Research Article Bayesian Estimation of Inequality and Poverty Indices in Case of Pareto Distribution Using Different Priors under LINEX Loss Function

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1 Advaces i Statistics Volume 2015, Aticle ID , 10 pages Reseach Aticle Bayesia Estimatio of Iequality ad Povety Idices i Case of Paeto Distibutio Usig Diffeet Pios ude LINEX Loss Fuctio Kamaljit Kau, Sageeta Aoa, ad Kalpaa K. Mahaja Depatmet of Statistics, Pajab Uivesity, Chadigah , Idia Coespodece should be addessed to Kamaljit Kau; kamaljitk010@gmail.com Received 29 August 2014; Accepted 7 Jauay 2015 Academic Edito: Kathik Devaaja Copyight 2015 Kamaljit Kau et al. This is a ope access aticle distibuted ude the Ceative Commos Attibutio Licese, hich pemits uesticted use, distibutio, ad epoductio i ay medium, povided the oigial ok is popely cited. Bayesia estimatos of Gii idex ad a Povety measue ae obtaied i case of Paeto distibutio ude cesoed ad complete setup. The said estimatos ae obtaied usig to oifomative s, amely, uifom ad Jeffeys, ad oe cojugate ude the assumptio of Liea Expoetial (LINEX) loss fuctio. Usig simulatio techiques, the elative efficiecy of poposed estimatos usig diffeet s ad loss fuctios is obtaied. The pefomaces of the poposed estimatos have bee compaed o the basis of thei simulated isks obtaied ude LINEX loss fuctio. 1. Itoductio The Paeto distibutio is a skeed, heavy-tailed distibutio that is used to model the distibutio of icomes ad othe fiacial vaiables. It as itoduced by Paeto [1]hich has a pobability desity fuctio of the fom f (x) = { m { x +1, m x< ; m,>0, { 0, otheise, ad cumulative distibutio fuctio is F (x) = { 1 ( m { x ), x m, { 0, otheise. The paamete m i (2) epesets the miimum icome i the populatio ude study ad assumed to be ko, hile the othe paamete isassumedtobeuko. The aveage icome fo Paeto distibutio is M= (1) (2) m ( 1), > 1. (3) I the cotext of icome iequality ad povety, Gii idex ad Povety measue head cout atio ae to most popula idices [2, 3]. Gii idex is geeally defied as G=1 tice the aea ude the Loez cuve 1 =1 2 L(p)dp, 0 0 p 1, hee L(p) = (1/μ) p 0 F 1 (t)dt is the equatio of the Loez cuve ad μ= 1 0 F 1 (t)dt is the mea of the distibutio. Equivaletly,Giiidexcaalsobedefiedas (4) G= Δ 2μ, (5) hee Δ= x y f(x)f(y)dx dy is populatio Gii 0 0 mea diffeece. The Povety idex head cout atio P 0 is simply the cout of the umbe of households hose icomes ae belo the povety lie divided by the total populatio. I tems of cotiuous distibutio, 0 P 0 = f(y)dy=f( 0 ), (6) 0 hee, 0 (> m) is called Povety Lie.

