Estimation of Lomax Parameters Based on Generalized Probability Weighted Moment

Transcription

1 JKAU: Sci., Vol. No., pp: 7-84 ( A.D./43 A.H.) Doi:.497 / Sci. -.3 Etimatio of Lomax Paamete Baed o Geealized Pobability Weighted omet Abdllah. Abd-Elfattah, ad Abdllah H. Alhabey Depatmet of Statitic, Faclty of Sciece, Kig Abdlaziz Uieity, Jeddah, Sadi Aabia Abtact: Pobability weighted momet method, itodced ad ecommed ealie a a alteate method to the claical momet, fo fittig tatitical ditibtio to data; i thoght to be le affected by amplig aiability ad be moe efficiet at podcig obt paamete etimate i cae of mall ample. I thi pape, the geealized pobability weighted momet method i applied fo etimatig the paamete of Lomax ditibtio.. Itodctio Geewood et al. [] itodced a ew method fo etimatig the paamete of ditibtio, which i called the method of pobability weighted momet (PW) which ca be applied to the pobability ditibtio fctio which hae iee fom, ch a Weibll, geealized Paeto, Log logitic. Ladwehe et al. [,3] compaed the pefomace of the etimatio, that hae bee obtaied by pobability weighted momet, ad the maximm likelihood method, ad the method of momet fo the paamete of Gmbel ditibtio ad fod that the method of pobability weighted momet gie bette elt compaed to the othe. Hokig et al. [4] oted that the ew defiitio of pobability weighted momet decibed by Geewood et al. [] ca be ed a the taditioal omet. oeoe, Sog ad Dig [5] tated the pobability weighted ditibtio of the ditibtio ca be applied to the ditibtio that ca ot be expeed i 7

2 Abd-Elfattah, A.. ad Alhabey, A. H. 7 iee fom, ad they hae applied thi method to Peao type III, Iee gamma, ad Log omal ditibtio. Rame [6] itodced a ew cla of pobability weighted momet a a geealizatio method which i called the geealized pobability weighted momet (GPW). Rame [6] applied the geealized pobability weighted momet to the geealized Paeto ditibtio ad oted that thi method gae the bet etimatio compaed to othe method. Alo, he compaed the pefomace of etimatio that hae bee obtaied by pobability weighted momet ad by the method of momet ad fod that the method of geealized pobability weighted momet gie the leat qae eo of the method of momet. oeoe, Ahka ad ahdi [7] applied the geealized pobability weighted momet method to log-logitic ditibtio. They compaed betwee the pefomace of etimatio that hae bee obtaied thogh geealized pobability weighted momet ad that of the maximm likelihood method. They oted that the geealized pobability weighted momet gie bette etimatio tha the maximm likelihood method, epecially fo mall ample. Recetly, Allam [8] applied the method of patial pobability weighted momet to etimate the paamete of geealized expoetial ditibtio ad Elhao [9] ed the geealized pobability weighted momet to etimate the paamete of geealized expoetial ad Peao type III ditibtio. The Lomax ditibtio i kow a Paeto ditibtio of the ecod kid o Peao Type VI ditibtio. It ha bee ed i the aalyi of icome data, ad bie faile data. It may decibe the lifetime of a deceaig faile ate compoet a a heay tailed alteatie to the expoetial ditibtio. Zaga [] deal with the popetie of the Lomax ditibtio with thee paamete. The elatio betwee the Lomax ditibtio ad ome othe ditibtio ae diced i the liteate. Thee ditibtio ae Weibll ditibtio, Compod Weibll ditibtio which i called the thee paamete B type XII ditibtio, expoetial ditibtio, Rayleigh ditibtio, beta type II, beta type I ditibtio, ifom ditibtio, geealized ifom ditibtio, ad exteme ale ditibtio. The objectie of thi pape i to tdy the etimatio poblem fo the paamete of Lomax ditibtio ig the GPW method. Thi pape i ogaized a follow: I ectio, we dic, the defiitio of ample etimato of pobability weighted momet. The GPW ae diced i ectio 3. I ectio 4, we dic the GPW etimatio of Lomax ditibtio. I ectio 5, the appoximate aymptotic aiace ad coaiace matix of GPW fo Lomax ae ietigated.

3 73 Etimatio of Lomax Paamete Baed o Geealized. The Sample Etimato of Pobability Weighted omet Ladweh et al. [3] itodced biaed etimato of α ad β, whe ad ae oegatie itege, which ae baed o the odeed ample x x,..., x, () ( ) ( ) fom the ditibtio F ad ae gie by: ad ˆ = i α x() i, (.) ˆ = i= i β x() i (.) i= Ladweh et al. [4] itodced a biaed etimato ofα, which i cotcted i accodace with the cocept of plottig poitio ad i gie by: i + h W (), = x i i= (.3) whee h =.35. They metioed that it i ot eceay to pecify that i a itege, i amed oly to be oegatie. Hokig et al. [] itodced a etimato of β which i gie by: b = i= p i, x () i ad etimato of α i gie by: c = x()( i p i, ) i= (.4) (.5) ( ) whee P i, i a plottig poitio that i, a ditibtio fee etimate of F x() i. They metioed that eaoable choice of, ch a: P i, ( i a) P i,, < a < =, < a<, ( i a) Pi, = ( + a)

4 Abd-Elfattah, A.. ad Alhabey, A. H. 74 yield etimato b that ae aymptotically eqialet to coitet etimato of β ad α. βˆ, αˆ ad, theefoe, 3. The Geealized Pobability Weighted omet ethod Sog ad Dig [5] ad Dig et al. [] applied the method of pobability weighted momet (PW) fo etimatig the kow paamete of eeal ditibtio i expeible i iee fom. amog them they coideed the Peao type thee ditibtio. The method of geealized pobability weighted momet wa itodced by Rame [6] a a exteio of the oigial method of pobability weighted momet. It ed to etimate the paamete fo eeal ditibtio ch a geealized Paeto, Log-logitic ad Weibll ditibtio. Rame [6] popoed the GPW a a geealizatio fo PW method. All applicatio of the PW method hae bee coideed mall oegatie itege o the expoet. While the GPW ae ot eticted to mall oegatie itege o the expoet. The GPW of ode p = ad =, i gie by,,, theefoe: = E,, ( X ( F( X )) ) xf x f x dx, (3.) = ( ) ( ) whee, X i cotio aiable whoe ditibtio i beig etimated ad F i the cmlatie ditibtio fctio of X, ad ha eithe to be mall, o oegatie itege. The idea behid the PW ad GPW method i to obtai paamete etimato by eqatig PW (,, ) to the ample etimato of PW ad olig the eltig ytem of eqatio fo the ditibtio paamete. Fo example, if the ditibtio ha a two paamete, the taditio PW method, iole coideatio of = ad = i eqatio (3.). O the othe had, the GPW method coide = ad = whee ad ha eithe to be mall, o a o egatie itege. The ample etimate i thi cae i gie by:

5 75 Etimatio of Lomax Paamete Baed o Geealized () ( ) (3.) ˆ = x p p,,, i i i = i. 35 whee, p i =. The followig ae the tep fo GPW pocede fo etimatig the kow paamete of the ditibtio expeible i iee fom: Step (): Obtaiig the iee ditibtio fctio x ( F ) fo the gie ditibtio fctio F of the ditibtio, if poible. Step():Calclatig the theoetical GPW fom oe of the two fomla,, o,, whichee i poible, whee ad take ale,,... o,,... depedig o the mbe of the kow paamete. Geeal coideig a ditibtio ha thee paamete at mot, let =,, 3 the the fomla `,, take the followig fom,, ( X ( F X )) ) = x(, = E ( ) (3.3) ( X ( F X )) ) = x(, = E ( ) (3.4) ad = E X ( F X )) 3 3 ( ) = x(,, ) 3 ( (3.5) Step (3): obtaiig the paamete of iteet i tem of oe of the two theoetical fomla,, o,, ay of which i poible. Step(4): Replacig the theoetical GPW by oe of thei ample etimato ˆ,, o ˆ,,. The followig ae the tep fo GPW pocede fo etimatig the kow paamete of the ditibtio ca ot be expeed i iee fom: Step(): calclatig the theoetical GPW fom oe of the two fomla,, o,, which oe i poible.

6 Abd-Elfattah, A.. ad Alhabey, A. H. 76 ( X ( F X )) ) = x(,, = E ( ). x = x f () t dt f ( x)dx (3.6) ( ) = x( o = E X ( F( X )),, ). x = x f () t dt f ( x)dx (3.7) each of the two ale ad take ale,,... o,,... depedig o the mbe of the kow ditibtio paamete. Geeally coideig a ditibtio ha thee paamete. Let =,, 3 the the fomla,, take the followig fom, = E X ( F X )) ( ) = x(,, ) (, = x x f () t dt f ( x)dx (3.8), ( X ( F X )) ) = x(, = E ( ), = x x f () t dt f ( x)dx. (3.9) ad = E X ( F X )) 3 3 ( ) = x(,, ) 3 (, = x x f 3 () t dt f ( x)dx (3.)

