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1 Global Jounal of HUMAN SOCIAL SCIENCE Economics Volume 13 Issue Vesion 1.0 Yea 013 Type: Double Blind Pee Reviewed Intenational Reseach Jounal Publishe: Global Jounals Inc. (USA) Online ISSN: x & Pint ISSN: X On Fitting of Genealized Paeto Distibution By T. A. Raja & A. H. Mi Division of Agicultual Statistics, Skuast-K Shalima. (J&K) Abstact - The Paeto distibution is to model the income data set of a society. The distibution is appopiate to the situations in which an equilibium exists in distibution of small to lage. Thee exists many genealization appoaches to the distibution. In this pape an effot has been made to compae the applicability of genealized Paeto distibution with Picklands (1975) by using a eal life income data set. The model has povided consideable a good fit to the data set. Some well known distibutions has been deived as a special case of this model fo suitable choice of paametes. Keywods : paeto distibution, picklands (1975) genealized paeto distibution, a new genealized Paeto distibution, Income data set, Goodness of fit. GJHSS-E Classification : FOR Code: OnFittingofGenealizedPaetoDistibution Stictly as pe the compliance and egulations of: 013. T. A. Raja & A. H. Mi. This is a eseach/eview pape, distibuted unde the tems of the Ceative Commons Attibution- Noncommecial 3.0 Unpoted License pemitting all non-commecial use, distibution, and epoduction in any medium, povided the oiginal wok is popely cited.

2 On Fitting of Genealized Paeto Distibution Abstact - The Paeto distibution is to model the income data set of a society. The distibution is appopiate to the situations in which an equilibium exists in distibution of small to lage. Thee exists many genealization appoaches to the distibution. In this pape an effot has been made to compae the applicability of genealized Paeto distibution with Picklands (1975) by using a eal life income data set. The model has povided consideable a good fit to the data set. Some well known distibutions has been deived as a special case of this model fo suitable choice of paametes. Keywods : paeto distibution, picklands (1975) genealized paeto distibution, a new genealized Paeto distibution, Income data set, Goodness of fit. I. Intoduction a) Paeto Distibution (PD) T he Paeto distibution was poposed by an Italian bon Swiss economist named Vilfedo Paeto (1897) as a model fo the distibution of income. It is a skewed, heavy tailed distibution and is some times efeed as Badfod distibution. Paeto used this α, β>0 and x 0 Whee β is a scale paamete and α is a shape paamete. T. A. Raja α & A. H. Mi σ f(x, α, β, λ)= α/ β = 0 othewise whee λ<x< α, β>0, α>0 β is a scale paamete, α is a shape paamete and λ is the location. b) Genealized Paeto distibution (GPD) Like othe distibutions the Paeto distibution was genealized. The Genealized Paeto distibution = 0 othewise f(x, α, β)= 1/ β 1 distibution to descibe the allocation of wealth among individuals. A lage potion of wealth of many societies is owned by a smalle pecentage of the people in that society. This distibution s sometimes expessed moe simple as the Paeto pinciple o The 80-0 ule which says that 0% of the population owns 80% of the wealth. This distibution is not limited to descibing wealth o income distibution, but to many situations in which an equilibium is found in the distibution of the small to the lage. It is widely used and has played a vey impotant ole in explaining population occuence, natual esouces, insuance isk, business failues and has ecently been used to study the ozone levels in the uppe atmosphee. Wingo (198) discussed the unimodility of the conditional likelihood function of the Paeto distibution using multi censoed samples. Anold and Pess (1983) gave an extensive histoical suvey of its use in the content of income distibution. The pobability density function (p. d. f) of two paamete Paeto distibution is defined as ( x λ) + α +1) (GPD) was intoduced by Picklands (1975). The pobability density function (p.d.f) is defined as x α The ange of x is 0 x<α fo α 0 and 0 x β/α fo α>0 The GPD is heavy tailed, skewed and is used to model exteme values as investigated by Hoking and Autho α σ : Division of Agicultual Statistics, Skuast-K Shalima. (J & K). taiqaaja@ediffmail.com f(x, α, β)= α α +1) ( α / β ) (1.1) β The pobability density function (p. d. f) of thee paamete Paeto distibution is defined as 1 ( 1) α (1.) (.1) Well (1987), Smith (1989, 1990), Davison and Smith (1990). Smith (1990) gave a eview of two most widely used methods based on genealized exteme value distibution. Samia and Mohammad (1993) used five 9Yea Global Jounal of Human Social Science ( E D) Volume XIII Issue WII Vesion I 013 Global Jounals Inc. (US)

