The Extended Fréchet Distribution: Properties and Applications

In this paper, we study a new model called the Burr X exponentiated Frechet Distribution. The new model exhibits unimodal, unimodal then buthtab and buthtab hazard rates. Various properties of the new model are explored including moments, generating function, probability weighted moments, Stress-strength model and order statisics. The maximum likelihood method is used to estimate the model parameters. Simulation results to assess the performance of the maximum likelihood estimates are discussed. We compare the flexibility of the proposed model with other extensions of the Frechet distribution by means of two real data sets.


Introduction
The Fréchet distribution (Fréchet, 1942), also known as type II extreme value distribution, has many applications in extreme value theory as an important distribution in extreme value theory.The Fréchet distribution has applications in stochastic phenomena such as rainfall, floods, air pollution (see, Kotz and Nadarajah, 2000). Further, Harlow (2002) and Zaharim et al. (2009) applied Fréchet distribution in engineering applications and in analyzing wind speed data, respectively. To explore more informations about applications of Fréchet distribution see, e.g. Kotz and Nadarajah (2000) and Resnick (2013).
The survival function (SF) and probability density function (PDF) of the Fréchet (Fr) distribution are given (for > 0) by where ( ; , ) = 1 − ( ; , ) and > 0 is a scale parameter and > 0 is a shape parameter.
Our aim in this paper is to propose and study another extension of the Fréchet model called the Marshall-Olkin generalized Fréchet (MOGFr) distribution. Its main feature is that two additional positive shape parameters are inserted in Equation (1) to provide great flexibility for the generated model. Based on the Marshall-Olkin generalized-G (MOG-G) family (Yousof et al., 2018), we construct the fourparameter MOGFr distribution and provide some of its mathematical properties. We prove that the MOGFr distribution is capable of modelling various shapes of data using two data sets. It can provide better fits than other nested and non-nested model in two real data applications.
The remainder of this article is outlined as follows: In Section2, we define the MOGFr distribution and provide its special cases. We derive some mathematical properties of the MOGFr distribution including linear representation for its PDF, quantile and generating functions, ordinary and incomplete moments, mean residual life, mean waiting time and order statistics in Section 3. The maximum likelihood estimation method is discussed in Section 4. In Section 5, the MOGFr distribution is applied to two real data sets to illustrate its importance. Finally, in Section 6, we give some concluding remarks.

The MOGFr distribution
Consider the SF of a baseline model, ( ; ) = 1 − ( ; ), with a parameter vector , then the cumulative distribution function (CDF) MOG-G family is defined by where are two positive shape parameters representing the different patterns of the MOG-G family. The corresponding PDF and hazard rate function (HRF) of (2) are , ∈ ℝ, where ( ; ) is the PDF of a baseline model, ( ; ) = ( ; )/ ( , ) is the baseline HRF, and are positive shape parameters. The random variable with PDF (3) is denoted by ∼MOG-G( , , ). For = 1, we have the Marshall-Olkin-G family (Marshall and Olkin, 1997), for = 1, the MOG-G family reduces to the generalized-G family (Gupta et al., 1998), and for = = 1, we obtain the baseline distribution.
Combining (1) and (2), we obtain the CDF of the MOGFr distribution The PDF amd HRF of the MOGFr distribution are, respectively, given by  Table 1.

Properties of the MOGFr distribution
In this section, we derive some mathematical properties of the MOGFr distribution including linear representation for its PDF, ordinary and incomplete moments, mean residual life, mean waiting time, quantile and moment generating functions and order statistics.

Yousof et al. (2018) derived a useful linear representation of the CDF of the MOG-G family as
, where 0 = (2/ ) and for ≥ 1, we have Then, the PDF of the MOG-G family can also be expressed as where ℎ +1 ( ) denotes the exp-G density with positive power parameter . Hence, the MOGFr density can be rewritten as Then, the PDF of the MOGFr reduces to (6) where ( +1) ( ) is the Fr PDF with shape parameter and scale parameter ( + 1) 1/ . Equation (6) reveals that many properties of the MOGFr distribution can be derived from the Fr properties.
Let be a random variable with Fr distribution (1) with parameters > 0 and > 0. The th ordinary and incomplete moments of are given (for < ) by iis the lower incomplete gamma function.

