PROJECT TOPIC NO FOUR-PARAMETER ODD GENERALIZED EXPONENTIAL-PARETO DISTRIBUTION: ITS PROPERTIES AND APPLICATIONS
The Pareto Distribution has received some sizeable attention in the academia espe-cially 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 behaviour of the new distribution were derived. We also provide plots of the CDF, pdf, sur-vival function and hazard function for various values of distribution parameters. The plots for the pdf indicated that it is positively skewed and therefore more appropri-ate 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 endeavour comprising Biology, Physics, Actuarial Science, Geography, etc. Pareto made several important contributions to 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 popula-tion. 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 Augs-burg, 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 modelling 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 fi-asco or financial ruin (Klugman et al., 2004). Schroeder et al., (2010) utilized it in modelling 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.
FOUR-PARAMETER ODD GENERALIZED EXPONENTIAL-PARETO DISTRIBUTION: ITS PROPERTIES AND APPLICATIONS
1.2.1 Properties of Pareto Distribution From the definition provided in section (1.2), The Cumulative Distribution Function (CDF) of a Pareto random variable with parameters and k is given by:
|G(x; ; k) = 1 − ‹||•||f or ≤ x < ∞; k; > 0||(1.2.2)|
The Probability Density Function (pdf) was obtained by diﬀerentiating the CDF in equation (1.2.2) with respect to x.
dG(x; ; k)
g(x; ; k) = xk+1
- − k ‰−kx−k−1Ž = k kx−(k+1)
|f or ≤ x < ∞; k; > 0||(1.2.3)|
where > 0 is a scale parameter and k > 0 is the shape parameter.
The mean and variance of the Pareto distribution are respectively given as:
|E(X) =||k||f or||k > 1|
|and||k − 1|
|V ar(X) =||k 2||f or k > 2|
1.3 Generalized Exponential Distribution
Gupta and Kundu (1999) introduced the Generalized Exponential Distribution (GED) also known as Exponentiated Exponential Distribution (EED). It is a probability distribution having two parameters, and is very good in analyzing (especially) posi-tively skewed data because its Probability density function is positively skewed. The CDF of GED is given by:
G(x; ; ) = ‰1 − e− xŽ f or x; ; > 0
where is the shape parameter and is the scale parameter.
The corresponding pdf is:
g(x; ; ) = ‰1 − e− xŽ −1 e− x f or x; ; > 0
The survival function is:
S(x; ; ) = 1 − G(x; ; ) = 1 − ‰1 − e− xŽ
|The hazard function is;|
|h x; ;||g x; ;||1 e x −1 e||x|
|S x||;||1( −1||e x||−|
|(||) =||− (||− )|
(1.3.1) The distribution is found to be mostly applicable in analyzing and modeling life time data. Gupta et al., (2000) studied it and estimated the parameters of the GED using diﬀerent method of estimation and compare their performance through numerical simulation.
1.4 Statement of the Research Problem
Many lifetime models have been proposed or developed by diﬀerent reseachers using diﬀerent generalization procedures to analyze data for certain purposes like the flex-ibility of properties and performance of new distribution compared to the original ones, but still there are some real datasets that cannot be fitted with the already existing standard distributions. Several researchers proposed some generalizations to the existing distributions to fill in this gap, however, this served as room for us to propose a new distribution called the Odd Generalized Exponential-Pareto Distri-bution (OGEPD) using a probability distribution generator introduced by Tahir et al., (2015) known as the Odd Generalized Exponential (OGE) family of distribution.
Recently Tahir et al., (2015) introduced some special distributions that can be ob-tained from their generator which include the Odd Generalized Exponential Weibull (OGE-W), the Odd Generalized Exponential Frechet (OGE-fr) and the Odd Gen-eralized Exponential Normal (OGE-N) distributions. Also, El-Damcese (2015) used this same generator to introduce a new Odd Generalized Exponential Gompertz (OGE-G) distribution.
Tahir et al.,(2015) highlighted that the OGE family gen-erates distributions whose shapes are symmetrical, left-skewed, right-skewed and reversed-J. Also the kurtosis is more flexible. Due to this fact, the OGE family can be very essential to fit real dataset under various shapes. In view of the foregoing, we were motivated to study OGEPD based on the informa-tion available to us, that no any study was conducted on it since the introduction of the generator by Tahir et al.,(2015). More so, the available literature studied earlier highlighted that almost all generalized distributions (in which one or more parameters were added) performed well and have better presentation of data than their counterparts with less number of parameters.