2 2 Advaces i Statistics by I case of Paeto distibutio, Gii idex (G) [4, 5]isgive G= ad Povety measue (P 0 ) is 1 (2 1), > 1 2, (7) P 0 =F( 0 ) =1 ( m (8) ) =1 λ 0, 0 hee, 0 (> m) ad λ 0 =(m/ 0 ). Thus, 0 is pe capita aual icome epesetig a miimum acceptable stadad of livig ad P 0 epesets the popotio of populatio havig icome equal to o less tha 0. The estimatio of Gii idex (G) ad Povety measue (P 0 ) ad the associated ifeece usig classical appoach (paametic ad opaametic) is available i liteatue [5 8]. Hoeve, i the Bayesia setup, this has ot evoked the iteest of may eseaches [9, 10]. I the peset pape, ou focus ill be o the estimatio of iequality ad povety idices i the Bayesia setup. Whe the Bayesia method is used, the choice of appopiate distibutio plays a impotat ole, hich may be categoized as ifomative, oifomative, ad cojugate s [11, 12]. I the peset pape, thee s (to oifomative s ad oe cojugate ) ae used to estimate shape paamete, Gii idex, Aveage icome, ad Povety measue. The to oifomative s ae Uifom ad Jeffeys, hile cojugate is chose as Tucated Elag distibutio. I Bayesia estimatio, the citeio fo good estimatos fo the paametes of iteest is the choice of appopiate loss fuctio. I Bayesia estimatio, to types of loss fuctios commoly used ae Squaed eo loss fuctio (SELF) ad Liea expoetial (LINEX) loss fuctio. The simplest type of loss fuctio is squaed eo, hich is also efeed to as quadatic loss is give as L (θ) =( θ θ) 2, (9) hee θ is the estimato of θ. The usual squaed eo loss fuctio is symmetical ad associates equal impotace to the losses due to oveestimatio ad udeestimatio of equal magitude. Hoeve, such a estictio may be impactical; fo example, i estimatio of shape paamete of Classical Paeto distibutio, the oveestimatio ad udeestimatio may ot be of equal impotace as ove estimate of shape paamete gives a ude-estimate of iequality idex hich seems to be moe seious as compaed to ude estimate of shape paamete because e ae ofte iteested i educig icome iequality idex. This leads oe to thik that a asymmetical loss fuctio be cosideed fo estimatio of shape paamete hich associates geate impotace to oveestimatio. A umbe of asymmetical loss fuctios have bee poposed i statistical liteatue [13 16]. Vaia [16] poposedauseful asymmetical loss fuctio ko as Liea expoetial (LINEX) loss fuctio hich is give as L( θ θ)=e b( θ θ) b( θ θ) 1, b=0. (10) The posteio expectatio of the LINEX loss fuctio (10) is E(L( θ θ))=e b θe(e bθ ) b( θ E(θ)) 1, (11) hee E() deotes posteio expectatio ith espect to the posteio desity of θ. By a esult of Zelle [17] the Bayes estimato of θ deoted by θ ude the LINEX loss fuctio is the value hich miimizes posteio expectatio ad is give by θ = 1 b l [E (e bθ )], (12) povided that the expectatio E(e bθ ) exists ad is fiite [18]. I Figues 1(a) ad 1(b),values of L(θ) ae plotted fo the selected values of θ fo b=1ad b= 1.Itisseethat,fo b=1,thefuctioisquiteasymmeticithavalueexceedig the taget beig moe seious tha a value belo the taget. But, fo b= 1, the fuctio is also quite asymmetic ith a valuebelothetagetvaluebeigmoeseiousthaavalue exceedig the taget. Fo small value of b, the LINEX loss fuctio ca be expaded by Taylo s seies expasio as exp (b ( θ θ)) b( θ θ) 1 = = i=0 i=2 b i ( θ θ) i i! b i ( θ θ) i i! b2 ( θ θ) 2 2. b( θ θ) 1 (13) Thus, the LINEX loss fuctio is appoximately equal to squaed eo loss fuctio fo small values of b (see Figue 1(c)). This loss fuctio has bee cosideed by Zelle [17], Basu ad Ebahimi [19], ad Afify [20] fo diffeet distibutios. I the peset study, LINEX loss fuctio is used fo estimatig the shape paamete, Gii idex, Mea icome, ad a Povety measue i the cotext of Paeto distibutio usig oifomative s (Uifom ad Jeffeys ) ad oe cojugate (Tucated Elag distibutio) alog ith some assumptios egadig the sampled populatio. Bayesia appoach ith ad posteio distibutios alog ith samplig schemes i the cotext of Paeto distibutio is give i Sectio 2. ISectio 3, Bayesia estimatos of shape paamete, Gii idex, Mea icome, ad Povety measue usig diffeet s ude

