ON FOUR-PARAMETER ODD GENERALIZED EXPONENTIAL PARETO DISTRIBUTION: ITS PROPERTIES AND APPLICATIONS
The Pareto Distribution has received some size-able attention in the academia especially by adding some elements of flexibility to it through introduction of one or more parameters using generalization approaches. It plays a vital role in analyzing skewed dataset especially in reliability analysis. In this research, we proposed and study a four-parameter Odd Generalized Exponential-Pareto Distribution (OGEPD). Some statistical properties comprising moments, moment generating function, quantile function, reliability analysis, distribution of order statistics and limiting behavior of the new distribution were derived. We also provide plots of the CDF, pdf, survival function and hazard function for various values of distribution parameters. The plots for the pdf indicated that it is positively skewed and therefore more appropriate for fitting positively skewed datasets. The parameters of the distribution were estimated using method of maximum likelihood. Finally, the proposed distribution is applied to two real datasets to illustrate its fit as compared to other distributions.
1.1 Background of the study
The Pareto distribution is a widely known distribution in applied sciences as well as in Economics. It was introduced in order to explain the distribution of income in the society (Pareto, 1896). It was first proposed by a Professor of Economics, Vilfredo Pareto (1843-1923). The distribution was found while studying various distributions for modeling income in Switzerland. The various forms of the Pareto distribution are versatile and can usually be used to model uncertainties. Since that time its applicability spans diverse areas of human endeavor comprising Biology, Physics, Actuarial Science, Geography, etc. Pareto made several important contributions to the Economics, mostly in the study of income distribution and in the analysis of individuals choices.
Pareto found out that income approximately follows a Pareto distribution, which is considered as power law probability distribution. The Pareto principle was named after him and noted that 80% of the land in Italy was owned by 20% of the population. One of Pareto’s equations attained special importance and argument. He was captivated by problems of power and wealth. How do people get it? How is it spread around society? How do those who have it use it? The gap between rich and poor has always been part of the human condition, but Pareto resolved to measure it. He collected piles of data on wealth and income through different centuries, across different countries: the tax records of Basel, Switzerland, from 1454 and from Augsburg, Germany, in 1471, 1498 and 1512; contemporary rental income from Paris; personal income from Britain, Prussia, Saxony, Ireland, Italy, and Peru. What he discovered or thought he discovered was striking. When he plotted the data on a graph sheet, with income on one axis, and number of people with that income on the other, he observed similar scenario nearly everywhere in every era. Society was not a “social pyramid” with the percentage of rich to poor sloping gently from one class to the next. Instead it was more of a “social arrow” the bottom was very fat indicating where the mass of men live, and at the top was very thin indicating where the wealthy elite reside. Nor was this effect by chance; the data did not remotely fit a bell curve, as one would anticipate if wealth were randomly distributed. “It is a social law”, he wrote: something “in the nature of man”. At the bottom of the Wealth curve, he wrote, Men and Women starve and children die young. This reason makes Pareto to develop model for distribution of wealth.
The Pareto distribution was used to model prevalence of earthquakes, forest fire areas, oil and gas field sizes (Burroughs and Tebbens, 2001), as well as in online analytical processing (OLAP) by (Nadeau and Teorey, 2003) purposely to obtain meaningful information easily from large amount of data residing in a data ware-house. The Pareto distribution is a combination of exponential distribution with gamma mixing weights, some properties of the Pareto distribution shows that the distribution is a heavy tailed distribution. In insurance application, heavy tailed distribution are important tools for modeling extreme loss, principally for the more risky types of insurance like medical insurance. In financial applications, the study of heavy tailed distributions offers information about the potential for financial fiasco or financial ruin (Klugman et al., 2004). Schroeder et al., (2010) utilized it in modeling risk drive sector errors. Ever since, it plays a vital role in analyzing and dealing with skewed dataset as well as in reliability analysis.
The Pareto distribution has received more attention in the sense that many authors studied and added some elements of flexibility to it, by introducing one or more parameters to the distribution using some generalization approaches.
g(x; ; k) =
1.2 Pareto Distribution
If X is a random variable that follows a Pareto distribution, then the probability that X is larger than some number x, that is the survival function (also called tailed function) is given by:
G (x; ; k) = ‹ • f or ≤ x < ∞; k; > 0 (1.2.1)
where is the (necessarily positive) minimum possible value of x and k is a positive parameter. The Pareto distribution is characterized by a scale parameter and a shape parameter k, which is also called the tail index. When this distribution is used to model the distribution of wealth, then the parameter k is called the Pareto index.