Cubic Transformation Burr XII Distribution 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

Muntadher Jumaah Mahdi

Supervised By

Ass. Prof. Dr. Mushtaq Kareem Abd Al-Rahem

Abstract

The Burr XII distribution with two shape parameters (α , β) is one of the important continuous statistical distributions. This distribution gained real importance in the last three decades due to the importance of its use in practical cases، and this distribution was applied in the study of reliability، failure time display، quality control, acceptance modeling (Acceptance of the sample) in cases where the normal distribution is an inappropriate model. Based on the above, a new generalization studied for the Burr XII model, Named “the New Transformed Cubic Burr XII Distribution,” “NCTBXII”. The proposed model is a generalization of the Burr XII distribution. By adding two additional parameters to its distribution function. This new distribution is not founded in any previous research work.

The thesis studied its statistical and mathematical properties and estimated the parameters of the new distribution and survival function using five methods of estimation, namely (Maximum likelihood method, Least squares method, Weighted Least squares method, Fractional Estimates method and Minimum distance method using Cramer-Von Mises) through a detailed simulation study using the Monte Carlo simulation method, where different values ​​of distribution parameters were selected and 8 different cases were formed, as well as 4 different sample sizes (30,60,80,100), The estimators of these methods were compared based on the mean squares error criterion and according to sample sizes using the Ranks method. This work was performed by Mathematica 12.2 software packages. Finally, we used two real data models, one for a local sample size (107), which represents the survival times for patients infected with Covid-19 virus, and the other for a global sample size (76) represented by the survival times of the epoxy plate exposed to pressure until failure or breakage through applying the results Extracted from the experimental side in order to show that the proposed distribution is a suitable model for modeling these types of data more than the distributions of (Burr XII, Transmuted Burr XII, Cubic Transmuted Burr XII).