Generalization of Some Statistical Distributions Using Kumaraswamy Distribution

Abstract

The modeling and analysis of lifetime is an important aspect of statistical work in a wide variety of scientific and technological fields. The failure behavior of any system can be considered as a random variable due to the variations from one system to another resulting from the nature of the system. Therefore, it seems logical to find a statistical model for the failure of the system. In other applications, survival data are categorized by their hazard rate, For example, there are distributions with fixed hazard rate such as the exponential distribution. Other distributions are characterized by incremental hazard rate. Some have decreasing failure rate, and others combine the three kinds on different time periods to appear in the the form of bathtub. The Kumaraswamy distribution is similar in its simplest form to the beta distribution in terms of probability density function and cumulative distribution function. This distribution has an advantage to the beta distribution, because it is simpler to use especially in simulation studies. The Kumaraswamy distribution is applicable to many natural phenomena , such as the heights of individuals, scores obtained on a test, atmospheric temperatures, hydrological data and landslides. It can be a useful tool to analyze customer lifetime duration in marketing research and can be used quite effectively in analyzing real data. This distribution is in use in electrical, civil, mechanical, and financial engineering applications. The Kumaraswamy distribution arises depending on order statistics, and its form clearly does not depend on special functions, thus have a distinct role in ease of statistical modeling, it also has a tendency for application in educational uses. In recent years, generalized distributions have been widely studied in statistics as they possess flexibility in applications. This is justified because the traditional distributions often do not provide good fit in relation to real data set studied. In this thesis, according to Kumaraswamy distribution we propose a new class of generalized distributions called the Exponentialed Kumaraswmay Lindley (EKumL) that is capable of modeling bathtub-shaped hazard function. The beauty and importance of this distribution lies in its ability to model monotone and non-monotone failure rate function, which are quite common in lifetime data analysis and reliability. The thesis consists of three chapters as follows: