On A New Two Parameter Fréchet Distribution with Applications

A new flexible extension of the Fréchet model is proposed and studied. Some of its fundamental statistical properties are derived. The importance of the new model is shown via two applications to real data sets. We assess the performance of the maximum likelihood estimations of the new distribution with respect to sample size n. The assessment was based on a simulation study. The new model is much better than other important competitive models.


1.Introduction
The aim of this paper is to introduce first the generalization of the one parameter Fréchet (Fr) distribution using the one parameter Lomax-G (Lx-G) family originally introduced by Cordeiro  .
The LxFr density can be "right-skewed", whereas the LxFr HRF can be "upside" down (see Figure 1) . Hereafter, we denote by ∼ LxFr ( , , ), a RV having density function (5). The CDF (4) of can be expressed as First, we consider two power series 1 and 1 Applying (7) for ( ) in (6) gives Second, using the binomial expansion, the last equation can be expressed as Third, applying (8) for ( ) in the last equation we get where , , = (  Second, using (5) and the last equation, we have Applying (7) for ( ) in the last equation, we obtain Third, using the binomial expansion for ( ) , the last equation can be rewritten as | ( ≤ and < ) .

Residual life and reversed residual life functions
The n ℎ moment of the residual life, say Then, the n ℎ moment of the reversed residual life of is

Order statistics
Let 1 , ⋯ , be a random sample (RS) from the LxFr and let 1 : , ⋯ , : be the corresponding order statistics. The PDF of the i ℎ order statistic, say : , is given by where (•,•) is the beta function, then we can write where , , ( + −1) is as defined before. So, the PDF of : becomes using the last expression as Then, the density function of the LxFr order statistics is a linear combination of the Fr density. Based on this equation, the properties of : can be easily determined from those properties of the Fr density. Then The q ℎ ordinary moment of : say ( : ) , is determined from (12) as

3.Numerical analysis for the ( ), Var( ) , Ske( ) and Ku( ) measures
Numerical analysis for the ( ), Var( ) , Ske( ) and Ku( ) are calculated in Table 2 and 3 using the well-known relationships for some selected values of parameters using the R software. Based on Tables 2 and 3 we note that, the skewness of the LxFr distribution can range in the interval (−272.68, 61.43), whereas the skewness of the Fr distribution varies only in the interval (1.2, 3.5). Further, the spread for the LxFr kurtosis is ranging from 3.932 to 5058.65, whereas the spread for the Fr kurtosis only varies from 5.7 to 48.1 with the above parameter values.

4.Maximum likelihood estimation
be a RS from the LxFr model with parameters and . For determining the MLE, we have the following log-likelihood function The score vector is given as . Setting the nonlinear system of equations ( ) = 0 and ( ) = 0 and solving them simultaneously yields the MLE. To solve these equations, it is usually more convenient to use nonlinear optimization methods such as the quasi-Newton algorithm to numerically maximize ℓ .

5.Simulation studies
We simulate the LxFr model via taking =20, 50, 150, 500 and 1000. Then, repeating this process = 1000 times and calculate the averages of the estimates (AEs), mean squared errors (MSEs). Table 2 gives all simulation results. The numerical results in Table 1   HRFs for the proposed model for the 1 st and 2 nd data. These plots reveal that the proposed distribution yields a better fit than other nested and non-nested models for both data sets.