A NEW DISTRIBUTION FOR MODELING EXTREME VALUES

In this work, we introduce a new distribution for modeling the extreme values. Some important mathematical properties of the new model are derived. We assess the performance of the maximum likelihood method in terms of biases and mean squared errors by means of a simulation study. The new model is better than some other important competitive models in modeling the repair times data and the breaking stress data.


Introduction
The extreme value theory (EVT) is very popular in the statistical literature, it is devoted to study of stochastic series of independent and identically distributed random variables (iid RVs). In EVT, we study the behavior of EVs even though these values have a very low chance to be occur, but can turn out to have a very high impact to the observed system. Fields such as finance and insurance are the best fields of research to observe the importance of the EVT. The study of EVT started in the last century as an equivalent theory to the central limit theory (CLT), which is dedicated to the study of the asymptotic distribution of the average of a sequence of RVs. The CLT states that the sum and the mean of the RVs from an arbitrary distribution are normally distributed under the condition that the sample size ( ) is sufficiently large. However, in some other studies we are looking for the limiting distribution of maximum (max) or minimum (min) values rather than the average. Assume that 1 , 2 , . . ., is a sequence of iid RVs distributed with CDF denote ( ) . One of the most interesting statistics in a research is the sample maximum Then if ( ) is a non-degenerate distribution function then it will belong to one of the three following fundamental types of classic extreme value family, the Gumbel distribution (Type I); the Fréchet distribution (Type II); the Weibull distribution (Type III). A RV is said to have the Fréchet (Fr) distribution if its probability density function (PDF), cumulative distribution function (CDF) are given by and The more flexible version of the Fr model is the exponentiated Fréchet (EFr) distribution, with PDF and CDF are given by (for ≥ 0) and respectively, where > 0 is a scale parameter and , > 0 is a shape parameters, respectively.
The rest of this article is outlined as follows: In Section two, we introduce the genesis of the new model, Section 3 introduces a motivation and justification. A useful representation is given in Section 4. Some Mathematical properties are derived in Section 5. Section 6 shows the estimation method. Section 7 display the simulations results. Four applications are provided in Section 8. Finally, Section 9 deals with some concluding remarks.

The genesis of the new model
In this paper we will use the Transmuted Topp Leone G (TTL-G) family introduced by compiled the two families to obtain a more flexible one. We used the TTL-G to establish a new extension of the Fr model. To this end, we can write the PDF of the TTLEFr model (for ≥ 0) as the PDF corresponding to (5) is where , > 0 and | | ≤ 1. The HRF for the new model can be expressed as

Motivation and justification
Suppose " 1 and 2 " are two independent RVs with CDF (3). Define Then, the CDF of is given by (5).

Useful representations
Equation (5) can be expanded as and finally (  2 ) and ( ; , ) is the CDF of the Fr distribution with scale parameter ( ) 1 and shape parameter . The corresponding TTLEFr density function is obtained by differentiating (9) where ( +1) ( ; , ) is the PDF of the Fr model with scale parameter [ ( + 1)] 1 and shape parameter . So, the new density (6) can be expressed as a double linear mixture of the Fr density. Then, several of its structural properties can be obtained from Equation (10) and those properties of the Fr model.

Moment generating function
Here, we will introduce two methods for getting the moment generating function (MGF) of the new model. The 1 one: The MGF ( ) = ( ) of can be derived from equation (9) as The 2 method: First, we determine the generating function of (1). Setting = −1 , we can write this MGF as By expanding the first exponential and calculating the integral, we have where the gamma function is well-defined for any non-integer . Consider the Wright generalized hypergeometric function defined by Combining expressions (10) and (13), we obtain the MGF of , say ( ), as

Moments of order statistics
Let 1 , … , be a random sample from the TTLEFr distribution and let 1: , … , : be their corresponding order statistics. The PDFof i ℎ order statistic, : , can be written as where (⋅,⋅) is the beta function. Substituting (5) and (6) in equation (13) and using a power series expansion, we have The PDF of : can be expressed as Based on the last equation, we note that the properties of : follow from those of Fr density. So, the moments of : can be expressed as

Residual life and reversed residual life functions
The n ℎ moment of the residual life

Simulation studies
Using the inversion method, we simulate the TTLEFr model by taking =20, 50, 200 and 500. For each sample size, we evaluate the MLEs of the parameters using the optim function of the R software. Then, we repeat this process 1000 times and compute the biases (Bias) and mean squared errors (MSEs). Table 1 gives all simulation results. The values in Table 1 Table 2 and the MLEs and corresponding standard errors are given in Table 3. We note from the values in Table 2 that   the TTLEFr model has the lowest values of  The parameters of the above densities are all positive real numbers except for the TFr distribution for which | | ≤ 1.   Figure 3: Estimated PDF, estimated CDF, P-P plot, estimated HRF and TTT plot of the repair times data.

Application 2: Breaking stress data
The 2 set is an uncensored data set consisting of 100 observations on breaking stress of carbon fibers (in Gba) given by Nichols and Padgett (2006) and these data are used by Mahmoud and Mandouh (2013) Table 4 and the MLEs and corresponding standard errors are given in Table 5. We note from Table 4 that the TTLEFr gives the lowest values the * , * , AIC, BIC, and K-S statistics (for the 2 data set) as compared to further models, and therefore the new one can be chosen as the best one. The histogram and other related important plots of the 2 data are displayed in Figure   4.

Concluding remarks
In this work, we introduce a new distribution for modeling the extreme values. Some important mathematical properties of the new model are derived. We assess the performance of the maximum likelihood method in terms of biases and mean squared errors by means of a simulation study. The new model is better than some other important competitive models in modeling the repair times data and the breaking stress data. We hope that the new model will attract a wider application in areas such as survival and lifetime data, engineering, meteorology, hydrology, economics and others.