On The Exponential Power Distribution
Mathematical Theory and ModelingISSN 2224-5804 (Paper)ISSN 2225-0522 (Online)Vol.8, No.8, 2018www.iiste.orgOn the Exponential Power DistributionBismark Kwao Nkansah1, Michael Manford2*, Nathaniel Haward11. Department of Statistics, University of Cape Coast, Cape Coast, Ghana2. Directorate of Academic Planning and Quality Assurance, Cape Coast Technical University, P. O. BoxAD50, Cape Coast, Ghana* E-mail of the corresponding author: mykemanford@yahoo.comAbstractThe paper examines the nature of the exponential power distribution (EPD) in terms of its location, Β΅, scale, Ξ²and shape, Ο, parameters. It establishes conditions under which the distribution is legitimate and reliable. Itderives among others the moment and kurtosis of the distribution as well as the maximum likelihood estimatorsof the parameters. It then uses data on health to assess the departure of the distribution from normality. Threemain softwares are used, namely; EasyFit, MATLAB and Minitab.In the application, we find that the EPD, for some values of π½, significantly fits Weight, Height and Body MassIndex out of seven variables covered. We deduce that the EPD would be inappropriate for fitting asymmetricaldatasets, since the variables which are not significant are found to be highly skewed.Keywords: Exponential power distribution, Kurtosis, Legitimacy, Statistical Distributions1. IntroductionThe defining characteristics of statistical distributions are their dependence on parameters and the incorporationof stochastic terms. The properties of the distributions and the properties of quantities derived from them arestudied in a long-run, average sense through expectations, variances, skewness and kurtosis. The fact that theparameters of the distribution are estimated from the data introduces a stochastic element in applying a statisticaldistribution. This is because the distribution is not deterministic but includes randomness. Parameters and relatedquantities derived from the distribution are likewise random.A statistical distribution of a variable is an approximate representation of its population distribution which maybe parametric or non-parametric. A theoretical parametric distribution generally provides a simple parsimonious(and usually smooth) representation of the population distribution. It can be used for inference of the centiles (orquantiles) and moments of the population distribution and other population measures. Inference of the momentscan be particularly sensitive to misspecification of the theoretical distribution and especially to misspecificationof the heaviness of the tail(s) of the population distribution (Forbes, Evans, Hastings & Peacock, 2011).Over the last two decades, many researchers have developed interest in the construction of flexible parametricclasses of statistical distributions that are more flexible than the normal distribution. Many practical applicationsrequire models of data exhibiting a skewed or peaked distributions, and some researchers suggest the use ofdistributions which are more robust for such data. Some of these applications are in areas that include health,environmental and finance. There are several parametric classes of distributions to choose from. Rigby,Stasinopoulos, Heller and Bastiani (2017) have reviewed many of them. Subbotin (1923) introduced a class ofdistribution called the exponential power distribution (EPD) which is