3 Advaces i Statistics L(θ) L(θ) (a) θ θ (b) L(θ) (c) Figue 1: (a) LINEX Loss fuctio he θ = ad b=1. (b) LINEX Loss fuctio he θ = ad b= 1. (c) LINEX Loss fuctio he θ = ad b = 0.1. θ the assumptio of LINEX loss fuctio ae obtaied. Fially, i Sectio 4, simulatio is doe to compae the efficiecy of thee diffeet appoaches usig thee s ad loss fuctios. The obustess of the hypepaametes is give i Sectio 4.1 though simulatio study. Sectio 5 pesets the coclusio of the study. 2. Pelimiay about Samplig Scheme, Pios, ad Posteio Desities The Bayesia aalysis of the Paeto distibutio (2) is based o the folloig cesoed samplig scheme o pesoal icome data. It is assumed that aual icomes of the pesos ae ude study but exact figues x 1,x 2,x 3,...,x ae available oly fo those idividuals hose aual icome does ot exceed a pescibed aual icome 0 (> m),adfo the emaiig ( ) idividuals, the exact icome figues ae uko but e do ko that thei aual icome exceed the pescibed figue 0. Befoe the aival of the sample data o pesoal icomes, is pedetemied but ot, hich is a adom. This cesoig scheme used is efeed as ight cesoed samplig scheme. The likelihood fuctio L() fo complete sample i case of Paeto distibutio [4]is L () = m ( i=1 x i ) (+1). (14) I case of cesoed data, the likelihood fuctio fo ay distibutio [21]is! L (X; ) = f (x; ) [1 F(x; )]. (15) ( )! i=1

4 4 Advaces i Statistics The likelihood fuctio fo Paeto distibutio i cesoed sample is L () = m λ ( ) ( i=1 x i) (+1) e Z, (, ), (16) hee Z = l(m P ) is poduct icome statistics [22]ad P = ( i=1 x i). Bayes estimatos of Gii idex ad Aveage icome illotbecovegetitheiteval[0, 1/2] ad [0, 1], espectively, ad the method ill fail to ok. Hece, this difficulty is emoved by assumig >1, to obtai diffeet Bayes estimatos. The ad posteio desities fo oifomative s (Uifom ad Jeffeys ) ad cojugate ae explaied belo. (i) Uifom Pio. I pactice, the ifomative s ae ot alays available; fo such situatios, the use of oifomative s is ecommeded. Oe of the most idely used oifomative, due to Laplace [23], is a uifom. Theefoe, the uifom has bee assumed fo the estimatio of the shape paamete of the Paeto distibutio. Uifom fo is g u () 1. (17) Combie likelihood fuctio (16) ith the desity (17) by usig Bayes theoem to obtai the posteio desity as g u () = = L () g() L () g() d (Z ) +1 Γ(+1,Z ) e Z, (18) hee Γ(a, y) = y ua 1 e u du, y > 0 is the uppe icomplete gamma fuctio ad posteio desity g u () is left tucated Gamma distibutio. (ii) Jeffeys Pio. Aothe oifomative has bee suggested by Jeffeys [24] hich is fequetly used i situatios hee oe does ot have much ifomatio about the paametes. This is defied as the distibutio of the paametes popotioal to the squae oot of the detemiats of the Fishe ifomatio matix, that is, g() I(), hee I() = E[( 2 / 2 ) log L( x)] is Fishe s ifomatio of the give distibutio. I case of Paeto distibutio, g j (). (19) A motivatio fo Jeffeys is that Fishe s ifomatio (I()) is a idicato of the amout of ifomatio bought by the model (obsevatios) about. The posteio desity is obtaied as g j () = (Z ) Γ(,Z ) 1 e Z, ( < ), (20) hich is left tucated Gamma distibutio. Note: Extesio of Jeffeys Pio.Jeffeys isapaticula case of extesio of Jeffeys poposed by Al-Kutubi ad Ibahim [25], defied as g () [I ()] c, (21) hee c is a positive costat. Fo c = 0.5,iteducesto Jeffeys. I case of Paeto distibutio, this is g e () ( 2 )c. (22) The posteio distibutio by usig extesio to Jeffeys is obtaied as g e () = (Z ) 2c+1 Γ( 2c+1,Z ) 2c e Z, ( < ). (23) (iii) Cojugate Pio. The cojugate as itoduced by Raiffa ad Schlaife [26], hee the ad posteio distibutios ae fom the same family, that is, the fom of the posteio desity has the same distibutioal fom as the distibutio. Fo the existece of Gii idex ad Mea icome fo the Paeto distibutio, e must take ito accout a tucated distibutio sice the adom vaiable is defied i (, ),heethecostat>1is assumed to be ko. Let have Tucated Elag distibutio [22] g c () = (β) l Γ(l,β) l 1 e β TED (β,l;), (<<, >1, β>0, l=1,2,...), hee β ad l ae the hypepaametes. The posteio desity fo is g c () = (β + Z ) +1 Γ(+l,(β+Z )) +l 1 e (β+z ) TED ((β + Z ),(+l) ;). (24) (25) The posteio desity (g c ()) follos Tucated Elag distibutio ith paametes (β + Z ) ad ( + l).

