A NEW GENERALIZED PARETO DISTRIBUTION. Abstract

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1 NEW GENERLIZED PRETO DISTRIBUTION bd Elfattah,. M. * Elshepieny, E.. * Hussein, E.. * a_afattah@hotmail.com ahmedc55@hotmail.com bstact In this pape, we intoduce a new genealized Paeto distibution and studied its popeties. Some well known distibutions can be deived as a special cases as well as some deived distibutions. 1 Intoduction The Paeto distibution was fist poposed by Paeto (1897) as a model fo the distibution of income. It can be used to epesent vaious othe foms of distibutions (othe than income data) that aise in human life. It has played a vey impotant ole in the investigation of city population sizes, occuence of natual esouces, insuance-isk, and business failues. nald (1983) gives an etensive histoical suvey of its use in the contet of income distibution. andom vaiable T is said to have a thee paamete Paeto distibution if its pobability density function (p.d.f) is given by α f( t, αβλ,, ) = β t λ 1 + β ( 1) α+ () 1 whee λ < <, β > 0, α > 0, whee λ is the location paamete, β is the scale paamete and α is the shape paamete. In this pape, we will genealized the thee paametes Paeto distibution (1) by adding anothe shape paamete. The popeties of the suggested distibution will be investigated as well as its elations with othe well known distibutions. Fou Paametes Paeto Distibution. Suppose X is distibuted fom Paeto distibution, the p.d.f of X will be α f( ; αβλ,,,) = β λ β c1 1 + β λ ( ) whee λ < <, β > 0, α > 0, > 0, λ is the location paamete, β is the scale paamete and ( α, ) ae the shape paametes. * Institute of Statistical Studies & Reseach, aio Univesity, Egypt

2 The paticula case c=1, the p.d.f. () educes to the thee paamete Paeto distibution (1) with paametesα, β andλ. Figue (1) illustates the shape of () fo selected values of (α =1, β =1, λ =0, c=.5, 1, 5, 10). nothe some possible shapes of () fo ( α =3, β =1, λ =0, c=.5, 1, 5, 10) ae shown in Figue (). The effect of the paamete c can be seen easily. 3 4 c=10, c=5, c=.5, c=1 f () f1() f() f3() 1 c=10, c=5, c=.5, c=1 f () f1() f() f3() Figue (1) Figue () The cumulative distibution function F( ), suvival S( ) and hazad ate H ( ) functions of () is given by and H ( ) = ( ;, β, λ ) F α, = α β S ( ) = λ β 1 λ 1 1+ β λ 1 + β λ 1 + β The hazad ate function is an impotant quantity fo chaacteizing life phenomena. Some possible shapes of (3) fo (α =1, β =1, λ =0, c=.1,.5,.9, ) ae shown in Figue (3). 6 α 1 α, (3) c=, c=.5, c=.1, c=.9 h ( ) h1( ) h( ) h3( ) Figue (3)

3 The th moment about mean of the genealized Paeto distibution( is given by: ) µ = µ = λ Ε ( ) ( ) f ( ; α, β, λ, ) = αβ ( 1) j = 0 () µ d 1 1 j j Γ α Γ 1+ Γ α Γ 1+ j j Γ( α) Γ whee Γ. is defined as the gamma function, and = 1,,3,..., Theefoe, the mean, vaiance and coefficient of vaiation of X ae given by j µ = β 1 1 Γα Γ 1 + Γ ( α ) + λ, and σ = = β Γα Γ 1 + Γα ' Γ( α ) Γ( α ) Γ σ. V ( X ) = 100 µ ( ) Γ Γα Γ α Γ α 1 1 Γ 1 + Γα + λγ Γ ( α ) 1 α 1 3 Some Special ases The 4-paametae genealized Paeto distibution () can be specialized to diffeent known distibutions such as: (i) Fo 1, educes to a thee paamete Paeto distibution (1). (ii) When = ( ) ( ) β =, educes the 4-paamete compound Weibull gamma distibution with p.d.f ( ;,, λ ) f α, = α ( ) 1 λ ( λ) 1 + λ < <,,, α > 0 3

