The Exponentiated Marshall-Olkin Fréchet Distribution

A new five-parameter model called the exponentiated Marshall-Olkin Fréchet distribution is studied. Various of its mathematical properties including ordinary and incomplete moments, quantile and generating functions and order statistics are investigated. The proposed density function can be expressed as a linear mixture of Fréchet densities. The maximum likelihood method is used to estimate the model parameters. The flexibility of the new distribution is proved empirically using two real data sets.


Introduction
The Fréchet distribution, also known as type II extreme value distribution, is one of the important distributions in extreme value theory and it has wide applicability in extreme value theory.This distribution was proposed by Maurice Fréchet (1924), who investigated it as one possible limit distribution for a sequence of maxima.The Fréchet distribution is widely used in applications involving stochastic phenomena such as rainfall, floods, air pollution (Kotz and Nadarajah, 2000), material properties in engineering applications (Harlow, 2002), analyzing wind speed data (Zaharim et al., 2009) and advanced mathematical results on point processes and regularly varying functions (Resnick, 2013), among others.Further details about the Fréchet distribution and its applications can be explored in Kotz and Nadarajah (2000).
The statistical literature contains many extended forms of the Fréchet distribution.For example, Nadarajah and Kotz (2003) pioneered the exponentiated Fréchet, Nadarajah and Gupta (2004)  In this article, we define and study a new five-parameter model called the exponentiated Marshall-Olkin Fréchet (EMOFr) distribution and provide some of its properties.We prove, by means of two applications, that the EMOFr distributions can give better fits than most of the above mentioned distributions.
The new model is generated by applying the exponentiated Marshal-Olkin-G (EMO-G) family (Dias et al., 2016) The probability density function (pdf) of the EMO-G is given by where  > 0,  > 0 and  ∈ (−∞, 1) are shape parameters.
The rest of this chapter is organized as follows.In Section 2, we define the EMOFr distribution, provide its special cases and some plots for its pdf and hazard rate function (hrf).In Section 3, we provide a useful mixture representation for its pdf.In Section 4, we derive some of its mathematical properties.Maximum likelihood estimation of the model parameters is addressed in Section 5.In Section 6, we provide two applications to real data to illustrate the importance and flexibility of the new distribution.

The EMOFr Distribution
In this section, we define the EMOFr distribution.The new model generated by applying the exponentiated Marshall-Olkin transformation to the Fréchet distribution.
The cdf of the Fréchet distribution is given by (for  > 0) The corresponding pdf is given by where  > 0 is a scale parameter and  > 0 is a shape parameter, respectively.Now, we proceed to define the new EMOFr distribution.By inserting the cdf of the Fréchet distribution in equation ( 1), we obtain the cdf of the EMOFr The corresponding pdf of the EMOFr is given by where  > 0,  > 0,  ∈ (−∞, 1) and  > 0 are shape parameters and  > 0 is a scale parameter.Henceforth, we denote by ~EMOFr(, , , , ) a random variable having pdf (4).
The hrf of  is given by  The EMOFr is a flexible model which contains 15 special models are listed in Table 1.

Linear Representation
In this section, we derive a useful linear mixture representation for the cdf and pdf of the EMOFr distribution.The cdf of the EMOFr in (3) can be expressed as    Applying the binomial expansion defined by Then, the cdf of the EMOFr reduces to Using the generalized binomial expansion defined by we can write Then, equation ( 6) reduces to .
Using (7), we have Then, the cdf of the EMOFr reduces to where and (;  1/ , ) is the cdf of the Fréchet distribution with scale parameter  1/ and shape parameter .
By integrating equation ( 8), we obtain where (;  1/ , ) is the Fréchet density with scale parameter  1/ and shape parameter .Equation ( 9) reveals that the EMOFr density can be written as a linear combination of Fréchet densities.So, several of its mathematical properties can be obtained from those of the Fréchet distribution and equation ( 9).

