Fuzzy Bayesian Estimation of the(Topp – Leone Kumaraswamy) distribution With Application

A Thesis
Submitted to the council of the College of Administration & Economics – University of Karbala as a partial fulfillment of  the  Requirements for  the degree of Philosophy of Doctorate in statistics science

By
Laith Ali Mohammed Ai-Hayali

Supervised
Prof .Dr. Abdul hussian. H.Habeab.Altai

ABSTRACT :-The process of mixing distributions is one of the vital and important processes that contribute to increasing the flexibility and efficiency of the basic distributions. The thesis was based on a recent idea, which is to build a new probability distribution function using the function generating the distributions (Topp – Leone), by placing the single probability distribution (Coomaraswamy) in the base The generated function and then deriving the statistical characteristics of the new resulting distribution (Kumaraswamy- Topp Leone), which is characterized by efficiency and flexibility in representing the data, as the greater the parameters of the distribution, the greater the accuracy, reliability, and preference for this distribution. In the long term, the focus is on reliability. Failure times were studied, which It is often a mixture of randomness and fuzziness, and this is expressed in fuzzy numbers, which in turn leads us to estimate the fuzzy reliability function (FuzzyReliability) under certain ranges of belonging to the fuzzy group.

The Monte Carlo simulation method was used using the MATLAB language in order to evaluate the performance of parameter estimators and the reliability function for all estimation methods using different sample sizes, large (n=70, n=100), medium (n=40), small (n=20), and when Different cutoff parameter levels (τ-Cut=0.2,0.4,0.6,0.8) and different values of the parameters at each sample size. The parameters of the proposed distribution and the reliability function for the experimental data were estimated using several estimation methods, including standard Bayesian information methods under different loss functions, including: Ordinary methods such as the maximum likelihood method, including algorithms such as the gray wolf optimization algorithm and the genetic algorithm. By applying the (MSE) measure, the superiority of the standard Bayesian information method was reached under a weighted quadratic linear loss function (〖BWLS〗_10) and for all cutting levels. As for the side Applied: The TLKUM distribution was applied practically to real data representing the failure time (malfunctions) affecting the engines, which were obtained from the Ministry of Electricity/Diesel Power Station – Al-Jadriya, which numbered (70) engines, in order to estimate the expected future failure times. The study concluded The proposed distribution achieved an advantage over the basic distribution through the comparison criteria used in the study.

Fuzzy Bayesian Estimation of the(Topp – Leone Kumaraswamy) distribution With Application

A Thesis
Submitted to the council of the College of Administration & Economics – University of Karbala as a partial fulfillment of  the  Requirements for  the degree of Philosophy of Doctorate in statistics science

By
Laith Ali Mohammed Ai-Hayali

Supervised
Prof .Dr. Abdul hussian. H.Habeab.Altai

ABSTRACT :-The process of mixing distributions is one of the vital and important processes that contribute to increasing the flexibility and efficiency of the basic distributions. The thesis was based on a recent idea, which is to build a new probability distribution function using the function generating the distributions (Topp – Leone), by placing the single probability distribution (Coomaraswamy) in the base The generated function and then deriving the statistical characteristics of the new resulting distribution (Kumaraswamy- Topp Leone), which is characterized by efficiency and flexibility in representing the data, as the greater the parameters of the distribution, the greater the accuracy, reliability, and preference for this distribution. In the long term, the focus is on reliability. Failure times were studied, which It is often a mixture of randomness and fuzziness, and this is expressed in fuzzy numbers, which in turn leads us to estimate the fuzzy reliability function (FuzzyReliability) under certain ranges of belonging to the fuzzy group.

The Monte Carlo simulation method was used using the MATLAB language in order to evaluate the performance of parameter estimators and the reliability function for all estimation methods using different sample sizes, large (n=70, n=100), medium (n=40), small (n=20), and when Different cutoff parameter levels (τ-Cut=0.2,0.4,0.6,0.8) and different values of the parameters at each sample size. The parameters of the proposed distribution and the reliability function for the experimental data were estimated using several estimation methods, including standard Bayesian information methods under different loss functions, including: Ordinary methods such as the maximum likelihood method, including algorithms such as the gray wolf optimization algorithm and the genetic algorithm. By applying the (MSE) measure, the superiority of the standard Bayesian information method was reached under a weighted quadratic linear loss function (〖BWLS〗_10) and for all cutting levels. As for the side Applied: The TLKUM distribution was applied practically to real data representing the failure time (malfunctions) affecting the engines, which were obtained from the Ministry of Electricity/Diesel Power Station – Al-Jadriya, which numbered (70) engines, in order to estimate the expected future failure times. The study concluded The proposed distribution achieved an advantage over the basic distribution through the comparison criteria used in the study.