5 Advaces i Statistics 5 3. Bayesia Estimatio ude Liea Expoetial (LINEX) Loss Fuctio Usig Diffeet Pios 3.1. Bayesia Estimatos Usig Uifom Pio. Bayesia estimato of usig uifom (17) ad posteio desity (18), ude the assumptio of the LINEX loss fuctio (ef. (12))isobtaiedas Theeby, G u = 1 b log ( (Z ) 1 e Z /2 2 Γ(+1,Z ) ( (j+1)/2 j )(2b ) K Z j+1 ( 2bZ )). u = 1 b log E[e b ], E[e b ]= e b g u () d (Z = ) +1 Γ(+1,Z ) e (b+z) d Theefoe, (Z = ) Γ(+1,Z ) = Γ(+1,(b+Z )) Γ(+1,Z ) u = 1 b log (Γ(+1,(b+Z )) Γ(+1,Z ) Γ(+1,(b+Z )) (b + Z ) ( Z +1 ). b+z The Bayes estimato G of G, usig uifom is G u = 1 b log E[e bg ], (26) ( Z +1 ) ). (27) b+z The Bayes estimato M of M, usig uifom is M u = 1 b log E[e bm ], E[e bm ]=E[e bm/( 1) ] puttig t= 1 (Z = ) +1 Γ(+1,Z ) e (bm/( 1)+Z) d = (Z ) +1 e (bm+z ) Γ(+1,Z ) ( j ) t j e (bm/t+tz) dt 1 = (Z ) +1 e (bm+z ) Γ(+1,Z ) (30) (31) E[e bg ]=E[e b/(2 1) ] (Z = ) +1 Γ(+1,Z ) e (b/(2 1)+Z) d puttig t=2 1 =( Z +1 2 ) e Z /2 Γ(+1,Z ) ( j ) 2 1 t j e (b/t+tz /2) dt (By Biomial expasio) = (Z ) 1 e Z /2 2 Γ(+1,Z ) ( (j+1)/2 j )(2b ) K Z j+1 ( 2bZ ) (28) (29) (usig fomula (9) of 3.471, page 368 of Gadshtey ad Ryzhik [27] x ] 1 e β/x γx dx = 2(β/γ) ]/2 K 0 ] (2 βγ), [Re β>0, Re γ>0]hee K ] () is modified Bessel fuctio of thid kid). ( (j+1)/2 j )2(bm ) K Z j+1 (2 bmz ) (usig fomula (9) of 3.471, page 368 of Gadshtey ad Ryzhik [27] x ] 1 e β/x γx dx = 2(β/γ) ]/2 K 0 ] (2 βγ), [Re β>0, Re γ>0]hee K ] () is modified Bessel fuctio of thid kid) M u = 1 b log ((Z ) +1 e (bm+z ) Γ(+1,Z ) ( (j+1)/2 j )2(bm ) K Z j+1 (2 bmz )). The Bayes estimato P 0 of P 0, usig uifom, is P 0u = 1 b log E[e bp 0 ], E [e bp 0 ] =E[e b(1 λ 0 ) ] (32)

6 6 Advaces i Statistics (Z = ) +1 Γ(+1,Z ) e b(1 λ 0 ) e Z d, P 0u = 1 b log ( (Z ) +1 Γ(+1,Z ) e b(1 λ 0 ) 1 e Z d). (33) 3.2. Bayesia Estimatos Usig Jeffeys Pio. I case of Jeffeys (19) ad usig posteio desity (20), the Bayesia estimatos of, G, M,adP 0 ude the assumptio of the LINEX loss fuctio ae obtaied as follos: j = 1 b log E[e b ] b log ( e b g j () d) = 1 b log (Γ(,(b+Z )) Γ(,Z ) ( Z ) ), b+z Note. The expessio fo extesio of Jeffeys ca be obtaied ith some modificatios i Jeffeys ad ae listed belo: e = 1 b log (Γ( 2c+1,(b+Z )) Γ( 2c+1,Z ) G e = 1 b log ( (Z ) 2c+1 e Z /2 2 2c Γ( 2c+1,Z ) 2c M e = 1 b log ((Z ) 2c+1 e (bm+z ) Γ( 2c+1,Z ) 2c ( Z 2c+1 ) ), b+z ( 2c )( 2b (j+1)/2 ) K j Z j+1 ( 2bZ )), ( 2c )2( bm (j+1)/2 ) j Z K j+1 (2 bmz )), G j = 1 b log E[e bg ] b log ( e b/(2 1) g j () d) = 1 b log ( (Z ) e Z /2 2 1 Γ(,Z ) P 0e = 1 b log ( (Z ) 2c+1 Γ( 2c+1,Z ) e b(1 λ 0 ) 2c e Z d). (35) 1 M j = 1 b log E[e bm ] ( 1 (j+1)/2 j )(2b ) K Z j+1 ( 2bZ )), 3.3. Bayesia Estimatos Usig Cojugate Pio. Usig the Bayesia posteio desity (25),theBayesestimatosof, G, M,adP 0, ude the assumptio of the LINEX loss fuctio ae b log ( e bm/( 1) g j () d) = 1 b log ((Z ) e (bm+z ) Γ(,Z ) 1 P 0j = 1 b log E[e bp 0 ] ( 1 (j+1)/2 j )2(bm ) Z K j+1 (2 bmz )), b log ( e b(1 λ 0 ) g j () d) = 1 b log ( (Z ) Γ(,Z ) e b(1 λ 0 ) 1 e Z d). (34) c = 1 b log E[e b ] b log ( e b g c () d) = 1 b log (Γ(+l,(b+β+Z )) Γ(+l,(β+Z )) G c = 1 b log E[e bg ] b log ( e b/(2 1) g c () d) = 1 b log ( (β + Z )+l e (β+z )/2 2 +l 1 Γ(+l,(β+Z )) +l 1 ( β+z +l ) ), b+β+z ( +l 1 )( 2b (j+1)/2 ) j β+z K j+1 ( 2bZ )),

7 Advaces i Statistics 7 M c = 1 b log E[e bm ] b log ( e bm/( 1) g c () d) = 1 b log ((β + Z )+l e (bm+β+z ) Γ(+l,(β+Z )) +l 1 ( +l 1 )2( bm (j+1)/2 ) j β+z K j+1 (2 bmz )), P 0c = 1 b log E[e bp 0 ] b log ( e b(1 λ 0 ) g j () d) = 1 b log ( (β + Z ) +l Γ(+l,(β+Z )) e b(1 λ 0 ) +l 1 e (β+z) d). (36) Note: Case of Complete Sample. The Bayesia estimatos fo complete sample ca be obtaied usig oifomative s ad cojugate by simply substitutig =i the above estimatos. 4. Simulatio Study I ode to assess the statistical pefomace of these estimatos of shape paamete, Gii idex, Mea icome, ad Povety measue usig LINEX loss fuctio, a simulatio study is coduced. The estimated losses ae computed usig geeated adom samples fom Paeto distibutio of diffeet sizes. These estimated losses ae computed fo sample sizes = 20 (20), = 2.5 (1) 4.5, b = 1, = 1.5, ad m = 450. Thevalueof 0 = should be take fom Povety lie give by the Govemet of Idia i fo uba people. Fo the cojugate, the values of hypepaamete ae take as β = 0.5, l = 2; β = 2, ad l=2. The estimated losses of, G, M, adp 0 ith LINEX loss fuctio by usig oifomative (Uifom ad Jeffeys ) ad cojugate s ae tabulated i Tables 1, 2, 3,ad4,espectively. It is obseved fom the above simulatio study (ef. Tables 1, 2, 3,ad4)that (i) Bayesia estimatos ith cojugate (hypepaamete β = 0.5, l=2) pefom bette as compaed to oifomative s as it has smalle estimated loss fo, G, M,adP 0 ; (ii) i case of oifomative s, Jeffeys has less estimated loss tha uifom, hich implies that Bayesia methods ith Jeffeys ae bette; Table 1: Estimated loss fuctios fo usig LINEX loss fuctio Uifom Jeffey s Cojugate β = 0.5 β = Table 2: Estimated loss fuctios fo G usig LINEX loss fuctio Uifom Jeffey s Cojugate β = 0.5 β = (iii) a chage i the value of β o highe side does esult i a icease i the loss; the loss emais uaffected bythechageithevalueofl. I Table 5 simulatio study is take to fid estimated loss fo, G, M, adp 0 ude the assumptios of SELF usig diffeet s by cosideig small as ell as lage samples fo compaisos pupose ith the LINEX loss fuctio. Fom Table 5 ad its compaiso ith LINEX loss fuctio (ef. Tables 1, 2, 3, ad4), it is obseved that LINEX loss fuctio gives smalle loss i compaiso ith SELF fo

