تقدير دالة المخاطرة للتوزيع المركب (جمبرتز – الأسي) مع تطبيق عملي

Rp-Estimation of the Hazard Rate Function of a Compound distribution (Exponential-Gomper.pdf.

Abstract
In this thesis, a new probability distribution was found, which is (E-GD) Exponential-Gompertz Distribution using the (Compound Distributions method.) as a random variable that follows the exponential distribution to obtain the new probability distribution (Exponential Gompertz). Also, the statistical characteristics of the new distribution were derived, its features were estimated and the hazard function was determined using three methods of estimation, which are the (Maximum Likelihood method) (MLE),(weighted least squares method) (WLS), and (Kramer von Mess method) (CVM).In order to obtain the best results, the study made a comparison between the estimation methods by applying the (Monte Carlo simulation method) using the (Wolfram Mathematica 12.2) program.We conducted several experiments, and in this thesis, these experiments were repeated (1000) times for each experiment with different sample sizes (small, medium, large) in order to reach a better level of homogeneity. The results showed the preference of the Maximum Likelihood method in estimating the hazard function for the exponential Gompertz distribution for all sample sizes and then the weighted least squares method, while Kramer von Meese’s method was not the best for any of the sample sizes.
In order to show the efficiency of the distribution in representing a sample of real data, it was applied to a large random sample of patients, represented by the duration of their survival until death or recovery for patients with breast cancer, and by applying the best methods, it was found that the exponential Gompertz distribution fits these data better and explains their behavior compared to Gompertz distribution or the exponential distribution alone