believed to be more flexible than thenormal distribution in terms of kurtosis.Subbotin in his study on the Law of Frequency of Error formulated an axiom which states that the probability ofa random error π depends only on the absolute value of the error itself and can be expressed by a functionπ(π) having continuous first derivative almost everywhere. Based on this axiom, Subbotin obtained a densityfunction called Subbotinβs family of distributions given byπ(π) πβππ₯π{ βπ π π },12Ξ ( )π(1)with π , β 0 and π 1. This class of distributions is said to be symmetric, but with variationin kurtosis. It was noted that this distribution has many structural properties close to the normal distribution.There is also a link in the axiom considered by Subbotin and that of Gauss, as Gauss used similar axiom to121
Mathematical Theory and ModelingISSN 2224-5804 (Paper)ISSN 2225-0522 (Online)Vol.8, No.8, 2018www.iiste.orgderive the usual normal distributionπ(π₯) 1π 2π1 π¦ π 2ππ₯π { (2π) },(2)with two parameters; π , (mean or location parameter), and π 0 (standard deviation or scaleparameter). Several researchers (Coin, 2017; Giller, 2005; Nadarajah, 2005; PogΓ‘ny & Nadarajah, 2009; Tahir,Cordeiro, Alizadeh, Mansoor, Zubair & Hamedani, 2015) have introduced various classes of distribution relatingto the Subbotinβs family of distributions. Some studies have used the name the Generalized GaussianDistribution, Generalized Normal Distribution or Generalized Error Distribution to refer to the ExponentialPower Distribution.Giller (2005) in his study expressed the EPD as12π 1 π 11 π¦ π ππ(π¦ π, π, π) ππ₯π { },Ξ(π 1)2 π(3)Giller stated that if π 1/2, then π(π¦ π, π, π) π(π, π 2 ) (Normal) and if π 1, thenπ(π¦ π, π, π) πΏ(π, 4π 2 ) (Double Exponential or Laplace). In the limit as π 0, π(π¦ π, π, π) π(π π, π π)(Uniform).PogΓ‘ny and Nadarajah (2009) modified the EPD and expressed it as1ππ 1π¦ π ππ(π¦ π, π, π) ππ₯π { }.1π2Ξ ( )π(4)This distribution has three parameters given as π, π and π which represent the location, scale (or dispersion)and shape of the distribution, respectively. They further noted that π(π¦ π, π, 1) is Laplace (or DoubleExponential) and π(π¦ π, π, 2) π(π, π 2 /2 ) . Also, the pointwise ππππ π(π¦ π, π, π) coincides with thedensity function of uniform distribution, π(π π, π π).Mineo and Ruggieri (2005) expressed their EPD asπ(π¦ π, π, π) π11 π¦ πππ₯π { },1π ππ12qq Οπ Ξ (1 )π(5)Mineo and Ruggieri explained that the parameter q determines the shape of the curve; in this way, it is linked tothe thickness of the tails, and thus to the kurtosis, of the distribution. In fact, by changing the parameter q, theEPD describes both leptokurtic (0 π 2) and platikurtic (π 2) distributions.Purczyriski and Bednarz-Okrzyriska (2014) adopted a class of EPDs of the formπ(π¦) ππππ₯π{ π¦ π π },12Ξ ( )π(6)where Ξ(1/π) is an Eulerβs gamma function. For π 1, the EPD turns into the Laplace distribution(bi-exponential), and for π 2, a Normal distribution is obtained given π 1/π 2.More generally, the EPD can be deduced asπ(π¦ π, π, π) ππ 1 ππ₯π { π π¦ π π },π(7)for πππ π¦ , π , π 0 and π 0. This distribution function is characterized by alocation parameter π, a scale parameter π, and a shape parameter π, where π is the normalizing constant and πis a constant which may depend on π. The Normal distribution is obtained from this distribution when π 2,whereas heavier (or lighter) tail distributions are produced for π 2 (ππ π 2). In particular, we obtain thedouble exponential distribution for q 1 and the uniform distribution for π .122
