Estimating parameters Gumbel Pareto Distribution

The proposed method for generating a new distribution, depends on the Cumulative Distribution Function (CDF) of two distributions, namely, Gumbel distribution and pareto distribution.We obtain a new compound, which is called (Gumble – Pareto distribution GPD). In this research we work on deriving the formula for the new distribution, and all other additional properties as well as introducing different methods to estimate the four parameters (μ, θ , α , B ) By different Method like Maximum likelihood, and method of Moments estimator and also, we derive Percentiles estimator and least squares and also weighted least square. Then the Comparison is done through simulation.


Introduction
The Gumbel Pareto is well known as probability distribution for its ability to model different types of data.and also has many applications in risk analysis and quality.Many papers on the distribution of extremes appeared in the literature.Gumbel (1958)   ] … (7) In this article, derived some properties of GPD such as moments and additional properties including the mean deviations and modality are studied.
uch as moments and additional properties including the mean deviations and modality are studied.

Estimation of parameters
For the four parameters of GPD defined in equation ( 2), which are two shape parameters (α,θ) and two scale parameters (B,  * ), these four parameters are estimated using different methods maximum likelihood method, and moments method.also we derived to obtain Central Moment and apply numerical method on the statistical measure like mean, variance.

Percentiles Estimation (PE)
Kao in (1959) initially discovered this method via the graphical approximation to the best linear unbiased estimators.The estimators might be found by fitting a straight line to the theoretical points determined from the distribution function, and the sample percentile points.In the case of a GP distribution, it is probable to use the same idea to determine the estimators of α, B, and  * based on PE, because of the structure of its distribution function.Since G(x) distinct in (1).
First of all, we find numerically the value of x where x= G -1 (x, α, B,  * ), since Pi is the estimate of G (X(i) , α , B ,  * ) . is the most used estimator of G(X(i)).

Least squares Estimation (LSE) and Weighted least squares estimators (WLSE)
This method was initially proposed by Swain et al in 1988, to estimate the parameters of Beta distribution.However, suppose x1, x2, ….xn is a r.s. of size n with distribution function G(x), uses the distribution of G(x(i)).For a sample of size (n) we have in [5].We achieved extensive simulations to contrast the performances of the various methods, which were stated in section (3), mainly with respect to their MSE for various sample sizes, and for various parameters values.

E(G(x(i)) =
gave detailed results on extremes value theory in his book statistics of Extremes.Furthermore, Gumbel has been agree with Johnson et al. (1995), as the first to bring attention to the possibility of using the Gumbel distribution to model extreme value of random data.Kotz and Nadarajah (2000), and Beirlant et al, (2006).Alzaatrch, Lee and Famoye (2013) proposed a method for generating new distribution, Al-Aqtash et al (2014) proposed some properties of the Gumbel -Weibull distribution the mean deviations and modes are studied.Tahir et al (2015) proposed the introduction of a new four-parameter model named the Gumbel -Lomax distribution, stand up from the Gumbel -x generator recently which was proposed by Al-Aqtash (2013).

Table ( 1
) Empirical MSE to Estimate parameters of GP Distribution with Different Sample Size Methods of Estimation and Different Values of parameters α, B, µ

Table 1 :
Empirical MSE to Estimate parameters of GP Distribution with Different Sample Size Methods of Estimation and Different Values of parameters α, B, µ