On Six-Parameter Fréchet Distribution: Properties and Applications

This paper introduces a new generalization of the transmuted Marshall-Olkin Fréchet distribution of Afify et al. (2015), using Kumaraswamy generalized family. The new model is referred to as Kumaraswamy transmuted Marshall-Olkin Fréchet distribution. This model contains sixty two sub-models as special cases such as the Kumaraswamy transmuted Fréchet, Kumaraswamy transmuted Marshall-Olkin, generalized inverse Weibull and Kumaraswamy Gumbel type II distributions, among others. Various mathematical properties of the proposed distribution including closed forms for ordinary and incomplete moments, quantile and generating functions and Rényi and -entropies are derived. The unknown parameters of the new distribution are estimated using the maximum likelihood estimation. We illustrate the importance of the new model by means of two applications to real data sets.


Introduction
The procedure of expanding a family of distributions for added flexibility or to construct covariate models is a well-known technique in the literature.In many applied sciences such as medicine, engineering and finance, amongst others, modeling and analyzing lifetime data are crucial.Several lifetime distributions have been used to model such kinds of data.The quality of the procedures used in a statistical analysis depends heavily on the assumed probability model or distributions.Because of this, considerable effort has been expended in the development of large classes of standard probability distributions along with relevant statistical methodologies.However, there still remain many important problems where the real data does not follow any of the classical or standard probability models.The Fréchet distribution is one of the important distributions in extreme value theory and it has been applied to data on characteristics of sea waves and wind speeds.Further information about the Fréchet distribution and its applications were discussed in Kotz and Nadarajah (2000).
Recently, some extensions of the Fréchet distribution are considered.The exponentiated Fréchet (Nadarajah and Kotz, 2003), beta Fréchet (Nadarajah and Gupta, 2004 In this article we present a new generalization of the TMOF distribution called Kumaraswamy transmuted Marshall-Olkin Fréchet (Kw-TMOF) distribution based on the family of Kumaraswamy generalized (Kw-G) distributions introduced by Cordeiro and de Castro (2011).The main motivation for this extension is that the new distribution is a highly flexible life distribution which contains as sub models sixty two well known and unknown distributions, admits different degrees of kurtosis and asymmetry and the Kumaraswamy transmuted Marshall-Olkin Fréchet (Kw-TMOF) distribution provides a superior fit to real data than its sub models and non-nested models.

Definition 1.
A random variable is said to have Kw-G distribution if its cdf is given by ( ) , ( )where and are two additional parameters whose role is to introduce skewness and to vary tail weights.The corresponding pdf is given by where ( ) and ( ) are the cdf and pdf of the baseline distribution respectively.Clearly when , we obtain the baseline distribution.
Providing a new class of distributions is always precious for statisticians.Thus, the aim of this paper is to study the Kw-TMOF distribution.The fact that the Kw-TMOF distribution generalizes existing commonly used distributions and introduces new lifetime models is an important aspect of the model.Further, we demonstrate that the proposed model provides a significant improvement compared to some existing lifetime models and it is also a competitive model to the gamma extended Fréchet (da Silva et al., 2013) and beta Fréchet (Barreto-Souza et al., 2011) distributions.In addition, we investigate some mathematical properties of the new model, discuss maximum likelihood estimation of its parameters and derive the observed information matrix.
The rest of the paper is outlined as follows.In Section 2, we demonstrate the subject distribution and the expantions for the pdf and cdf.The statistical properties include quantile functions, random number generation, moments, moment generating functions, incomplete moments, mean deviations and Rényi and -entropies are derived in Section 3. The order statistics and their moments are investigated in Section 4. The characterization of the Kw-TMOF in terms of a truncated moment of a function of the random variable is given in Section 5.In Section 6, We discuss maximum likelihood estimation of the model parameters.In Section 7, the Kw-TMOF distribution is applied to two real data sets to illustrate the potentiality of the new distribution for lifetime data modeling.Finally, we provide some concluding remarks in Section 8.

The Kw-TMOF Distribution
The Kw-TMOF distribution and its sub-models are presented in this section.A random variable is said to have Kw-TMOF with vector parameters where ( ) if its cdf is defined (for ) by ( ) { * ( ) ./ ( ) ./ + * ( ) ./ + } where are two additional shape parameters.The corresponding pdf of the Kw-TMOF is given by A physical interpretation of Equation ( 4) is possible when and are positive integers.Suppose a system is made up of independent components in series and that each component is made up of independent subcomponents in parallel.So, the system fails if any of the components fail and each component fails if all of its subcomponents fail.
If the sub-component lifetimes have a common Kw-TMOF cumulative function, then the lifetime of the entire system will follow the Kw-TMOF distribution (4).
From another view; suppose a system consists of independent sub-systems functioning independently at a given time and that each sub-system consists of independent parallel components.Suppose too that each component consists of two units.If the two units are connected in series then the overall system will have Kw-TMOF distribution with whereas if the components are parallel then the overall system will have Kw-TMOF distribution with .
Furthermore, we can interpret the system from the redundancy view.Redundancy is a common method to increase reliability in an engineering design.Barlow and Proschan (1981) indicate that, if we want to increase the reliable of a given system, then redundancy at a component level is more effective than redundancy at a system level.
That is, if all components of a system are available in duplicate, it is better to put these component pairs in parallel than it is to build two identical systems and place the systems in parallel.
The proposed Kw-TMOF model is very flexible model that approaches to different distributions when its parameters are changed.The flexibility of the Kw-TMOF is explained in Table 1 where it has sixty two sub-models when their parameters are carefully chosen.Figure 1 provides some plots of the Kw-TMOF density curves for different values of the parameters and .Some plots of the hrf of the Kw-TMOF are displayed in Figure 2.