Mathematical Theory and ModelingISSN 2224-5804 (Paper)ISSN 2225-0522 (Online)Vol.8, No.8, 2018www.iiste.orgIn Equation (7), if we considerπ 2,1 Ξ²(8)(Elsalloukh, 2010), then the EPD is given as2π(π¦ π, π, π½) ππ 1π¦ π 1 π½ππ₯π { π }.π(9)This will be the basis for the study. The rationale for the expression for q is intended to enable us track thechanges in the characteristics of the distribution as a result of the values of π½. Also, the constant c will be chosento ensure that it does not affect the variability or the scale parameter of the distribution. This would ensure theflexibility of the tails whether heavy or thinner tails. If π½ 0, the distribution becomes a Normal distribution; ifπ½ 1, the distribution becomes a Laplace distribution, but if π½ 1, then the distribution turns to arectangular or uniform distribution.Figure 1: Exponential Distribution for some values of π½Figure 1 presents a typical example of EPD for various values of the parameter, π½. In Figure 1, it can beobserved that for a small value of π½ (e.g., π½ 0.99), the EPD has a flat top, whiles for large values of π½(e.g., π½ 1 and π½ 2) the EPD has a pencil-like top. Thus, for a decreasing values of π½, the EPD approachesuniform or a rectangular distribution.Many researchers have adopted different forms of EPD, by adopting different values of the constant π inEquation (9) which affect the scale parameter of the distribution as well as the normalizing constant π. For manyof the research, the choice of the constant π depends on the shape parameter π. Vianelli (1963) developedEPDs with the constant π deduced as π 1/π. Vianelli called the distribution βA Normal Distribution ofOrder πβ. Rahnamaei, Nematollahi and Farnoosh, (2012) adopted the distribution proposed by Vianelli in theirdata modelling. Giller (2005) adopted a case whereby π 1/2 which does not depend on π. Olosunde (2013)argued that there are limitation on using such family of EPDs, explaining that EPD exhibits thinner tails and careneeds to be taken to ensure that the tails are not affected by the choice of c. Due to Olosundeβs interest inanalysing data from heavy-tailed distribution, he adopted EPD with π as a constant function, π(π), which heexplained to regulate the tail region of the distribution. This study will adopt the approach of Olosunde toestimate the constant π, but will ensure that it is estimated to make the variance of the EPD the same as the scaleparameter, π.Although, most research have addressed the characteristics of the EPDs of different kinds, information is rarelyprovided to address the reliability of the density function of their adopted distribution. Thus, the study willexamine the flexibility and the characteristics of EPD of various kinds and address the issue of its legitimacy andreliability. In the process, we will derive and estimate the parameters of the distribution with respect to a datasetand examine its fitness.123