8 8 Advaces i Statistics Table 3: Estimated loss fuctios fo M usig LINEX loss fuctio Uifom Jeffey s Cojugate β = 0.5 β = Table 4: Estimated loss fuctios fo P 0 usig LINEX loss fuctio Uifom Jeffey s Cojugate β = 0.5 β = oifomative s ad cojugate fo small as ell aslagesamplesizes.whesamplesizeiceasesestimated loss deceases i all cases Choice of Hypepaametes. Siha ad Holade [28] suggested that a Bayes estimate is obust ith espect to its hypepaamete if it leads to a high (mi / max) idex of the estimate fo the vayig values of those hypepaamete. To check esults, simulatios ae doe by takig diffeet values Table 5: Estimated loss fuctios fo,g,m,adp 0 usig diffeet s ude the assumptios of SELF. Fo Fo G Fo M Fo P 0 Uifom Jeffey s Cojugate β = 0.5 β = of hypepaamete ad keepig ad fixed (ef. Tables 6 ad 7). The atio (mi / max) i case of both Gii idex ad Povety measue is close to 1 fo diffeet combiatios of l ad β idicatig theeby the Bayes estimates ae obust ith espect to hypepaametes, hich justifies the use of hypepaametes i simulatio study. 5. Coclusio Thesimulatiostudy ascaiedoutisectio 4suggests that Bayesia estimatos usig cojugate (hypepaamete β = 0.5, l = 2) pefom bette tha to oifomative s (Uifom ad Jeffeys ) i geeal. It is also obseved that LINEX loss fuctio esults i smalle loss tha the SELF fo both small ad lage samples iespective of the choice of the s take fo the Bayesia estimatos. Hece, the combiatios of cojugate ad LINEX loss esults i smalle loss tha the choice of othe to s ad squaed eo loss fuctio. Oe ca futhe ife that as sample size iceases the expected loss fuctio deceases fo all cases.

9 Advaces i Statistics 9 Table 6: Bayes estimate of Gii idex usig cojugate ( =, =3.5). β l (Mi / Max) β (Mi / Max) l Table7:BayesestimateofPovetymeasueusigcojugate( =, =3.5). β l (Mi / Max) β (Mi / Max) l Coflict of Iteests The authos declae that thee is o coflict of iteests egadig the publicatio of this pape. Ackoledgmets The authos ae thakful to the aoymous efeees ad the edito fo thei valuable suggestios ad commets. Refeeces [1] V. Paeto, Cous D Ecoomic Politique Pais,Rougeadcie,1897. [2] C. Gii, Vaiability ad Mutabiltity,C.Cuppii,Bologa,Italy, [3] J. Foste, J. Gee, ad E. Thobecke, A class of decomposable povety measues, Ecoometica, vol.52,o.3,pp , [4] B. C. Aold ad S. J. Pess, Bayesia ifeece fo Paeto populatios, Ecoometics,vol.21,o.3,pp , [5] T. S. Moothathu, Samplig distibutios of Loez cuve ad Gii idex of the Paeto distibutio, Sakhya (Statistics), Seies B,vol.47,o.2,pp ,1985. [6] P. K. Se, The hamoic Gii coefficiet ad affluece idexes, Mathematical Social Scieces, vol. 16, o. 1, pp , [7]P.M.Dixo,J.Weie,T.Mitchell-Olds,adR.Woodley, Bootstappig the Gii coefficiet of iequality, Ecology, vol. 68,o.5,pp ,1987. [8] P. Basal, S. Aoa, ad K. K. Mahaja, Testig homogeeity of Gii idices agaist simple-odeed alteative, Commuicatios i Statistics: Simulatio ad Computatio,vol.,o.2, pp , [9] E. I. Abdul-Satha, E. S. Jeevaad, ad K. R. M. Nai, Bayes estimatio of Loez cuve