ) ( ; )

Quantile Function
Starting with the well-known definition of the 100 q th quantile it is clear that ( ) ( ;M) ; 0, 0 1 Quantiles of the Kw Kw W distribution can be obtained by equating the cdf from (5) to q: Easily, substituting q with 1/2 (the second quantile) yields the median

The Mode
The nature logarithm of ( 6) is Then, differentiating the nature logarithm of ( 6) with respect to x and equating it to zero yields: The last equation is a nonlinear equation and does not have an analytic solution with respect to x.Therefore, we have to solve it numerically.If x 0 is a root for the last equation it must be

The Mean Deviation
Generally, the mean deviation about the mean and about the median for a random variable X respectively can be given from Easily, we can obtain them respectively from we must obtain the following integration of J(.), where J(.) is called the incomplete first moment

The Survival and Hazard Function
Generally, the survival function of a random variable X can be given from Generally, the Hazard function can be given from ;  so the hazard function of the Kw Kw W distribution can be given by substituting ( 6) and ( 10) into last equation

Order Statistics
A simple random sample X 1 , X 2 ,.., X v given from the Kw Kw W distribution where X´s are i.i.d random variables, has the density f u:v (x u:v ) of the u th order statistic, for u =1,2,...…,v as follows, see Arnold, et al.(1992 Substituting (15) and ( 7) into (14) yields the density f u:v (x u:v ) of the u th order statistic

Moments of order statistics
Generally, the moments of order statistics can be given from Substituting (16) into last equation gives , , , ,  , , , 0 0 , , , 0 L-moments are linear functions of expected order statistics where L-moments is defined as, see Hosking (1990) Where,
6.The Asymptotes of F(X) and f (X) In this section, we shall obtain the asymptotes of the cdf and pdf of the Kw Kw W distribution

The Asymptotes of cdf
First: when x converges to zero Using Maclaurin series expansion of the cdf gives Using Maclaurin series expansion of the cdf gives and using only first and second terms of cdf binomial expansion gives approaches the standard normal distribution where n  In this section, we will obtain some properties of the asymptotes of the extreme values

The Maximum Likelihood Estimation
We now determine the maximum likelihood estimates (MLEs) of the parameters of the Kw Kw W distribution from complete samples.Let x 1 ,x 2 ,…,x n be a random sample of size n from the Kw Kw W (x;M) distribution, where The log-likelihood function for the vector of parameters (a, b, α, β, θ and λ) can be written as The score functions for the parameters a, b, α, β, θ and λ are given by e a e e e ab e e e e e a e e ab x e e e x e e e a e x e e e e a b e x e e e x a e x e e e e x a b The MLEs of the unknown parameters are obtained by solving the nonlinear likelihood equations from (21) to (26) but They cannot be solved analytically, so we shall use a statistical software to solve the equations numerically.We can use iterative techniques such as a Newton-Raphson algorithm to obtain the estimate.

The Variance-Covariance Matrix
Let θ is the vector of the unknown parameters (a, b, α, β, θ, λ), The element of the 6 ×6 information matrix I (a, b, α, β, θ, λ) can be approximated by: and I -1 (a, b, α, β, θ, λ) is the variance-covariance matrix of the unknown parameters, where second derivatives can be easily driven via Mathcad, Maple, Matlab, or R.

Illustration Study
An experiment is given to illustrate new results of the Kw Kw W (a, b, α, β, θ, λ) distribution, This study is about MLEs of parameters of the Kw Kw W distribution.The algorithm of obtaining parameters estimates is described in the following steps: Step (1): Generate a random sample of size n as follows: u 1 ,u 2 ,..,u n , by using the uniform distribution (0,1) Step (2): transform the uniform random numbers to random numbers of the Kw Kw W distribution by using the quantile function of the Kw Kw W distribution: Step (3): Solve the 21 to 26 by iteration to get the maximum likelihood estimators via iterative techniques and repeat it many times.
In this experiment 10, 15, 30 and random numbers were generated using Mathcad package, then we obtained MLEs the Kw Kw W(a, b, α, β, θ, λ) distribution, where we started with parameters values: a =2, b = 5, α = 4, β = 3, θ=2 and λ=3 for 1000 times, then we used the conjugate gradient iteration method.Finally, we get the following results We see that the more sample size increases the more Biases and RMSEs decrease.

Conclusions
The Kw Kw W distribution is an important distribution in world of data sets because of its flexible properties and its generalization of some important distributions as the E Kw W distribution, the Kw W distribution, the EG W distribution, the E W distribution and the W distribution.We encourage researchers to do more researches and applications on the Kw Kw W in univariate and multivariate cases.

Figure 1 :
Figure 1: Plot of the Kw Kw W pdf for some parameter values We see from the last Figure that the sex parameters give a high degree of flexibility for the Kw Kw W distribution.

Figure 2 :
Figure 2: Plot of the Kw Kw W Hazard for some parameters values

6 . 2
The Asymptotes of the pdfFirst: when x converges to zero.

Figure 3 .
Figure 3. Probability density functions for different distributions

2016 pp165-184 167 2. The New Kumaraswamy Kumaraswamy Weibull Distribution Since
the cdf and pdf of the Weibull distribution respectively are Minimum value: this section we give a real data to illustrate an example for one distribution of the new family of Kw Kw distributions so called Kw Kw W distribution to see how the new model works practically and we shall use the Mathcad package version 15 to do that.
kumaraswamy Weibull (Kw W) distribution [derived from the Kw family], the exponentiated generalized Weibull (EG W) distribution [derived from the E G family], the exponentiated Weibull (E-W) distribution [derived from the E family] and the Pak.j.stat.oper.res.Vol.XII No.1 2016 pp165-184 182

Table 2 : The log-likelihood function, The likelihood ratio test statistic and p-values
Note that the log likelihood of the Kw Kw W (a,b,α,β,θ,λ) = 49.052 *