Mathematical Theory and ModelingISSN 2224-5804 (Paper)ISSN 2225-0522 (Online)Vol.8, No.8, 2018www.iiste.org2. Methodology2.1 The Gamma DistributionThe gamma function is very important in mathematical statistics. It is a continuous extension to the factorialfunction, which is only defined for the non-negative integers. The gamma function (or gamma integral) is givenby Ξ(s) y s 1 e y ππ¦ ,s 0,(10)0or sometimes 2Ξ(π ) 2 π¦ 2π 1 π π¦ ππ¦ ,π 0.(11)0Also, the function Ξ(s, t) π¦ s 1 e y ππ¦,(12)π‘for all s 0and y 0 is the incomplete gamma function. Now, if s 1, then Ξ(π ) (π 1)Ξ(π 1). Forany non-negative integer, the logarithmic derivative of π(π ) is the psi or digamma function denoted π(π )and given asπΞβ² (π )π(π ) (ln(Ξ(π ))) ,(13)ππ Ξ(π )and expressed as π π¦π π π¦(14)π(π ) ( ) ππ¦ .π¦1 π π¦0While there are other continuous extensions to the factorial function, the gamma function is the only one that isconvex for positive real numbers. Figure 2, presents a typical gamma function in the plane.Figure 2: Gamma function [Ξ(π )] in the whole complex planeFrom Figure 2, the function is defined for non-negative values of s, but undefined (discontinues) for somenegative values of s. The gamma function however plays a major role in the characteristics and properties ofEPD, as we will demonstrate in following Sections.2.2 The Legitimacy of the Exponential Power DistributionLet Y be a continuous random variable. The function π(π¦) is said to be a proper or legitimate probabilitydensity function (pdf) of the continuous variable Y if π(π¦) is positive for all values of y within βπ¦ , that isπ(π¦) 0 for π¦ β (non-negativity) and if β π(π¦)ππ¦ 1.(15)π¦124
Mathematical Theory and ModelingISSN 2224-5804 (Paper)ISSN 2225-0522 (Online)Vol.8, No.8, 2018www.iiste.orgThus, the EPD is a proper pdf of the continuous variable Y if π(π¦ π, π, π½) is for all values of y within βπ¦ andif 2 π(π¦ π, π, π½) ππ¦ ππ 1 π¦ π 1 π½ππ₯π { π } ππ¦ 1.π(16)Also, π(π¦ π, π, π½) 0 for all π¦ in the real line β not in βπ¦ . We note that probabilities are given by areasunder π(π¦ π, π, π½) asππ(π π π) π(π¦ π, π, π½)ππ¦ .(17)ππWe also note the peculiarity that π(π π) π π(π¦ π, π, π½)ππ¦ 0, for any arbitrary value π. This can becircumvented by defining the probability on a small interval (π π¦, π π¦) around π, where π¦ has a smallvalue. Thenπ π¦π(π (π π¦, π π¦)) π(π¦ π, π, π½)ππ¦ ,(18)π π¦is properly defined.Given the two conditions for the legitimacy of the EPD, it can be deduce that if the integral πΌ π(π¦ π, π, π½) ππ¦,(19) exists and is finite and strictly positive, then 1 πΌ is called the normalizing constant. In this paper, the coefficientk in Equation (9) is the normalizing constant so that the area under the graph of the EPD is 1. This ensures thelegitimacy of the EPD.We will now derive an expression for the normalizing constant k.2.2 Deducing the Normalizing ConstantThe normalizing constant k, can be deduced by ensuring that π¦ π πππ 1 ππ₯π { π } ππ¦ 1.π In relation to the absolute term 0ππ 1 { ππ₯π [ π ( Letπ₯ π (for whichπ¦ π ππππ¦ππ₯π¦ ππ1π , Equation (20) can be expressed as π¦ π ππ¦ π π) ] ππ¦ ππ₯π [ π () ] ππ¦} 1.ππ0) so that π 1 π₯ (π¦ π ππ1 1 ππ₯ π) and π π¦ ππ(21). Thus, we can deduce that π¦ ππ 11 π π₯ π 1 . ππ (20)Thus, we will have Equation (21) as0 111111 1 1ππ 1 [ π π₯ ππ π π₯ π ππ₯ π π₯ ππ π π₯ π ππ₯ ] 1,ππ 00 1111 1 1ππ 1 ππ π [ π₯ π π π₯ ππ₯ π₯ π π π₯ ππ₯ ]π 0 1,0 111 1 1 1π π π [ π₯ π π π₯ ππ₯ π₯ π π π₯ ππ₯ ]π 0 1.(22)Since the two integrals in Equation (22) are symmetrical, we should have 11 1 12π π π [ π₯ π π π₯ ππ₯ ] 1.π0(23)From Equation (10), the integral in Equation (23) are a family of gamma function given as1251 1 ππ₯ π π,