7 77 Etimatio of Lomax Paamete Baed o Geealized Step(): obtaiig the paamete of iteet i tem of oe of the two theoetical fomla,, ad,, whichee i poible. Step(3): Replace the theoetical GPW,,,,,, ad, 3, by thei ample etimato, ˆ,, ˆ,, ad ˆ, 3,. The fomla ˆ,, i baed o the ode complete ample x < x... < x of ize. It i gie by; ˆ () ( ),, = x i pi pi (3.) i= whee x () i i the i th obeatio i the odeed ample. i. 35 ad p i =. Fom eqatio (3.), the ample etimato of ˆ,, ae gie by: ˆ ˆ i.35 = xi i= i.35 = xi i=. (3.) (3.3) ad ˆ 3 i.35 = xi i= 3 (3.4) 4. GPW Etimato fo Lomax Ditibtio The GPW etimato fo paamete of the Lomax ditibtio will be obtaied. A metioed befoe, two et of GPW of the fom,,. Fo the Lomax ditibtio the GPW of the fom,, will be ed becae of it ha a imple aalytical tcte tha,,. The pocede of GPW etimatio fo the Lomax ditibtio i mmaized i the followig tep. Step (): The iee cmlatie ditibtio fctio of the Lomax ditibtio i gie by:

8 Abd-Elfattah, A.. ad Alhabey, A. H. 78 x ( F) = λ ( F) α, α, λ, x > (4.) whee the pdf ad CDF of Lomax ditibtio ae epectiely, α x ( ) = ( + ) λ λ ( α + ) f x, >, λ, α > x ad x λ α F ( x) = ( + ). Step (): Obtai the theoetical pobability weighted momet, fo the Lomax ditibtio, that i:,, = x F df, (4.) take the ale,... accodig to the mbe of paamete., Sbtittig the fom of the iee ditibtio fctio of the Lomax ditibtio ito eqatio (4.) yield: α,, = λ ( F) F df (4.3),, = λ β( +, ) α +, whee β ( m, ) i the beta fctio. Sice the Lomax ditibtio ha a two paamete; the oly the fit two GPW,,, ad,,, ae eeded fo paamete etimatio. They ae gie by: = λ β( +, ), (4.4) α +,, ad = λ β( +, ) α +,, (4.5) Step (3): Obtai the paameteα,λ i tem of,, ad,, ig eqatio (4.4)ad (4.5). Fom eqatio (4.4) the;

9 79 Etimatio of Lomax Paamete Baed o Geealized λ β ( +, ) α + =,, Th fom eqatio (4.5) λ β ( +, ) α + =,, Sbtittig eqatio (4.6) ito eqatio (4.5) gie: (4.6) (4.7),, = β ( +, ),, α β ( +, ) α +, +,, β( +, ) =,, β( +, ) α + α +,, β( +, ),, β( +, ) = α + α + Step (4): Replace the theoetical GPW,,, ad,, by thei ample etimato, ˆ,, ad ˆ,,. The fomla ˆ,, itodced by Hokig (986) will be ed hee. The fomla ˆ,, i baed o the ode complete ample x < x... < x of ize. It i gie by; ˆ = x () ( ) i pi pi,, (4.8) i= whee x () i i the i th obeatio i the odeed ample;

10 Abd-Elfattah, A.. ad Alhabey, A. H. 8 i. 35 ad p i =. ˆ Fom eqatio (3.9), the ample etimato of,, ae gie by: ˆ i.35 = xi i=. ad ˆ i.35 = xi i=. Theefoe the etimatio of GPW fo the hape paameteα, ay αˆ will be, ˆ ˆ,, β( +, ),, β( +, ) = α + α + (4.9) Eqatio (4.9) will be oled fo oe kow which i the hape paamete etimate αˆ fom etimatio of GPW. The oltio of thi eqatio eqie iteatie techiqe. Oce αˆ i fod, detemiatio of the cale paamete etimated fom GPW, λˆ ad the etimatio GPW fo λ, ay λˆ will be; λˆ β ( +, ) α + = ˆ,, 5. Vaiace Coaiace atix fo the Etimato The aymptotic theoy ally poide a good appoximatio to the paamete of ditibtio fo ample of ize 5 ad alo fo mall ample of ize =,. Alo, it poide ome pelimiay iight ad motiatio fo the beqet compte imlatio expeimet. Aymptotic aiace of GPW aoid the e of exteie compte imlatio becae it baed o aalytical expeio. The aymptotic aiace of λ ad α ae appoximated by ig the aymptotic aiace coaiace of the GPW etimato ˆ,, ad ˆ,,,.