Useful Expansions
Expansions for Equations ( 3) and ( 4) can be derived using using the series expansion The cdf of the Kw-TMOF in Equation ( 3) can be expressed in the mixture form The pdf of the Kw-TMOF in (4) can be expressed in the mixture form where The Kw-TMOF density function can be expressed as a mixture of Fréchet densities.Thus, some of its mathematical properties can be obtained directly from those properties of the Fréchet distribution.Therefore Equation ( 4) can be also expressed as where ( )denotes to the Fréchet pdf where ( ) .

Quantile Function
The quantile function (qf) of is obtained by inverting (3) as Simulating the Kw-TMOF random variable is straightforward.If is a uniform variate on the unit interval ( ) then the random variable ( ) follows (4), i.e.Kw-TMOF ( )

Ordinary and Incomplete Moments
The th moment, denoted by , of (for ) is given as by Sitting , we get the mean of .The skewness and kurtosis measures can be calculated from the ordinary moments using well-known relationships.
Corollary 1.Using the relation between the central moments and non-centeral moments, we can obtain the th central moment, denoted by of a Kw-TMOF random variable as follows where ( ) is the on-central moments of the Kw-TMOF ( ) Therefore the th central moments of the Kw-TMOF ( ) if is given by The moment generating function ( ) of say ( ) ( ) (for ) is given by The th incomplete moments, denoted by ( ) of is given by Using Equation ( 6) and the lower incomplete gamma function, if we obtain The first incomplete moment of ,denoted by, ( ) is immediately calculated from Equation (8) by setting .

Rényi and -Entropies
The Rényi entropy of represents a measure of variation of the uncertainty.The Rényi entropy is defined by Therefore, the Rényi entropy of a random variable which follows the Kw-TMOF ( ) is given by ) and where ( ) We can find the estimates of the unknown parameters by setting the score vector to zero, ( ̂) and solving them simultaneously yields the ML estimators ̂ ̂ ̂ ̂ ̂and ̂.These equations cannot be solved analytically and statistical software can be used to solve them numerically by means of iterative techniques such as the Newton-Raphson algorithm.For the five parameters Kw-TMOF distribution all the second order derivatives exist.Setting these above equations to zero and solving them simultaneously also yield the MLEs of the six parameters.
For interval estimation of the model parameters, we require the observed information matrix ( ) * +( ) given in Appendix A. Under standard regularity conditions, the multivariate normal ( ( ̂) ) distribution can be used to construct approximate confidence intervals for the model parameters.Here, ( ̂) is the total observed information matrix evaluated at ̂ Therefore, Approximate ( ) confidence intervals for and can be determined as: where is the upper th percentile of the standard normal distribution.

Data Analysis
In this section, we provide two applications of the Kw-TMOF distribution to show its importance.We now provide a data analysis in order to assess the goodness-of-fit of the new model.For the two real data sets we shall compare the fits of the Kw-TMOF model with six of its sub models: the KMOIE, KMOIR, TMOF, MOF, TF and Fréchet distributions to show the potential of the new distribution.Moreover, we shall compare the proposed distribution with two non-nested models: gamma extended

Conclusions
In this paper, We propose a new six-parameter distribution, called the Kumaraswamy transmuted Marshall-Olkin Fréchet (Kw-TMOF) distribution, which extends the transmuted Marshall-Olkin Fréchet (TMOF) distribution (Afify et al., 2015).We provide some of its mathematical and statistical properties.The Kw-TMOF density function can be expressed as a mixture of Fréchet densities.We derive explicit expressions for the ordinary and incomplete moments, Rényi and -entropies.We also obtain the density function of the order statistics and their moments.We discuss maximum likelihood

Figure 1 :
Figure 1: Plots of the Kw-TMOF density function for some parameter values.

Figure 3 :
Figure 3: The estimated pdf and cdf of the Kw-TMOF model for data set I.

Figure 4 :
Figure 4: The estimated pdf and cdf of the Kw-TMOF model for data set II.