On Maximum Likelihood Estimates for the Shape Parameter of the Generalized Pareto Distribution

The general Pareto distribution (GPD) has been widely used a lot in the extreme value for example to model exceedance over a threshold. Feature of The GPD that when applied to real data sets depends substantially and clearly on the parameter estimation process. Mostly the estimation is preferred by maximum likelihood because have a consistent estimator with lowest bias and variance. The objective of the present study is to develop efficient estimation methods for the maximum likelihood estimator for the shape parameter or extreme value index. Which based on the numerical methods for maximizing the log-likelihood by introduce an algorithm for computing maximum likelihood estimate of The GPD parameters. Finally, a numerical examples are given to illustrate the obtained results, they are carried out to investigate the behavior of the method.


Introduction
The GPD are the only "stable" ones, i.e. the only ones for which the conditional distribution of an exceedance is a scale transformation of the original distribution. The GPD distribution has many good properties, (for more details see, for instance, Emberchts, Klüppelberg and Mikosch, 1997, Section 3.4, [4], and Reiss and Thomas, Section 1.4, [11]).
The maximum likelihood estimate of the general Pareto distribution (GPD) parameters are the values which maximize the likelihood function which is defined as below formula (6).
The asymptotic behavior properties of maximum likelihood estimator of the GPD parameter have been studied in many articles including the important works of Davison [2] and R.L Smith [12], the maximum likelihood estimators have a consistent estimator of the variance and he used it to replace the asymptotic variance of unknown parameters. The maximum likelihood estimates must be derived numerically for the GPD because there is no obvious simplification of the nonlinear likelihood equation as we defined in (8).
From a statistical perspective, the threshold is loosely defined such that the population tail can be well approximated by an extreme value model (e.g., the generalized Pareto distribution), and can be used the GPD maximum likelihood estimates for estimating extreme value index.
There is many numerical techniques for computes the GPD maximum likelihood estimates for the shape parameters ∈ γ have been proposed in many articles including the important works like Hosking and Wallis [10] for the parameter space to ( ) ( ) 1/ 2 1/ 2 < < − γ and Grimshaw [7] for extreme value index, the shape parameter 1 ≤ γ , and we propose an algorithm, to estimate the extreme value index, the shape parameter 1 ≥ − γ . For this we introduce the approach maximum likelihood of the extreme value index. Then, in section (2) we gives the numerical techniques for computes the GPD maximum likelihood estimates. And in section (3) we will be given a numerical example is based on two real data found in the literature, he is in order to illustrate the problem with estimation of the shape parameter ∈ γ in small samples by the maximum likelihood estimator.
Let 1 2 , , , n X X X L be a sequence of independent and identically distribution (i. i. d For an i.i.d. sample of size n , we denote the ascending order statistics by 1 The work of Fisher and Tippett (1928, [5]), Gnedenko (1943, [6]) and de Haan (1970, [8]) answered the question on the possible limits and characterized the classes of distribution functions F having a certain limit in (2).
This convergence result is our main assumption. Up to location and scale, the possible limiting dfs ( ) given by the so-called extreme value distributions G γ , defined by: Then it is well known [see Balkema and de Haan (1974, [1]) and Pickands (1975, [14])] that up to scale and location transformations with the shape parameter the generalized Pareto d.f. given by n k n n k n n k n k n n n k n where in the asymptotic setting where Proposition 1: If the random variable X has a generalized Pareto distribution, then the conditional distribution of X t − given X t ≥ is also generalized Pareto, with the same shape parameters γ .
The log-likelihood is given by The range for σ is 0 > σ for 0 > γ and : If no local maximum is found, then there is no GPD maximum likelihood estimate and the alternative estimators given by Hosking and Wallis (1987, [10]) are recommended.
To obtain a finite maximum of the GPD log-likelihood, the constraint 1 ≥ − γ must be imposed. Therefore, computing the GPD maximum likelihood estimates is an optimization on the constrained space There are two values of ( ) , γ σ that must be investigated to compute the GPD maximum likelihood estimate. The first is the local maximum of the log-likelihood on the space A .
The second is at the boundary of A , where 1 = − γ . The likelihood equations from (7) are then given in terms of the partial derivatives have been studies in many articles including the important works of Hann and Ferreira [9]; p 91 and Drees et al [3]. The resulting likelihood equations in terms of the excesses ( ) 2. Compute γ by ( ) Now, we go to found the solution of which is on (9) so must be computed numerically on the space .
The following theorem states several properties of that are useful in formulating an algorithm for determining her zeros with p is a natural number, required for modified Bisection algorithm to search the multi-roots of ( ) k ψ θ ; which we present in section (2) which rely with the numerical method that proposed by Tanakan, (2013, [13]), and best of them as well. For this in section (3) we use a two real data utilized the first collected from the exceedance of the threshold given by Grimshaw [7], and the second we use model extreme value by the real Danish fire data.

Modified Bisection Algorithms
A modified bisection algorithms is much more efficient than the bisection method. Furthermore, it is faster than the Newton method, and don not count the derivative of a function at the reference point, which is not always easy. In the practice, the initial solution is really important for the Newton method. But some initial solutions can make the method Newton diverges. Hence, by the intermediate value A modified bisection algorithms can reduced the number of iterations which less than the iteration number of the bisection method and nearby to the iteration number of Newton method, for numerical results see Tanakan ([13], section [3]).
In this work took the error are less than the tolerance, which is linked to condition Step 5: Compute,   (13) and for Modified bisection agrees Tanakan [13] we have also We can show that: And it's easy to prove the theorem (2) by using a mathematical induction. The result (13) and result (14) proved that the sequence respectively.

An Algorithm for the GPD Maximum Likelihood Estimates
In the data sets used investigating the GPD maximum likelihood estimates, it appears that there exists either no zero or two zeros on each interval. For this we preferred an algorithm that computes the GPD maximum likelihood estimates for search the zero of the function given on (9) by modified bisection algorithm. The algorithm that computes the GPD maximum likelihood estimates is given by the following: 1. Choose an ε , for example let

Numerical Examples
In the first example, we use the tensile-strength fiber data presented by Grimshaw (1993, [7]  We go to the interval ; U     ε θ . By the modified Bisection algorithm as we set in section (2), we will find that he converged to two roots are   . For a real second example application we use model extreme value. Let us consider the real Danish fire data, these data describe large fire insurance claims in Denmark from Thursday 3rd January 1980 until Monday 31st December 1990. A numeric vector containing 2167 observations. They offer many possibilities for modeling and have been used by many researchers to illustrate their methods, see McNeil (1997) and Resnick (1997). These data can be found in evir package of the R software (Ihaka and Gentelman cite R).
In this study, we are concerned with confidence bounds for the 73 monthly maximum losses during the mentioned from the given 2167 observations, listed in an increasing order.