1.6 Significance of the Study
The main importance of this research is to increase the flexibility of the Pareto dis-tribution, there by making it an appropriate model for modelling data of various shapes that cannot be adequately fitted with the conventional probability distribu-tions. Also the proposed distribution can be used in reliability analysis as well as in analyzing skewed datasets. Comparison between the proposed distribution and the existing distributions by using a real-life data will identify a better distribution.
1.7 Aim and Objectives of the Study
The aim of this research is to propose and study an Odd Generalized Exponential Pareto Distribution (OGEPD), its properties and as well application to real datasets. In order to achieve the stated aim, the following objectives shall be attained; by
- deriving the cumulative density function (CDF) as well as the probability density function (pdf) of the proposed distribution;
- studying some of the statistical properties of the new distribution comprising limit of the pdf, moments, moment generating function, quantile function, order statistics and reliability analysis;
- estimating the parameters of the new distribution using method of Maximum likelihoodcomparing the performance of the OGEPD with other related distributions using real datasets.
1.8 Scope of the Study
The study is focused only on extending research on Pareto distribution, by proposing a new four-parameter distribution (OGEPD) and then deriving its pdf, CDF, the asymptotic behaviour, survival function, hazard function, quantile function (which is useful for calculating the median, skewness as well as kurtosis and for generation of random number), density functions for the minimum and maximum order statistics, moment generating function. The parameters of the distribution are estimated using the method of Maximum Likelihood Estimation (MLE) procedure.
1.9.1 Probability distribution A probability distribution or model is a mathematical description that approxi-mately agrees with the frequencies or probabilities of possible events of a random variable. Probability distribution for a random variable describes how the probabil-ities are distributed over the values of a random variable.
1.9.2 Distribution Function
This is a function that gives the total probability of a random variable X which extend over all values within its ranges that are less than or equal to x. It is otherwise known as cumulative distribution function (CDF). If X is a continuous random variable with pdf f (x), then the distribution function is defined as;
|F (x) = p (X ≤ x) = S−∞ f (t) dt f or − ∞ < x < ∞||(1.9.1)|
The following properties must be satisfy for any continuous distribution function, properties (2) , (3) and (4) uniquely characterized the CDF and violating any one or more of these three properties regard the cdf invalid.
|1.||0 ≤ F (x) ≤ 1||−∞ < x < ∞|
|2.||F||′||x||dF x||f||x||0||F||x||is non-decreasing function of x|
|(||) =||( )||=||(||) ≥||⇒||(||)||−∞ f|
|3.||F||(−∞) =||xlim||F||(||x||) =||xlim||∫||x||f||(||x||)||dx||=||(||x||)||dx||=||0 and||F||(+∞) =|
|lim F||x||→−∞||x||f||∞||f||x dx||−∞|
|x||→∞||x||→∞ ∫−∞ ( )||∫|
|( ) =||= ∫−∞ ( )||=|
- F (x) is a continuous function of x on the right.
- The discontinuity of F (x) are at the most countable
- P (a ≤ X < b) = ∫ab f (x) dx = ∫−b∞ f (x) dx − ∫−a∞ f (x) dx = F (b) − F (a)
1.9.3 The Probability Density Function
The probability density function or simply density function is an opposite of distri-bution function. For any continuous random variable X, the Probability Density Function (pdf) denoted by f (x) is defined as follows;
|(||) =||(||) =||dF||(||x||)|
with the following properties
- f (x) ≥ 0 −∞ < x < ∞
- ∫−∞∞ f (x) dx = 1
- For any event A, the probability P (A)is given by
P (A) = S f (x) dx
Moment plays a vital role in statistics, most especially in applications were it can be used to derive the most important features and characteristics of a distribution such as measures of location, spread, skewness and kurtosis.For a continuous random variable X with pdf f(x), the rth moment about the origin can be defined as:
|(||) = S−∞||)|
1.9.5 Moment Generating Function
The process or technique used in obtaining or generating all the moments of a prob-ability distribution into one mathematical function is called the moment generating function. The moment generating function (mgf) of a continuous random variable
- having the pdf (f (x)) can be obtained as;
|MX||t||) =||E||‰||etX||∞ etxf||(||x||)||dx||(1.9.4)|
|(||Ž = S−∞|
where t is a real parameter and it is being assumed that the right-hand side of (1.9.4) is absolutely convergent for some positive numbers h such that −h < t < h .Thus
MX (t) = E (etX ) = E 1 + tX + t2X2 + ::: + trXr + :::
In other words, the mgf generates the moments of X by diﬀerentiation i.e., for any real number say r, the rth derivative of MX (t) evaluated at t = 0 is the rth moment ′r of X . And its only exist if the integral in equation (1.9.4) converges
1.9.6 Order Statistics
Order Statistics are used in detection of outliers, robust statistical estimation, and characterization of probability distributions, goodness of fit tests, entropy estima-tion, and analysis of censored samples, reliability analysis, quality control and strength of materials. It can also be used in determining the distribution of the smallest (minimum) order statistic X1 and largest (maximum) order statistic Xn of distribution respectively. Definition; let X1; X2; :::; Xn be n random variables from a continuous distribution with pdf f (x) and CDF F (x) . Then the pdf fi n(x) of the ith order statistic is defined as;
|n!||i−1 [1 − F (x)]n−i|
|fi n(x) = (i − 1)!(n − i)!f(x) [F (x)]||(1.9.6)|
1.9.7 Quantile Function
Quantile function is used for finding the median, skewness, kurtosis and for simula-tion of random numbers.
