The four-parameter Fréchet distribution: Properties and applications

In this article, we propose a new four-parameter Fréchet distribution called the odd Lomax Fréchet distribution. The new model can be expressed as a linear mixture of Fréchet densities. We provide some of its mathematical properties. The estimation of the model parameters is performed by the maximum likelihood method. We illustrate the good performance of the maximum likelihood estimates via a detailed numerical simulation study. The importance and usefulness of the proposed distribution for modeling data are illustrated using two real data applications.


Introduction
The Fréchet distribution (Fréchet, 1924) is one of the important distributions in extreme value theory and it has many applications in accelerated life testing, rainfall, earthquakes, floods, horse racing, wind speeds and sea waves. For more details about the Fréchet distribution and its applications see, e.g., Kotz and Nadarajah (2000) and Mubarak (2011). (1) The corresponding probability density function (PDF) of (1) is where > 0 is a scale parameter and > 0 is a shape parameter.
where and are positive shape parameter.
The OLx-G PDF is .
The rest of this article is organized as follows. In Section 2, we define the OLxF distribution and provide some plots for its PDF and hazard rate function (HRF). Section 3 is devoted to the derivation of a useful linear representation for the OLxF PDF. We derive some mathematical properties of the OLxF distribution including ordinary and incomplete moments, moment generating function and order statistics in Section 4. The maximum likelihood estimates of the OLxF parameters are provided in Section 5 and a numerical simulation study is conducted to assess the performance these estimates. The flexibility of the OLxF distribution is proved empirically by two real data applications in Section 6. We provide some conclusions in Section 7.

The OLxF distribution
In this section, we define the OLxF distribution. By inserting the CDF (1) into equation (3), we obtain the CDF of OLxF distribution (for > 0) The PDF corresponding to (5) where > 0 is a scale parameter and > 0, > 0 and > 0 are shape parameters representing the different patterns of the OLxF distribution. Henceforth, a random variable having PDF (6) is denoted by ~OLxF( , , , ).
The HRF of is given by The quantile function (QF) of the OLxF model follows as where ~ (0, 1) is a uniform random number. Figure 1 shows some plots of the OLxF density for selected values of , , and . The plots of the PDF indicate that the OLxF distribution can be reversed-J shaped, left-skewed or right-skewed. The plots of the HRF of the OLxF model are given in Figure 2 which reveals that the HRF can be increasing, decreasing or unimodal.

Linear representation
In this section, we provide a useful expansion for the OLxF PDF in terms of Fréchet densities.
Using the following two power series given by Cordeiro et al. (2019) provided a useful representation of the PDF of the OLx-G family as where , = (−1) and ℎ + +1 ( ) = ( + + 1) ( ) ( ) + denotes the exponentiated-G (Exp-G) class density with power parameter + + 1 > 0. Then, the PDF of the OLx-G class can be expressed as a double linear combination of Exp-G densities.
Using the CDF and PDF in (1) and (2) in Equation (8) Hence, the PDF of the OLxF model can be expressed as and + +1 ( ) is the PDF of the Fréchet distribution with shape parameter and scale parameter ( + + 1) 1/ .
Equation (9) reveals that the OLxF density can be written as a linear combination of Fréchet densities. Then, several of its properties can be obtained from those of the Fréchet distribution and Equation (9).
Let be a random variable having the Fréchet distribution (1) with parameters and . For < , the th ordinary and incomplete moments of are given by

The properties of the OLxF distribution
In this section, we derive some mathematical properties of the OLxF distribution including ordinary and incomplete moments, moment generating function and order statistics.