Ordinary and incomplete moments
The th ordinary moment of is given by The mean of follows by setting = 1 in (7).
In Table 2 we provide numerical values for the mean, variance, skewness and kurtosis of the MOGFr distribution, for some selected parameter values of , and with = 1, to illustrate their effects on these measures. Table 2 shows that, for fixed and , the mean, variance, skewness and kurtosis are decreasing functions of . For fixed and , the mean and variance are increasing functions of , whereas the skewness and kurtosis are decreasing functions of . Further, for fixed and , the mean, variance, skewness and kurtosis are decreasing functions of . One can see, from Table 2, that the MOGFr distribution can be left skewed or right skewed. Further, it can be platykurtic (kurtosis < 3) or leptokurtic (kurtosis > 3). Hence, the MOGFr model is a flexible distribution and can be used in modeling skewed data. The th incomplete moment of the MOGFr distribution follows using Equation (6) as Setting = 1 in Equation (8), we obtain the first incomplete moment of which has important applications related to the mean residual life, mean waiting time, Bonferroni and Lorenz curves.

Mean residual lifetime and mean waiting time
The mean residual life (MRL) (or life expectancy at age ) represents the expected additional life length for a unit, which is alive at age t and it is defined by ( ) = ( − | > ), > 0. The MRL of , can be defined as ( + 1) 1/ (1 − 1 , ( + 1) ( ) ).

Quantile and generating functions
The quantile function (QF) of is obtained by inverting (4) as The median of follows by setting = 0.5 in (12). The MOGFr random variable can be Simulated if is a uniform variate on the unit interval (0,1), then the random variable = at = ollows Equation (5).
The moment generating function (MGF) of follows from (6)  Using the exponential series for exp( / ) and after some simplifications, we have The Wright generalized hypergeometric function is defined by The MGF of the MOGFr distribution follows, by combining Equations (6) and (13), as

Order statistics
Let 1 , … , be a random sample of size n from the MOGFr distribution and let 1: , … , : be the corresponding order statistics. Then, the PDF of the of th order statistic, : , is defined by Then, the PDF of the th order statistic of the MOGFr distribution reduces to Hence, the PDF of the first order statistic 1: follows from (14) with = 1, as The PDF of the largest order statistic : is given by

Maximum likelihood estimation
The estimation of the MOGFr parameters from complete samples only is considered by the maximum likelihood method. Let 1 , … , be a random sample of the MOGFr distribution with parameter vector = ( , , , ) ⊺ The log-likelihood function for is ℓ = log + log + log + log − ( The maximum likelihood estimators (MLEs) can be obtained by maximizing (15) either by using the different programs such as R, SAS or by solving the nonlinear likelihood equations obtained by differentiating (15). The score vector elements, (Θ) = ℓ = ( ℓ , ℓ , ℓ , ℓ ) ⊺ , are .

Two applications
In this section, we ilustrate the flexibility and importance of the MOGFr distribution empirically by two real data applications. The first data set contains 101 observations with maximum stress per cycle 31,000 psi. The data refer to the fatigue life of 6061-T6 aluminum coupons (Birnbaum and Saunders, 1969). The second data set consists of 128 observations of bladder cancer patients which represents the remission times (in months) (Lee and Wang, 2003). Table 3 lists the competitive models of the MOGFr distribution which will be compared with it.  (2017) We shall consider the minus log-likelihood ( −l ), Kolmogorov Smirnov ( ) statistic, its P-value ( ), Cramér-von Mises ( * ) and Anderson-Darling ( * ) statistics to compare the fitted distributions. Tables 4 and 5 list the values of the MLEs and their corresponding standard errors (in parentheses) of the MOGFr parameters and other fitted models parameters. These tables also show the values −l, , , * and * statistics for both data sets. In Tables 4 and 5, we compare the MOGFr model with the TEFr, WFr, KFr, MOFr, BXFr, EFr, MFr and Fr distributions. We note that the MOGFr model gives the lowest values for the −l, , * and * statistics and the largest value of the among all fitted models. Hence, the MOGFr model could be chosen as the best model to explain both data sets.  Estimates −l * * Figure 5 shows the TTT plots of the MOGFR distribution for both data sets. The TTT plot for fatigue life data is concave which indicates that it has an increasing hazard rate, whereas the TTT plot for the cancer data is concave then convex which indicates an upside down bathtub hazard rate. Hence, the MOGFr distribution is a suitable for modeling both data sets.

Conclusions
In this paper, we propose a new four-parameter model called the MarshallOlkin generalized Fréchet (MOGFr) distribution, which contains the Fréchet, Marshall-Olkin Fréchet and exponentiated Fréchet distributions, among others as special cases. The MOGFr density function can be expressed as a linear mixture of Fréchet densities. Explicit expressions for some of its mathematical quantities including the quantile and generating functions ordinary and incomplete moments, mean residual life, mean waiting time and order statistics are derived. The MOGFr parameters are estimated by the maximum likelihood method. The proposed distribution provides better fits than some other nested and non-nested models by using two real data sets.