On some properties of Generalized Transmuted - Kumaraswamy Distribution

This article introduces a new lifetime model called the generalized transmuted-Kumaraswamy distribution which extends the Kumaraswamy distribution from the family proposed by Nofal et al., (2017). We provided hazard and survival functions of the proposed distribution. The statistical properties of the proposed distribution are provided and the method of Maximum Likelihood Estimation (MLE) was proposed in estimating its parameters.


Introduction
The Kumaraswamy probability distribution was pioneered by Ponnambalam Kumaraswamy (often known as Poondi Kumaraswamy, 1980) is a family of continuous probability distributions defined on the interval [0, 1] suitable for physical variables that have lower and upper bounded of both its probability density function and cumulative distribution function. The Kumaraswamy (1980) distribution is a similar to Beta distribution, but much simpler to use especially in simulation studies. This distribution has a wide range of applications in hydrology which faithfully fit hydrological random variables such as daily rainfall, daily stream flow, etc. and many natural phenomena. For more details see Sundar and Subbiah (1989), Fletcher and Ponnambalam (1996), Seifi (2000), Ponnambalam (2001) and Ganji (2006). The kumaraswamy probability density function has the same basic shape properties as the beta distribution (Jones, 2009) and Cordeiro (2010Cordeiro ( , 2012 but it depends on the values of its parameters: it is unimodal for c > 1 and d > 1; uniantimodal for c < 1 and d < 1; increasing for c > 1 and d≤ 1; decreasing for c ≤ 1 and d > 1 and constant for c = d = 1. Jones (2009), made some similarities and differences between the kumaraswamy and beta distributions and also highlighted several advantages of kumaraswamy distribution over beta distribution such as a simple normalizing constant; simple explicit formulae for the distribution and quantile functions which do not involve any special functions; a simple formula for random variate generation; explicit formulae for moments of order statistics. The cumulative distribution function (cdf) and probability density function (pdf) of the Kumaraswamy distribution are respectively given by 11 00 ( ; , ) ( ; , ) and 11 ( ; , ) where , cd are positive shape parameters. (2007) proposed a new transmuted family using Quadratic rank transmutation map which was proposed and studied by Arya (2011), Elbatal (2013), Ahmad (2015) and many others have also been defined and studied in the literatures. In this research we introduce a new lifetime model which is generalized two-parameter Kumaraswamy distribution, called the generalized transmuted -Kumaraswamy (GT-Kw) distribution which extends the quadratic rank transmutation map pioneered by Shaw (2007) by adding two additional shape parameters to generate more flexible distributions based on Generalized Transmuted -G ( GT -G) family pioneered by .

Let ( , )
Gx and ( , ) gx denote the cumulative distribution and density functions of baseline model with parameter vector  . The cdf and pdf of GT -G family defined by Nofal (2017) are given by: and where a, b are positive shape parameters and 1   is transmuting parameter. Now, equation (4) can also be written as: which is a mixture of two E-G densities with two power parameters a and (a+b), where G family (Gupta, 1998) and if 1 ab ==, the GT -G family corresponds to T -G family (Shaw, 2007).

Expansion for the distribution and density functions
A random variable X is said to have Generalized Transmuted -Kumaraswamy (GT-Kw) distribution denoted by ( respectively. Using (7), the proposed density function can also be written as a mixture of two E-G densities defined as Plots of the cdf and pdf of the proposed distribution for some parameters are displayed in figure 1.

Survival and Hazard functions 3.1 Survival function
The survival function of the GT-Kw distribution with parameters , , , 0 a b c d  and 1   is given as where F(x) is defined in (6)

Hazard function
The hazard function of the proposed distribution is given by where , , , 0 a b c d  are shape parameters, 1   is the transmuting parameter and f(x), S(x) are density and survival functions defined in (7) and (12) respectively. Plots of the survival and hazard functions of the proposed GT-Kw distribution are presented in Figure 2.

Statistical Properties of the GT-Kw distribution
In this section, some properties of the proposed distribution were studied such as moments, moment generating function, characteristic function, incomplete moment and order statistics.
Inserting (17) in (16) Finally, the r th moments of the proposed GT-Kw distribution is obtained by substituting (18) and (19) in (14) as given in (20)  which complete the proof.

Characteristics Function 5
The characteristics function of the proposed GT-Kw distribution is given by   (7) can also be written as ( ) The sample statistic that maximizes the likelihood function ()  is called the maximum likelihood estimator of  and is denoted by  . The ML Estimator of the GT-Kw model will be obtained after differentiating (37) with respect to each parameter a, b, c , d ,  and setting the result to zero.

Conclusion
In this paper, we proposed and study a new model, called generalized transmuted -Kumaraswamy model based on the GT -G family studied by . We provided an explicit expression for the survival and hazard functions. The statistical properties including moments, moment generating function, incomplete moments, characteristics function and order statistics were derived. We discuss maximum likelihood estimation by using maximum method. We hope that the proposed model will attract wider application especially in hydrology to model hydrological data and related areas.