Ordinary and incomplete moments
The four-parameter Fréchet distribution: Properties and applications 253 The th ordinary moment of is given by For < , we obtain ′ = ∑ ∞ , =0 , Using Equation (10), we have the mean of with = 1.
The skewness and kurtosis measures can be evaluated from the ordinary moments using well-known relationships.
The th incomplete moment of the OLxF distribution is defined by Using Equation (9), we can write Then, we have (for < ), The first incomplete moment, 1 ( ), follows from the above equation with By setting = −1 , we can write Using the exponential series for the first exponential Calculating the integral, we obtain

Consider the Wright generalized hypergeometric function defined by
Hence, we can write ( ; , ) as Using Equations (9) and (11), the MGF of , ( ), reduces to

Order statistics
Let 1 , … , be a random sample of size from the OLxF distribution and (1) , … , ( ) be the corresponding order statistics. Then, the PDF of the th order statistic : , : ( ), is defined by where Taking ( ) and ( ) in Equation (12) to be the CDF and PDF of the Fréchet distribution, then Equation (12) can be rewritten as The last equation reduces to where ( + +1) ( ) denotes the density function of the Fréchet model with shape parameters and scale parameter ( + + 1) 1/ . Hence, the PDF of the OLxF order statistics is a linear mixture of Fréchet PDFs. Based on Equation (13), we can easily derive the th moment of : is given (for < ) by ,s ( + + 1) Γ (1 − ).

Estimation and simulation
In this section, we provide the estimation of the OLxF parameters from complete samples only by maximum likelihood estimation method. We investigate the MLEs of the parameters of the OLxF( , , , ) model. Let 1 , … , be a random sample from the OLxF model with parameter vector = ( , , , ) .
The log-likelihood function for , ℓ = ℓ( ), is We can obtain the estimates of the unknown parameters by setting the score vector to zero, (̂) = . By solving these equations simultaneously gives the maximum likelihood estimates ̂,̂ and ̂. These estimates can be obtained numerically using iterative techniques such as the Newton-Raphson algorithm.
Now we study the performance and behavior of maximum likelihood estimators (MLEs) of OLxF parameters by generating 10,000 samples of the OLxF distribution using the QF in Equation (7)

Application of OLF model
In this section, we provide two applications of the OLxF distribution using two real data sets. The first data set refers to the exceedances of flood peaks (in m 3 /s) of the Wheaton River near Carcross in Yukon Territory, Canada. The data consist of 72 exceedances for the years 1958-1984, rounded to one decimal place (Choulakian and Stephens, 2001).     The second data set represents the remission times (in months) of 128 bladder cancer patients (Lee and Wang, 2003). The parameters of the above densities are all positive real numbers except for the TEF distribution for which | | ≤ 1.
We compare the fitted distributions using the following goodness-of-fit measures namely, the minus maximized loglikelihood (−l), Cramér-Von Mises ( * ), Anderson-Darling ( * ) statistics, Kolmogorov-Smirnov (KS) statistic and its p-value (PV). Tables 6 and 7 provide the values of −l, * , * KS, PV, the MLEs and their corresponding standard errors (in parentheses) for both data sets respectively. The plots of the fitted OLxF PDF, CDF, survival function (SS) and probability probability (PP) plots for both data sets are displayed in Figures 3 and 4, respectively.
Its noted, from Tables 1 and 2, that the OLxF model has the lowest values for the −l, * , * and KS statistics and largest value for the PV among all fitted models. Then, the OLxF model can be chosen as the best model for Wheaton river data. The HRF plots of the OLxF distribution for the two data sets are shown in Figure 5. One can see that the HRF of the OLxF model is unimodal for both data sets.

Conclusions
In this paper, we propose and study a new extension of the Fréchet model called the odd Lomax Fréchet (OLxF) distribution, which extends the Fréchet distribution. The OLxF PDF can be expressed as a linear mixture of Fréchet densities. We derive explicit expressions for its ordinary and incomplete moments, generating function and order statistics. The model parameters are estimated by maximum likelihood. Further, we conduct a numerical Monte Carlo simulation study which illustrates that the maximum likelihood approach performs very well in estimating the OLxF parameters. The proposed model provides better fits than some other well-known competitive models using two real data applications.