Estimating the Triple fuzzy distribution based on the Quantile Function
Submitted to the council of the college of Administration &Economics\ University of Karbala as partial fulfillment of the requirements for the Master degree in Statistics Sciences

By
 Shams Najy Elaiwy

Supervision
Prof.  Dr. Mahdi Wahab Nea’ama

monotonically decreasing) data usually refers to a probability distribution function or probability density function that is non-increasing, meaning that it either decreases or remains constant as you move from one point to another in the domain it is defined as. The thesis aimed to generalize the One Parameter Inverse Lindley Distribution for the purpose of expanding the basic distribution properties to fit monotonically descending data using the quantile function principle based on the T-R{Y} distribution class proposed by (Alzaatreh et al., 2014) to generalize the distributions for the purpose of finding the T-IR{Y} distribution class as well as finding a new distribution from this class considering that the distribution of the first variable T follows the inverse exponential distribution with one parameter (Inverse Exponential Distribution) and the variable R has an inverse Lindley distribution with one parameter and the variable Y has an exponential distribution with one parameter, so the resulting expanded distribution is Inverse Exponential- Inverse Lindley- Exponential under the theory of fuzzy sets by converting the resulting distribution to fuzzy based on a formula proposed by (Ali and Nima, 2022) as the resulting distribution is a fuzzy triangular distribution based on the quantile function, which is abbreviated as (FEILIE). The distribution parameters were estimated using the Maximum Likelihood and Maximum Prduct Spaces methods using Monte-Carlo simulation experiments, as well as applying it to real data to demonstrate the feasibility of the new distribution. The superiority of the (MPS) method over the Maximum Likelihood method was found. The proposed distribution was also applied to a group representing the survival times of women with breast cancer, as the estimates of the (MPS) method were inconsistent with the real values of each of the functions (probability density – clustering density – reliability), as the values estimated by this method are closest to the real values of the proposed inverse exponential – inverse Lindley – fuzzy exponential distribution. We note that when the survival time is six months and eight days, the probability of tumor reduction is (96%). When the patient’s survival time is one month and ten days, the probability of tumor reduction is (1.3%).