Statistical Properties & Different Methods Of Estimation Of A New Extended Weighted Frechet Distribution

: In this paper, we introduce a new distribution is called the extended weighted Frechet distribution, which we obtain by applying the Azzalini method and deduced some statistical properties such as mean, variance, coefficients of variation, coefficient of skewness, and coefficient of kurtosis. The parameters of the new distribution were estimated by the following estimation methods: Maximum Likelihood Method (MLE) and percentile method. We used the Monte Carlo simulation to compare the performances of the proposed estimators obtained through methods of estimation.


Azalini's Method
Mathematician Azzalini, A. (1985) introduced this method by inserting an additional parameter into the normal distribution to obtain a new distribution called the skew-normal distribution to achieve more flexibility in the normal distribution function as it is an extension of it. After that, many researchers inserted the shape parameter into non-symmetric distributions such as the T-skew, Skew-Cauchy by (Gupta et al., 2002), and Skew-logistic distribution by (Nadarajah 2009).
from eq. (5) we found the cumulative function to the new distribution by integration: By (1) in the (7), we get: The r th moment about the origin of extended weighted Frechet distribution is: At r=1, we obtain the mean :

3.2: Variance
The Variance can be obtained by finding The r th moment about the mean when the value of r=2: E(x − μ) r = ∫ (x − μ) r f(x)dx ∞ 0 … (9) By (1) in the (9), we get: The r th moment about the mean of extended weighted Frechet distribution is: At r=2, we obtain the variance:

3.3: Coefficients of Variation
The mathematical formula for the coefficient of variation is:

3.4: Coefficient of Skewness
The mathematical formula for the Coefficient of Skewness is:

3.5: Coefficient of Kurtosis
The mathematical formula for the Coefficient of Kurtosis is:

Estimation 4.1: Maximum Likelihood Method (MLE)
If x1,x2...xn are random variables distributed in the extended weighted Freight distribution, then: To estimate the parameter (): To estimate the parameter (): The MLE ,̂ ̂ Can be obtained by solving the likelihood eqs.

4.2: Maximum product of spacing estimator method
The MLE ,̂ ̂ can be obtained by solving the likelihood eqs. Clearly, it is difficult to solve the equations (13) , (14), and (15) therefore applying Newton-Raphson's method.

4.3:Method of Cramer-Von Mises Minimum
The CVME, as a type of minimum distance estimator, has less bias than the other minimum distance estimators The MLE ,̂ ̂ can be obtained by solving the likelihood eqs. Clearly, it is difficult to solve the equations () , (), therefore applying Newton-Raphson's method.

Simulation Study
We ran a simulation to compare the behavior of the estimates with respect to mean squares of error (MSEs) using the rank method, as this method is based on selecting the lowest order for the sum of the total and partial ranks of all estimation methods for a set of imposed parameter values (,̂,̂). The results were as shown in Table (1).