Mathematical Theory and ModelingISSN 2224-5804 (Paper)ISSN 2225-0522 (Online)Vol.8, No.8, 2018www.iiste.org 11 1 π₯ π π π₯ ππ₯ Ξ ( ).π0Thus,1 π2ππ 1 1[ Ξ ( )] 1.π π(24)Thus, the normalizing constant k is given asπ 112π πNow, for π π 21 π½1[Ξ ( 1)]π.(25)so, we have1(1 π½)2π 23 π½[Ξ ()]2.(26)From Equations (25) and (26), the normalizing constant, k, is undefined for some values q and π½, respectively.This makes the EPD illegitimate. Figure 3 presents the graphical relationship between k and the shape parameter,π½. It can be observed that k is undefined for some negative values of π½, and for higher values of π½ (π½ 80).This makes the integral over β not equal to 1, and thus, the EPD is illegitimate for such values of π½.Figure 3: Relationship between the normalized constant, k and the shape parameter, π½2.3 The Central Moment of the Exponential Power DistributionThe ith central moment of a random variable Y for EPD function, π(π¦ π, π, π) is given by πΈ[(π¦ π)π ] (π¦ π)π π(π¦ π, π, π) ππ¦, ππ 1 (π¦ π)π ππ₯π { π π¦ π π } ππ¦.πSince the integral is symmetrical about the location parameter, π, we have π¦ π ππΈ[(π¦ π)π ] [1 ( 1)π ]ππ 1 (π¦ π)π ππ₯π { π () } ππ¦,ππso that πΈ[(π¦ π)π ] 0 for odd values of i.126(27)
Mathematical Theory and ModelingISSN 2224-5804 (Paper)ISSN 2225-0522 (Online)Vol.8, No.8, 2018By standardization, setting z (π¦ ππwww.iiste.org) so that π§ π (π¦ π)πππand π π π§ π (π¦ π)π . Now, πππ§ ππ¦ and πΈ(π§) 0.Substituting these deductions into Equation (27), we have πΈ[(π¦ π)π ] ππ 1 [1 ( 1)π ] π§ π π π ππ₯π{ ππ§ π } πππ§,0 ππ π [1 ( 1)π ] π§ π ππ₯π{ ππ§ π } ππ§,(28)0Now, we let π₯ ππ§ π , so that π 1 π₯ π§ π . Thus, π§ π 1 1 ππ₯ π ,πππ§ π π π π₯ π andππ§ππ₯1π11 1ππ₯π . π Equation (28) then givesπ1 ππΈ[(π¦ π)π ] ππ π [1 ( 1)π ]π π π 1 π 1 1 π₯ π₯ π π₯ π π ππ₯,π 0π 1 π 11 π 1ππ [1 ( 1)π ]π π π₯ π π π₯ ππ₯.π0(29)Equation (29) simplifies asπΈ[(π¦ π)π ] π 11 ππ 1 ππ [1 ( 1)π ]π π [Ξ ()] .ππMaking substitution for k, we obtainπΈ[(π¦ π)π ] 12π ππ 1 11π 1 π π [1 ( 1)π ]π π [ Ξ ()],1 1ππ[ Ξ ( )]π πwhich simplifies asπ 1)] π π [1 ( 1)π ]π( 1 ).12π π[Ξ ( )]π[Ξ (π]πΈ[(π¦ π) For q 21 π½(30)so, we have(π 1)(1 π½)[Ξ ()] π π [1 ( 1)π ]2π]πΈ[(π¦ π) ( 1 π½ ).1 π½22[Ξ ()]π2(31)We will now use the ith central moment to derive the mean, variance, skewness and kurtosis of the EPD. Thefollowing sections present the results.2.3.1 Mean and Variance of the Exponential Power DistributionIt can be deduced from Equation (30) that πΈ(π¦) π when π 1. In Equation (30) again, if π 2, then2 1[Ξ ()] π 2 [1 ( 1)2 ]π2]πΈ[(π¦ π) ( 1 ).12π π[Ξ ( )]πThus,3[Ξ ( )] π 2ππ£ππ(π¦) ( 1 ) ,(32)1 [Ξ ( )] π ππ127
Mathematical Theory and ModelingISSN 2224-5804 (Paper)ISSN 2225-0522 (Online)Vol.8, No.8, 2018π 21 π½3[Ξ ( )]ππ2 .1[Ξ ( )]π2 π Now, for π www.iiste.org(33)so, we haveπ 1 π½2π£ππ(π¦) 3(1 π½)[Ξ ()]2π 2.1 π½[Ξ ()]2(34)From Equation (33), it can be observed that, the π£ππ(π¦) is affected by the shape parameter, q, and the constant,c. We want to derive c such that π£ππ(π¦) π 2 .Let3[Ξ ( )]πβ .(35)1[Ξ ( )]πThus, if β 1, then π£ππ(π¦) π 2 . Therefore, we would find c such that23π π [Ξ ( )]π 1.(36)1[Ξ ( )]πThis gives2 ππ π 23π [Ξ ( )]ππ 21 [Ξ ( )]π,(37)or113(1 π½) 1 π½1 π½ 1 π½π [Ξ ()][Ξ ()].22(38)2.3.2 Skewness and Kurtosis of the Exponential Power DistributionThe third central moment given byπ3 πΈ[(π¦ π)3 ],(39)is used to determine the symmetry of the distribution. As we know, π3 alone is a poor measure of skewnesssince the size is influenced by the units used to measure the values of X. To make this measure dimensionless, weuseπ3 πΈ[(π¦ π)3 ] πππ(π¦)3,(40)which is a measure of lack of symmetry. Since πΈ[(π¦ π)3 ] 0, π3 0.Generally, the coefficient of kurtosis, also known as the fourth standardized comulant, is given byπ4 πΈ[(π¦ π)4 ] 3.π£ππ(π¦)2(41)In terms of EPD, the coefficient of kurtosis is given by (π¦ π)4 π(π¦ π, π, π) ππ¦π4 3.π£ππ(π¦)2(42)This measures the nature of the spread of the values around the mean. Thus, it is a measure of the peakedness ofEPD or how heavy the tails of EPD are. If a random population has kurtosis above or below zero (0), it cannot be128
