Comparison between Modified Weighted Pareto Distribution and Many other Distributions

distribution, one of which is the method of Maximum Likelihood estimator method used in this paper.. Nagatsuka et al 4 2021, they presented a new development of the inference for the generalized Pareto distribution operating on all values of the k-Abstract In 2020 one of the researchers in this paper, in his first research, tried to find out the Modified Weighted Pareto Distribution of Type I by using the Azzalini method for weighted distributions, which contain three parameters, two of them for scale while the third for shape.This research compared the distribution with two other distributions from the same family; the Standard Pareto Distribution of Type I and the Generalized Pareto Distribution by using the Maximum likelihood estimator which was derived by the researchers for Modified Weighted Pareto Distribution of Type I, then the Mont Carlo method was used–that is one of the simulation manners for generating random samples data in different sizes ( n= 10,30,50), and in different initial values for each Pareto distribution family being used in the research. A comparison was done by using Akaike Information Criteria, Corrected Akaike Information Criteria


Introduction
The aim of this research boils down to three goals, the first goal is to find out and derive what was carried out by the researchers about the Maximum likelihood estimator (MLE) to estimate the three parameters for Modified Weighted Pareto Distribution of Type I (MWPDTI).The second goal is to compare the new distribution with two other distributions of the same family by using Akaike Information Criteria (AIC),(AICc), and (BIC).While the third goal is to generate data by using the Mont Carlo method in order to generate random samples of different sizes; with different initial values to compare the same-Pareto family distribution.
Azzalini 1 , found a method for computing the probability density function for weighted distributions according to Eq.1.
() =  7 , used the Akaike information criterion to study some biological models to obtain the best model in terms of estimating highly accurate parameters.
Pham MH. et al 8 , estimated the parameters of the Generalized Pareto Distribution (GPD) and they tested the goodness of fit.Cavanaugh JE.et al 9 , they reviewed the Akaike criterion, its properties, its derivation, and its use in predicting the estimation of the parameters of the statistical model 9 .
Sahmran MA. 10 , used the Azzalini method to find a modified and weighted distribution of a Pareto type I (MWPDTI) according to Eq.2.
The survival function and the hazard functions are equations respectively And the other properties of the MWPDTI are mentioned in 10 .

Maximum Likelihood Estimation Method (MLE):
The maximum likelihood estimator is used to find the estimated values for the parameters of MWPDTI.

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The Three Criteria (AIC, AICc and BIC): The following statistical criteria are used (AIC, AICc and BIC) to compare the MWPDTI with two Pareto distributions (SPDTI, GPD).Where m is the number of parameters in the distribution,  ̂ is the maximum log-likelihood function value for the distribution and n represents the sample size.

The Simulation Technique:
The Monte Carlo method is the most popular simulation technique used to generate the observations (samples) for any distribution.The simulation method is flexible for the test of experiments by replications many times.

Applying the Monte Carlo method for Modified
Weighted Pareto distribution Type I (MWPDTI), using the cumulative distribution function as follows: By substituting () as , with a random number the  = (), that is  =  −1 ()     = √ 36   any Figures and images that are not ours have been included with the necessary permission for re-publication, which is attached to the manuscript.
-Ethical Clearance: The project was approved by the local ethical committee at the University of Baghdad.

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( 2 < 1 ) ( 1 )( 1 ) 1 2023, 20(3 Suppl.):1108-1115 https://dx.doi.org/10.21123/bsj.2023.8169P-ISSN: 2078-8665 -E-ISSN: 2411-7986 Baghdad Science Journal shape parameter, in which they extracted new properties that lead to obtaining new confidence intervals.Omekam et al 5 , they used the method of introducing new parameters on the distribution of Pareto of the first type, resulting in a new distribution, and a more flexible distribution with new types of data.Pho KH et al 6 , in studying the problem of traffic accidents, using the comparison between Akaike's criterion and Bayes' criterion to get the best estimate of the parameters of the problem model.Portet S.