ad Gii-idex fo classical Paeto distibutio i some eal data situatio, Applied Statistical Sciece,vol.17,o.2,pp ,2009. [10] S.K.Bhattachaya,A.Chatuvedi,adN.K.Sigh, Bayesia estimatio fo the Paeto icome distibutio, Statistical Papes,vol.,o.3,pp ,1999. [11] R. Kass ad L. Wassema, The selectio of distibutios by fomal ules, Ameica Statistical Associatio,vol. 91, o. 431, pp , [12] J. Bege, The case fo objective Bayesia aalysis, Bayesia Aalysis,vol.1,o.3,pp.385 2,2006. [13] J. Aitchiso ad I. R. Dusmoe, Statistical Pedictio Aalysis, Cambidge Uivesity Pess, Lodo, UK, [14] J. O. Bege, Statistical Decisio Theoy Foudatios, Cocepts ad Methods, Spige, Ne Yok, NY, USA, [15]R.V.Cafield, Abayesiaappoachtoeliabilityestimatio usig a lossfuctio, IEEE Tasactio o Reliability, vol. R-19, o. 1, pp , [16] H. R. Vaia, A bayesia appoach to eal estate assessmet, i Studies i Bayesia Ecoometics ad Statistics i Hoo of LeoadJ.Savage,S.E.FiebegadA.Zelle,Eds.,pp , Noth-Hollad, Amstedam, The Nethelads, [17] A. Zelle, Bayesia estimatio ad pedictio usig asymmetic loss fuctios, JoualoftheAmeicaStatisticalAssociatio,vol.81,o.394,pp ,1986. [18] R. Calabia ad G. Pulcii, A egieeig appoach to Bayes estimatio fo the Weibull distibutio, Micoelectoics Reliability,vol.34,o.5,pp ,1994. [19] A. P. Basu ad N. Ebahimi, Bayesia appoach to life testig ad eliability estimatio usig asymmetic loss fuctio, Statistical Plaig ad Ifeece,vol.29,o.1-2,pp , [20] W. M. Afify, O estimatio of the expoetiated Paeto distibutio ude diffeet sample schemes, Applied Mathematical Scieces,vol.4,o.8,pp.393 2,2010. [21] A. C. Cohe, Maximum likelihood estimatio i the Weibull distibutio based o complete ad o cesoed samples, Techometics, vol. 7, pp , 1965.

10 10 Advaces i Statistics [22] A. Gaguly, N. K. Sigh, H. Choudhui, ad S. K. Bhattachaya, Bayesia estimatio of the Gii idex fo the PID, Test, vol. 1, o.1,pp ,1992. [23] P. S. Laplace, Theoie Aalytique des Pobabilities, Veuve Coucie, Pais, Face, [24] H. Jeffeys, A ivaiat fom fo the pobability i estimatio poblems, Poceedigs of the Royal Society. Lodo, Seies A: Mathematical, Physical ad Egieeig Scieces, vol. 186, pp , [25] H. S. Al-Kutubi ad N. A. Ibahim, Bayes estimato fo expoetial distibutio ith extesio of Jeffey ifomatio, Malaysia Mathematical Scieces, vol.3,o.2,pp , [26] H. Raiffa ad R. Schlaife, Applied Statistical Decisio Theoy, Divisio of Reseach, Gaduate School of Busiess Admiistatio, Havad Uivesity, [27] I. S. Gadshtey ad I. M. Ryzhik, Tables of Itegals, Seies ad Poducts, Uited States of Ameica, 7th editio, [28] S. K. Siha ad H. A. Holade, O the samplig distibutios of Bayesia estimatos of the Paeto Paamete ith pope ad impope s ad associated goodess of fit, Tech. Rep. #103, Depatmet of Statistics, Uivesity of Maitoba, Wiipeg, Caada, 1980.

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