Estimation of Fuzzy Reliability Function for Alpha Power Kappa Distribution with Practical Application

A letter submitted to the Council of the College of Administration and Economics at the University of Karbala

It is part of the requirements for a master’s degree in Science of Statistics

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Maryam Sadeq Kadhum Al-Naqqash

Supervision

Prof. Dr. Shrooq Abdul Redha Al-Sabbah

Mr. Dr. Sada Fayedh Mohammed

Abstract

The process of constructing, mixing, composing, and transforming probability statistical distributions is one of the important mathematical operations whose use has increased significantly and significantly over the recent decades. The reason for this is that the probability distributions resulting from these operations are characterized as probability distributions with high ability and great flexibility in representing real data for the various phenomena desired. Study it more broadly and precisely, and the process of transforming distributions is one of these methods recently used in expanding and transforming classical probability distributions, especially the distributions used in reliability functions and failure functions In this thesis, the Alpha Power Transformed (APT) formula was used to build a new probability model called the Alpha Power Transformed Kappa distribution (APTk), as it is an expansion and transformation of the original Kappa distribution (APT). Kappa distribution), the goal of this transformation is to obtain a new probability distribution that is believed to be highly flexible in representing real data, by adding a new shape parameter to the base distribution, We extracted some of the basic mathematical properties of the distribution and estimated its parameters and fuzzy reliability function using three different estimation methods: (the Maximum Possibility Method, the Cramer-Von Misses Minimum Method, and the Partial Estimators Method). The simulation experiment was conducted using the (Mont- Carlo) to evaluate the performance of the estimators for the new distribution using the above-mentioned approved methods. The simulation was conducted with several experiments and with different sample sizes, small, medium, and large. Using the ranks method and relying on the values ​​of the statistical criterion, the mean square error (MSE), a comparison was made between the three estimation methods, and the maximum likelihood method proved superior to the rest of the methods in estimating the parameters of the proposed distribution and its fuzzy reliability function at all sample sizes and at the cutoffs (0.1) and (0.2). The new probability distribution (APTk) was applied to real data of (100) observations representing the working hours until the downtime of the communications towers (Internet), and the process of fuzzing was conducted for this data with cutoffs (0.1), (0.2), and (0.3), and using the criteria (BIC) and (AIC) and (AICc) were compared between the kappa distribution and the APTk distribution. The new distribution gave high flexibility and efficiency in representing real data and proved superior to the original kappa distribution. The fuzzy reliability function for the APTk distribution was estimated using the maximum potential method. Which outperformed other methods in estimating the reliability function for medium and large sample sizes. Among the practical results of the letter was that it gave the telecommunications company that it could rely on the towers at a reasonable rate.