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 

Saad Sajjad Hariz 

Supervised By

Asst. Prof .Enas Abdel-Hafez Mahmoud  

Abstract

The process of expanding probability distributions is one of the important processes that has increased significantly over the past few decades. This is due to the inability of classical distributions to represent real data in a broader and more accurate way. The process of constructing distributions using derived families and classes is one of the methods used recently in expanding distributions. In this thesis, the Burr X-G Family distribution was used and applied to the one-parameter Lindley Distribution and the two-parameter Lindley Distribution to obtain new, more flexible probability models called the BurrX-Lindley OneParameter Distribution and the BurrX-Lindley TwoParameter Distribution, which are more flexible than the basic distributions under study. Some of its statistical properties were studied, such as the probability density function, the cumulative density function, the survival and risk function, and the generation function for both proposed distributions. The parameters and survival function of the Lindley TwoParameter Distribution were estimated using three estimation methods. (The maximum likelihood method, the Kramer-von-Mies method, and the weighted least squares method were used to obtain the best method for estimating the parameters and the survival function. A brief simulation study was conducted using the Monte-Carlo method. Several experiments were conducted with small, medium, and large sample sizes (150, 100, 75, and 50) and five models using different values ​​for the unknown parameters. The statistical criterion, the mean square error (MSE), was used to compare the four estimation methods for the parameter estimates. The researcher concluded that the maximum likelihood method was superior in estimating the unknown parameters and the survival function for all sample sizes.

As for the practical aspect, the new probabilistic model (BurrX-LindleyTwoParameter distribution) was applied to real data of (150) observations from the Karbala Health Department, Al-Hussein Teaching Hospital, representing the survival times of people with breast cancer until death. The performance of the proposed distribution was compared with the basic Lindley distributions under study. The new model provided more Flexibility and efficiency in representing real data and proven superior in