11 8 Etimatio of Lomax Paamete Baed o Geealized whee ˆ ˆ λ = φ (,,,,, ), ˆ ˆ ad α = ϕ (,,,,, ) Hokig [] metioed thee ae two et of the aymptotic aiace. The fit, if X i adom aiable with cmlatie ditibtio fctio F ad PW i ha the ample PW,,, the the coaiace matix i;,, ( ) ˆ A = A, (5.) whee ad A = I + I (5.),,, I, i the aiace of PW which i I y = + { F( x) } F( y) { } F( y) { } dx dy, (5.3) y + ad I { F( x) } F( y) = { } F( y) { } dx dy, (5.4) The ecod if X i adom aiable with cmlatie ditibtio fctio F ad PW i,, ha the ample PW ˆ,,, the the coaiace matix i; ( ) A = A, (5.5) whee A = I + I (5.6),,, ad I I, i the aiace of PW which i + ( F( x) ) ( F( y) ) F( x) = dx dy (5.7) x< y + ad I ( F( x) ) ( F( y) ) F( x) = dx dy (5.8) x< y

12 Abd-Elfattah, A.. ad Alhabey, A. H. 8 Ackowledgmet The atho wold like to thak Deahip of Scietific Reeach, Kig Abdlaziz Uieity fo ppotig thi eeach poject mbe (3-67/49). Refeece [] Geewood, J.A., Ladweh, J.., atala, N.C. ad Walli, J.R. (979) Pobability Weighted omet: Defiitio Ad Relatio To Paamete Of Seeal Ditibtio Expeible I Iee Fom, Wate Reoce Reeach, 5(5): [] Ladweh, J.., atala, N.C. ad Walli, J.R. (979a) Pobability Weighted omet Compaed With Some Taditioal Techiqe I Etimatig Gmbel Paamete Ad Qatile, Wate Reoce Reeach, 5: [3] Ladweh, J.., atala, N.C. ad Walli, J.R. (979b) Etimatio of Paamete Ad Qatile Of Wakeby Ditibtio, Wate Reoce Reeach, 5(6): [4] Hokig, J.R.., Walli, J.R. ad Wood, E.F. (985) Etimatio of the geealized Exteme-Vale Ditibtio by the ethod of pobability weighted momet, Techometic, 7(3): 5-6. [5] Sog, D. ad Dig, J. (988) The applicatio of pobability weighted momet i etimatig the paamete of the Peao type thee ditibtio, Joal of hydology, : [6] Rame, P. () Geealized Pobability Weighted omet: Applicatio To The Geealized Pobability Weighted omet: Paeto Ditibtio, Wate Reoce Reeach., 37(6): [7] Ahka, F. ad ahdi, S. (3) Compaio of Two Fittig ethod Fo The Log logitic, Wate Reoce Reeach, 39(8), At. No. 7. [8] Allam, S.A. (7) Etimatio of the Geealized Expoetial Ditibtio by the ethod of Patial Pobability Weighted omet..sc. Thei, Caio Uieity, Egypt. [9] Elhao, N.. (9) Geealized Pobability Weighted omet Etimato fo Some Ditibtio..Sc. Thei, Caio Uieity. [] Zaga, A-A. (999) Lomax ditibtio ad it ole i eliability ad life tetig.sc. Thei, Kig Abdl-Alaziz Uieity, Sadi Aabia. [] Dig, J., Sog, D. ad Ho, Y. (989) Expeio Relatig Pobability Weighted omet to Paamete of Seeal Ditibtio Iexpeible i Iee Fom, Joal of Hydology, : [] Hokig, J.R.. (986) The theoy of pobability weighted momet, Reeach Repot RC, IB Reeach Diiio, Yoktow Height, NY.

13 83 Etimatio of Lomax Paamete Baed o Geealized - :.Weibll, geealized Paeto, Log logitic.

14 Abd-Elfattah, A.. ad Alhabey, A. H. 84..