Mathematical Properties
In this section, we derive some mathematical properties of the EMOFr distribution including ordinary and incomplete moments, quantile and generating functions and order statistics.
Let  be a random variable having the Fréchet distribution defined in Section 2. For  < , the th ordinary and incomplete moments of  are given by

Ordinary and Incomplete Moments
The th ordinary moment of  is given by (;  1/ , ).
For  < , we obtain The mean of  follows from the last equation with  = 1, that is, The skewness and kurtosis measures can be evaluated from the ordinary moments using well-known relationships.
Then, the th incomplete moment of the EMOFr distribution is given (for  < ) by The first incomplete moment, denoted by  1 (), is and it has an important applications related to the Bonferroni and Lorenz curves and the mean residual life (MRL) and the mean inactivity time (MIT).The Bonferroni and Lorenz curves are very useful in economics, demography, insurance, engineering and medicine.The MRL and MIT are defined by  1 () = [1 −  1 ()]/() −  and  1 () =  −  1 ()/(), respectively.

Quantile and Generating Functions
The quantile function of  is determined by inverting (3) as Simulating the EMOFr random variable is straightforward.If  is a uniform variate in the unit interval (0,1), the random variable  = () follows (4).Then, we obtain After some algebra, we can write Using the Wright generalized hypergeometric function defined by Thus, the mgf of the Fréchet distribution reduces to Based on equations ( 9) and ( 10), the mgf of , say (), is given by

Estimation
In this section, we consider the estimation of the unknown parameters of the EMOFr from complete samples only by maximum likelihood.We investigate the MLEs of the parameters of the EMOFr(, , , , ) model.Let  = ( 1 , … ,   ) be a random sample from this model with unknown parameter vector  = (, , , , )  .
The log-likelihood function for , say ℓ = ℓ(), is given by We can obtain the estimates of the unknown parameters by setting the score vector to zero, ( ̂) = .By solving these equations simultaneously gives the MLEs  ̂,  ̂,  ̂,  ̂ and .These estimates can be obtained numerically using iterative techniques such as the Newton-Raphson algorithm.For the EMOFr distribution, all the second-order derivatives exist.
For interval estimation of the model parameters, we require the 5 × 5 observed information matrix () = {  } for ,  = , , , ,  Under standard regularity conditions, the multivariate normal  5 (0, ( ̂)−1 ) distribution can be used to construct approximate confidence intervals for the model parameters.Here, ( ̂) is the total observed information matrix evaluated at  ̂.Then, approximate 100(1 − )% confidence intervals for the model parameters can be determined in the usual way of the first-order asymptotic theory.

Applications
This section is devoted to illustrate the importance of the EMOFr distribution empirically using two applications to real data sets.The first data set refers to the survival times, in weeks, of 33 patients suffering from acute Myelogenous Leukemia (Feigl and Zelen, 1965).The second data set represents the exceedances of flood peaks (in m 3 /s) of the Wheaton River near Carcross in Yukon Territory, Canada.
The fitted distributions are compared using the following criteria: the −2ℓ ̂ (Maximized Log-Likelihood), AIC (Akaike Information Criterion), CAIC (Consistent Akaike Information Criterion), BIC (Bayesian Information Criterion) and HQIC (Hannan-Quinn Information Criterion).Tables 2 and 4 provide the numerical values of goodness-of-fit statistics or the fitted models, whereas the values of the MLEs and their corresponding standard errors (in parentheses) of the model parameters are listed in Tables 3 and 5, respectively.
The plots of fitted densities for the EMOFr model and other models, for both data sets, are displayed in Figures 3 and 6.Figures 4 and 7 display the QQ plots for both data sets, respectively.The estimated cdfs, for both data sets, of the competitive models are given in Figures 5 and 8.
to the Fréchet distribution.Dias et al. (2016) defined the EMO-G family of continuous distributions with three extra shape parameters by cumulative distribution function (cdf)

.
Plots of the EMOFr pdf for some parameter values are displayed in Figure1.
Figure 2 displays some possible shapes of the hrf of the EMOFr model for selected parameter values.

Figure 1 :
Figure 1: Plots of the EMOFr density function for selected parameter values

Figure 3 :Figure 5 :
Figure 3: The fitted pdfs of the EMOFr model and other models for Leukemia data

Figure 6 :
Figure 6: The fitted pdfs of the EMOFr model and other models for Wheaton River data

Figure 7 :Figure 8 :
Figure 7: Q-Q plots for Wheaton River data

Table 1 : Special models of the EMOFr model
1/ and  and