The Transmuted Weibull Lomax Distribution: Properties and Application

A new five parameter model is proposed and stutied. The new distribution generalizes the Weibull Lomax distribution introduced by Tahir et al. (2015) and is referred to as transmuted Weibull Lomax (TWL) distribution. Various structural properties of the new model including ordinary and incomplete moments, quantiles, generating function, probability weighted moments, Rényi and q-entropies and order statistics are derived. We proposed the method of maximum likelihood for estimating the model parameters. The usefulness of the new model is illustrated through an application to a real data set.


Introduction
In fact, there are hundreds of continuous univariate distributions. However, in recent years, applications from the environmental, financial, biomedical sciences, engineering among others, have further shown that data sets following the classical distributions are more often the exception rather than the reality. Since there is a clear need for extended forms of these distributions a significant progress has been made toward the generalization of some well-The cumulative distribution function () cdf (for > 0) of the Weibull lomax distribution is given by ( , , , , ) = 1 exp where  is a scale parameter, , a  and b are shape parameters. The corresponding probability density function ( ) pdf is given by ( , , , , ) = 1 1 1 The aim of this paper is to provide more flixible extension of the Weibull Lomax (WL) distribution using the transmutation map technique introduced by Shaw  According to the Quadratic Rank Transmutation Map, (QRTM), approach a random variable X is said to have transmuted distribution if its cumulative distribution function () cdf is given by where () Gx is the () cdf of the base distribution, which on differentiation yields fx and ( ) gx are the corresponding pdfs associated with ( ) cdfs F x and () Gx respectively. For more information about the quadratic rank transmutation map is given in Shaw and Buckley (2007). Observe that at = 0, In this paper we provide mathematical and statistical properties of the exponentiated Weibull Lomax (TWL) distribution. The rest of the paper is outlined as follows. In Section 2, we define the subject distribution and provide the graphical presentation for its pdf and hrf . In Section 3, we provide a very useful expansions for the pdf and cdf of the new model. Section 4 provides statistical properties including quantile functions, random number generation, ordinary and incomplete moments, moment generating functions, mean deviations, probability weighted moments and Rényi entropy are derived. In Section 5, the order statistics and its moments are discussed. The maximum likelihood estimates (MLEs) and the asymptotic confidence intervals of the unknown parameters are demonstrated in Section 6. In section 7, the TWL distribution is applied to a real data set to illustrate its usefulness. Finally, some concluding remarks are given in section 8.

The TWL Distribution
The Transmuted Weibull Lomax (TWL) distribution and its sub-models are presented in this section. The random variable   The corresponding pdf of X is given by   1 1 ( , , , , , ) = 1 1 1 exp 1 1 where  is a scale parameter representing the characteristic life, ,  and  are shape parameters representing the different patterns of the TWL distribution and  is the transmuted parameter. The reliability function (rf), and cumulative hazard rate function (chrf) of the .
r v X are given by  

Mixture Representation
The TWL density function given in (4) can be rewritten as By inserting (1) and (2) in Equation (5) Equation (6)  Inserting this expansion in Equation (6) and, after some simplification, we obtain Appluing a power series expansion again, we get Equation (7) can be rewritten as a mixture of exponentiated Lomax (EL) densities , , , , , = ( ),

Statistical Properties
Established algebraic expansions to determine some structural properties of the TWL distribution can be more efficient than computing those directly by numerical integration of its density function. The statistical properties of the TWL distribution including quantile and random number generation, moments, factorial moments, cumulants, moment generating function, incomplete moments, mean deviations, probability weighted moments and Rényi and q entropies are discussed in this section.  (11) we can get the median of .
Simulating the TWL random variable is straightforward. If U is a uniform variate on the unit interval (0,1), then the random variable = q Xx follows (5), i.e.

Moments
The th moment, denoted by , using (7) (14) and cumulants ( n  ) of X are obtained from (13) The skewness and kurtosis measures can be calculated from the ordinary moments using well-known relationships.
The th descending factorial moment of ( is the Stirling number of the first kind.

Moment Generating Function
The moment generating function () mgf of the TWL distribution is given by the following theorem.

Incomplete Moments
The important application of the first incomplete moment refers to the Bonferroni and Lorenz curves. These curves are very useful in economics, reliability, demography, insurance and medicine. The answers to many important questions in economics require more than just knowing the mean of the distribution, but its shape as well. This is obvious not only in the study of econometrics but in other areas as well.
 is the first incomplete moments and M is the median of X .

Probability weighted moments
The probability weighted moments (PWMs) are used to derive estimators of the parameters and quantiles of generalized distributions. These moments have low variance and no severe bias, and they compare favorably with estimators obtained by the maximum likelihood method The ( ; ) s r th PWM of

Rényi and q-Entropies
The Rényi entropy of a random variable X represents a measure of variation of the uncertainty. The Rényi entropy is defined by Therefore, we can write and then by inserting (4) in equation (19), we obtain The L-moments are analogous to the ordinary moments but can be estimated by linear combinations of order statistics. They exist whenever the mean of the distribution exists, even though some higher moments may not exist, and are relatively robust to the effects of outliers. Based upon the moments in Equation (21), we can derive explicit expressions for the L-moments of X as infinite weighted linear combinations of the means of suitable TWL distributions. They are linear functions of expected order statistics defined by     One simply can obtain the  's for X from Equation (21) with = 1. q

Estimation
The Therefore the score vector is , , , A  , whilst the MLEs and their corresponding standard errors (in parentheses) of the model parameters are shown in tables 2, respectively. These numerical results are obtained using the MATH-CAD PROGRAM.

Conclusions
In this paper, We propose a new five-parameter model, called the transmuted Weibull Lomax (TWL) distribution, which extends the Weibull Lomax (WL) distribution introduced by Tahir et al. (2015). An obvious reason for generalizing a standard distribution is the fact that the generalization provides more flexibility to analyze real life data. We provide some of its mathematical and statistical properties. The TWL density function can be expressed as a mixture of exponentiated Lomax (EL) densities. We derive explicit expressions for the ordinary and incomplete moments, factorial moments, cumulants, generating function, probability weighted moments, Rényi and q-entropies. We also obtain the density function of the order statistics and its moments. We discuss maximum likelihood estimation. The proposed distribution was applied to a real data set; it was shown to provide a better fit than several other related models and non-nested models. We hope that the proposed model will attract wider application in areas such as engineering, survival and lifetime data, meteorology, hydrology, economics (income inequality) and others.