Structure Weighted Mirra Distribution to estimate the reliability function with Practical Application

A Thesis Submitted to

Council of  The Administration and Economics/ Karbala University as Partial fulfillment of the Requirements for the Degree of Master of Science in Statistics

Presented by researcher

Ali Hussein Noori Al-Awwadee

Supervised By

Prof. Dr. Shrook Abd-Al Rida Saeed Al-Sabbah

Abstract

The study seeks to use the theory of weighted distributions in constructing a new proposed probability distribution known as the Weighted Mirra Distribution with three parameters (α, θ, c), and Some of its properties have been studied, and Estimating its parameters and calculating the reliability function in four estimation methods (Maximum Likelihood Method, Least Squares method, Weighted Least Squares method, Percentiles Estimators Method), For the purpose of comparing the estimation methods for its reliability function, the Monte Carlo simulation method was employed using the Wolfram Mathematica 12.2 program To perform several experiments with different sample sizes (small, medium and large) and By means of a statistical standard Mean Integral Error Squares (IMSE) The results showed the preference of the method of Maximum Likelihood in calculating the reliability function of the new distribution at medium and large sample sizes, and the preference of the weighted least squares method at small sample sizes.

The distribution was applied to real data with (100) observations representing the working times of pivot irrigation devices until failure, by the Goodness of fit Tests   It has been proven that the weighted distribution is superior to the representation and description of this data compared to the Mirra distribution, Also, the reliability function of the real data was estimated using the best methods that were reached on the experimental side (Maximum Likelihood Method), We found that the average working times until failure of the devices is (3.233) months, and the value of the estimated average reliability function is (0.49962), meaning that the device can be relied upon by 50% within three months and seven days approximately.