4 (iii) Fo and, educes to the 3-paamete compound β = = ( ) Rayleigh gamma distibution with p.d.f ( ; α,,λ ) f = α ( λ ) λ 1 + λ < <, α, > 0 (iv) When β andα 1, = = ( ) Weibull eponential distibution with p.d.f educes to the 3-paamete compound ( ;,, λ ) f α, = 1 + ( λ) λ < <,, > 0 (v) Fo and 1, β = =α = ( ) eponential eponential distibution defined as educes to the paamete compound 1 f ( ;,λ ) = λ 1+ λ < <, > 0 (vi) Fo β 1and λ = 0, educes the paamete Bu XII distibution. = ( ) ( c) f ;α, = c α 1 ( 1 ) ( α + + 1) whee > 0, α > 0, c > 0.Whee c and α ae the shape paametes. (vii) Fo with p.d.f. = 1and λ = 0, ( ) educes to the paamete Loma distibution ( ;α, β ) f = β α 1 + β > 0 β, α > 0 ( = = 1 =0 ) ( ) (viii) Fo β and λ, p.d.f f ( ;α ) = α ( ) ( α ) educes to the beta type II distibution with > 0, α > 0 4 Some Tansfomed distibutions. The 4 paamete genealized Paeto distibution ( ) can be tansfomed to many diffeent known distibutions as follows: 4

5 1 λ (a) When y = ln λ, ( ) can be tansfomed to 3- β paamete Weibull distibution, i. e, the p.d.f of y will be 1 f ( y;,α, λ) = α ( y λ ) e α ( yλ ) λ < y <, α > 0 λ (b) When y = ln λ β paamete eponential distibution with the following p.d.f. f ( y;α,λ) = α α ( yλ ),( ) can be tansfomed to e < y < λ α > 0 λ (c) Fo y = β ln λ β type I genealized logistic distibution with p.d.f. defined as, ( ) can be tansfomed to 3 paamete (d) When p.d.f α f = β e β ( y; α, β, λ) λ y = β yλ yλ β 1+ e < y <, ( ) can be tansfomed to beta type II β (,α ) 1 with f ( y;α ) = α [ ] ( α ) y y > 0, α > 0 Refeences bd Elfattah,. M., Elshepieny, E.. and Hussein, E.. (007) Paametes estimation fo new genealized Paeto distibution. Submitted l-hussaini, E. K. (1991). chaacteization of Bu type XII distibution. ppl. Math. Lett., 4(1), nald, B.. and Pess, S.J ( 1983 ). Bayesian infeence fo Paeto populations. Jounal of Econometics, 1, Bu, I. W. and islak, P. J. (1968). On a geneal system of distibution: I. Its cuve-shape chaacteistics, II. The sample median. J. me. Statist. ssoc., 63, Ismail, S. (004) simple Estimato fo the Shape Paamete of the Paeto Distibution with Economics and Medical pplications, Jounal of pplied Statistics, Vol. 31, NO. 1,

6 Masoom, M.. and Saalees Nadaajah, (006) tuncated Paeto distibution, ompute ommunications Paeto, V. (1897). cous d'economic politique, Lausanee and Pais; Range and ie. Salem,. M. and Mead, M. (199 ) : Modified Maimum Likelihood Estimatos fo the thee paamete Paeto Distibution. The Scientific Jounal fo Economic and ommece, Faculty of ommece, in shames Univesity, Vanman, k. (1976 )."Estimatos Based on ode statistics fom a Paeto Distibution, "Jounal of meican Statistical ssociation, 71, pp , Wingo, D. R. (1983). Maimum likelihood methods fo fitting the Bu type XII distibution to the life test data. Biometical J., 5,