Mathematical Theory and ModelingISSN 2224-5804 (Paper)ISSN 2225-0522 (Online)Vol.8, No.8, 2018www.iiste.orgadequately represented by a normal distribution. From Equation (30),5[Ξ ( )] π 4π4]πΈ[(π¦ π) ( 1 ) .(43)1 [Ξ ( )] π ππThus, the coefficient of Kurtosis could then be deduced as5[Ξ ( )] π 4π1 ( 1 )[Ξ ( )] π πππ4 3.(44)π£ππ(π¦)2Substituting the expression for var(y) into Equation (44), we have5[Ξ ( )] π 4π1 ( 1 )[Ξ ( )] π πππ4 2 3,3[Ξ ( )]2ππ[1 (π 1 π ) ][Ξ ( )]πwhich simplifies asπ4 51[Ξ ( )] [Ξ ( )]ππ3[[Ξ ( )]]π2 3.(45)In terms of π½, we haveπ4 5(1 π½)1 π½[Ξ ()] [Ξ ()]223(1 π½)[[Ξ ()]]22 3.(46)Table 1: Estimation of the values of the constant, c, the normalized constant, k and the kurtosis for some valuesof π½π½qckKurtosis (π4 )-1.0 .257129
Mathematical Theory and ModelingISSN 2224-5804 (Paper)ISSN 2225-0522 (Online)Vol.8, No.8, 2018www.iiste.orgTable 1 presents some estimations of the normalizing constant, k, the constant c which ensures that π£ππ(π¦) π 2 and the coefficient of kurtosis of the EPD for some values of π½. The estimations were based on thederivations of π, π, π and π4 in Equations (8), (26), (38) and (46), respectively. The values were estimatedbased on the values of π½ of -1, -0.8, -0.6, , 2. From the table, it can be observed that for π½ 0, thecoefficient of kurtosis, 4, is zero (0), indicating a mesokurtic distribution with identical distribution as that of theNormal. Also, for negative values of π½, π4 is also negative, with exception of π½ 1 for which thedistribution is rectangular and hence π4 is undefined. For positive values of π½, π4 is also positive. For π½ 1,π4 is equal to 3, indicating a do
1. Department of Statistics, University of Cape Coast, Cape Coast, Ghana 2. Directorate of Academic Planning and Quality Assurance, Cape Coast Technical University, P. O. Box AD50, Cape Coast, Ghana * E-mail of the corresponding author: mykemanford@yahoo.com Abstract
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Silat is a combative art of self-defense and survival rooted from Matay archipelago. It was traced at thΓ© early of Langkasuka Kingdom (2nd century CE) till thΓ© reign of Melaka (Malaysia) Sultanate era (13th century). Silat has now evolved to become part of social culture and tradition with thΓ© appearance of a fine physical and spiritual .
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9-2 Exponential Functions Exponential Function: For any real number x, an exponential function is a function in the form fx ab( ) x. There are two types of exponential functions: Exponential Growth: fx ab b( ) x, where 1 Exponential Decay: fx ab b( ) , where 0 1
Unit 6 Exponential and Logarithmic Functions Lesson 1: Graphing Exponential Growth/Decay Function Lesson Goals: Identify transformations of exponential functions Identify the domain/range and key features of exponential functions Why do I need to Learn This? Many real life applications involve exponential functions.