3 On Fitting of Genealized Paeto Distibution Yea Global Jounal of Human Social Science ( E ) D Volume XIII Issue WII Vesion I modifications of moments to estimate the paametes of Paeto distibution. Abdel Ghaly etal (1998) obtained the pediction of the shape paametes. Choulakin and Stephens (001) obtained the goodness of fit fo the genealized Paeto distibution. Abd Elfattah etal (007) obtained a new genealized Paeto distibution and deived some well known distibutions as special cases. c) A Model of Genealized Paeto Distibution In this pape a model of genealized Paeto distibution as given by Abd Elfattab etal (007) by Whee τ (.) = γ ( x λ) x λ f(x; α, γ, β, λ)= α/ β 1+. β β whee λ<x< α, β>0, α>0 and γ>0 α and γ ae shape paamete and λ is the location and β is a scale paamete To pove that f(x) is a pobability density function, following conditions ae to be satisfied. i. f(x) 0 and ii. f ( x). = 1 αβ ( x µ ) µ = E( x µ ) =. f ( x; α, β, γ, λ). ( 1) j λ 1 1 j j τ ( α ). τ (1 + ) τ ( α ). τ (1 + ) γ γ γ γ.. τ ( α) τ ( + 1 j= 0 j α ) is defined as the gamma function. Hee can take any value =1,, 3., thee foe mean and vaiance of x can be defined as σ intoducing one moe shape paamete γ is applied it to eal life data set egading family income sample fom Kashmi (Jammu and Kashmi)-India. The pobability density function of the new genealized Paeto distibution is as j γ 1 λ<x< α, β>0, α>0 and γ>0 ae the paametes of the model. f(x) 0 fo all x which poves (i) clealy f(x) 0 establishes condition (ii) as f ( x). = 1 genealized Paeto distibution is 1 1 τ ( α ). τ (1 + ) γ γ µ β + λ, τ ( α) 1 1 τ ( α + ). τ (1 ) τ ( α + ). τ (1 ) = γ γ γ γ β τ ( α) τ ( α) = and (3.1) The th moment about mean of the d) Deivations of Some Distibutions Many distibutions can be deived fom 4- paamete genealized Paeto distibution fo diffeent choices of the paametes. (i) Fo γ = 1 the fou-paamete Paeto distibution educes to thee-paamete Paeto distibution with p. d. f as f(x, α, β, λ) = ( x λ) + whee λ<x< α, β>0, α>0 β is a scale paamete, α is a shape paamete and λ is the location 013 Global Jounals Inc. (US)

4 On Fitting of Genealized Paeto Distibution (ii) Fo γ =1 and λ=0 it educes to two paamete Lomax distibution with p. d. f as f(x, α, β ) = α/ β Whee x>0 and, β>0, α>0 (iii) Whee x>0 and, α>0 Class x + β Table 1 : Fitting of New Genealized Paeto Distibution Income (Rs) Xi Fo (γ = β =1 and λ=0) it educes to Beta type II distibution with p. d. f as f(x, α, ) = α ( ) 1+ x Obseved fequency (Oi) Similaly many moe distibutions can be deived fo suitable choice of paametes. e) Goodness of Fit As Paeto distibution povides a good fit to income data a lot of wok has been done on it. In this pape the new genealized Paeto distibution has been fitted to income data along with Picklands (1975) genealized Paeto distibution to thee hunded families fom Kashmi valley of Jammu and Kashmi-India. The sample has been selected at andom and statified andom sampling pocedue involving all the six disticts of Kashmi valley has been adopted fo the pupose. Expected Fequency (Ei) by Picklands (1975) Expected Fequency (Ei) 1 < 10, ,000-0, ,000-30, ,000-40, ,000-50, ,000 and above 1 Total The mean and standad deviation of the above data set has been found as mean= and standad deviation= The Chi-squae statistics fo new genealized Paeto distibution efeed by its p- value is (p=0.387) and Chi-squae statistics fo Picklands (1975) genealized Paeto distibution efeed by its p-value is (p=0.843) eveals clea nonsignificance in both the cases. Thus encouaging that the new genealized Paeto distibution also povides a good fit to the eal life data set. Refeences Réféences Refeencias 1. Abdel-Ghaly, A. A., Attia, A. F. and Aly, H. M. (1998), Estimation of The Paametes of Paeto Distibution and The Reliability Function Using Acceleated Life Testing with Censoing, Communications in Statistics, Simulation and Computation, 7(), Abd Elfattah, A. M. Elshepieny, E.A and Hussein, E.A (007). A new genealized Paeto distibution. J. of Intestat #001,Dec Anald, B.C and pess, S.J (1983). On a geneal system of distibution: I its cuved-shape chaacteistics, II. The sample median. J. Ame. Statistics. Association; 63, Chaoulakian, V. and Stephens, M.A (001). Goodness of fit tests on the genealized Paeto distibution, Technometics, 43(1), Davison, A.C and Smith, R.L (1990).Model fo exceedances ove high thesholds (with Comments), Jounal of the Royal Statistical Society, Se.B, 5, Hosking, J.R.M and walls (1987).Paamete and quantile estimation fo the genealized Paeto distibution Technometics, 9, Picklands, J. (1975).Statistical infeence using exteme unde statistics, Annals of statistics, 3, Samia A. S. and Mohamed M.M. (1993). Modified moment estimatos fo the thee paametes Paeto distibution. ISSR, Caio Univesity, Vol.(8), Pat (). 9. Smith R.L (1989).Thesholds methods fo sample extemes in Statistical extemes and applications, Ed. J. Tiago De Oliveia, Dodecht; Riedel Smith, R. L (1990).Exteme value analysis of time Seies, An application to tend detection in goundlevel ozone, Statistical Sciences, 4, Wingo, D.R (1983). Maximum Likelihood methods fo fitting the Bu Type XII distibution to the life test data. J of Biometical, 5, Global Jounals Inc. (US) Yea Global Jounal of Human Social Science ( E D) Volume XIII Issue WII Vesion I

5 On Fitting of Genealized Paeto Distibution Yea Global Jounal of Human Social Science ( E ) D Volume XIII Issue WII Vesion I This page is intentionally left blank 013 Global Jounals Inc. (US)