Definition; let Q(u) = F−1(u) be the Quantile Function (qf) of CDF F(x) for 0 < u
- By solving F (x) = u. The qf of X is given as;
|x = Q(u) = F −1(u)||(1.9.7)|
1.9.8 Scale Parameter
Define how spreads out the data are. A large scale value stretches the distribution, while a smaller scale value shrinks the distribution.
1.9.9 Shape Parameter
Define how data are distributed but does not aﬀect the location or scale of a distri-bution. A large shape value gives a left-skewed curve, whereas a small shape values gives a right-skewed curve.
1.9.10 Reliability Analysis
Survival or reliability analysis is the process of modelling time-to-event information, also known as transition data (or survival time data or duration data). It is also seen as a statistical techniques used to describe and quantify time-to-event data. This analysis was originally developed or introduced for the purpose of evaluating the treatment eﬃcacy of fatal condition like cancer. But also used in many other situation such as time for hand fracture to heal, excessive breast feeding and time to another pregnancy, time to exercise to maximum tolerance, time to breakdown for a machine, survival of patient after surgery, length of stay in hospital and so on. In survival analysis, we are interested in the time interval between entry into the study and an event. The outcome of interest is time-to-event. It has applicability in many areas of human endeavor including engineering, medicine, sciences, industry and etc.
1.9.11 Survival Time
This shows the length of time taken for failure to occur. The survival curve can be used to study times required to reach any well-defined endpoint.
1.9.12 Survival Function
Survival function S(t) is defined as the proportion of the population that has sur-vived to time t or the probability that a system will survive beyond a given time. However, S(t) can be plotted as a function of time to produce a survival curve, and the area S(t) under the curve to the right of time t will indicate the proportion of individuals in the population who have survived to time t. At time t equals to zero there will be no failure so S(t) will be equals to one. Mathematically, the survival function is defined as:
|S(t) = 1 − F (t)||(1.9.8)|
where F (t) is the failure function (CDF) indicating the cumulative proportion of the population that has died up to time t
1.9.13 Hazard Function
Hazard function h(t) is also known as conditional failure rate or instantaneous haz-ard and is defined as the instantaneous rate at which a randomly-selected individual known to be alive at time (t-1 ) will die at time t. In order word it is the probability that a system or individual will fail or die for an interval of time. Its gives the proportion of the population present at time t that fails per unit time. The hazard function is defined mathematically as;
where f (t) and F (t) are the instantaneous failure rate (pdf) and failure function (CDF), respectively of any baseline distribution when the variable under consider-ation is the length of time taken for an event to occur e.g. death
1.9.14 Maximum Likelihood Estimation
The method of maximum likelihood estimation was introduced by R.A. Fisher in 1922. It is the method that allows us to estimate the parameter that maximizes the likelihood function (joint probability density function). Definition; Let X1; X2; :::; Xn be n random variables from a population with sample values x1; x2; :::; xn having joint probability density function as f (x1; x2; :::; xn; ) where is an unknown parameter. Then the likelihood functionL ( ), of the random samples is defined as:
|L ( ; x1||; x2||; :::; xn) = f (x1||; x2||n||f(xi; )||(1.9.10)|
|; :::; xn; ) = L ( ) = i 1|
The sample statistic that maximizes the likelihood function L ( ) is called the max-imum likelihood estimator of and is denoted by Â.