A Bivariate Distribution with Conditional Gamma and its Multivariate Form

Transcription

1 Joural of Moder Appled Statstcal Methods Volue 3 Issue Artcle 9-4 A Bvarate Dstrbuto wth Codtoal Gaa ad ts Multvarate For Sue Se Old Doo Uversty, sxse@odu.edu Raja Lachhae Texas A&M Uversty, raja.lachhae@tauk.edu Norou Dawara Old Doo Uversty, dawara@odu.edu Follow ths ad addtoal works at: Part of the Appled Statstcs Coos, Socal ad Behavoral Sceces Coos, ad the Statstcal Theory Coos Recoeded Ctato Se, Sue; Lachhae, Raja; ad Dawara, Norou (4) "A Bvarate Dstrbuto wth Codtoal Gaa ad ts Multvarate For," Joural of Moder Appled Statstcal Methods: Vol. 3 : Iss., Artcle 9. DOI:.37/jas/ Avalable at: Ths Regular Artcle s brought to you for free ad ope access by the Ope Access Jourals at DgtalCoos@WayeState. It has bee accepted for cluso Joural of Moder Appled Statstcal Methods by a authorzed edtor of DgtalCoos@WayeState.

2 Joural of Moder Appled Statstcal Methods Noveber 4, Vol. 3, No., Copyrght 4 JMASM, Ic. ISSN A Bvarate Dstrbuto wth Codtoal Gaa ad ts Multvarate For Sue Se Old Doo Uversty Norfolk, VA Raja Lachhae Texas A&M Uversty - Kgsvlle, TX Norou Dawara Old Doo Uversty Norfolk, VA A bvarate dstrbuto whose argal are gaa ad beta pre dstrbuto s troduced. The dstrbuto s derved ad the geerato of such bvarate saple s show. Exteso of the results are gve the ultvarate case uder a jot depedet copoet aalyss ethod. Sulated applcatos are gve ad they show cosstecy of our approach. Estato procedures for the bvarate case are provded. Keywords: Gaa dstrbuto, Gaa fucto, Beta fucto, Beta dstrbuto, geeralzed Beta pre dstrbuto, coplete gaa fucto Itroducto The gaa ad beta dstrbutos are the two ost cooly used dstrbuto whe t coes to aalyzg skewed data. Sce Kbble (94), the bvarate gaa has gaed cosderable atteto. The ultvarate for of the gaa has bee proposed Johso et al. (997) ad by ay other authors, but there s o ufyg forulato. Eve the ultvarate expoetal faly of dstrbutos, there s o kow ultvarate gaa (Joe, 997 ). The splest of the ultvarate cases, the bvarate gaa dstrbuto, s stll rasg debates, ad has bee proposed Balakrsha ad La (9). The argal destes of the bavarate gaa ca soetes belog to other class of dstrbutos. A odfed verso of Nadarajah (9) bvarate dstrbuto wth Gaa ad Beta argals s cosdered, ad a codtoal copoet to the odelg s brought to accout. Kotz et al (4) proposed a bvarate gaa expoetal dstrbuto wth gaa ad Pareto dstrbuto as argals. I ths artcle, a bvarate gaa dstrbuto Sue Se s a Graduate Studet Assstat the Departet of Matheatcs ad Statstcs. Eal h at sxse@odu.edu. Dr. Lachhae s a Assstat Professor the Departet of Matheatcs. Eal h at raja.lachhae@tauk.edu. Dr. Dawara s a Assocate Professor the Departet of Matheatcs ad Statstcs. Eal h at dawara@odu.edu. 69

3 A BIVARIATE DISTRIBUTION WITH CONDITIONAL GAMMA wth gaa ad beta pre as argal dstrbutos s defed. By cludg the depedece structure, ore flexblty s added. Cosder two rado varables X, detfed as the coo easure, ad Y related to X, ad assug that X s a gaa rado varable wth paraeters α ad β ad the dstrbuto of Y X s a gaa rado varable wth paraeters a ad X. The frst secto followg ths troducto shows the bvarate dstrbuto wth the codtoal gaa. I the ext secto, Propertes, the a propertes of the bvarate codtoal gaa dstrbuto are gve. Exteso to the ultvarate settg s gve the ext secto, followed by a developet of coputatoal aspects the ferece. The calculatos ths paper volve several specal fuctos, cludg the coplete gaa fucto defed by x a, x t a e t dt, a ad the copleetary gaa fucto defed as Also, the beta fucto s defed as a t a, x t e dt, wth a a, x. a b b a a b Ba, b t t dt, for a, b, ad x postve real values. For x [,], α > ad β >, the beta dstrbuto ca be defed as f x x x B, Model Buldg ad Desty fuctos Let X be a gaa rv s wth shape ad rate paraeters deoted by α ad β, respectvely. The probablty desty fucto (pdf) of X s gve by 7

4 SEN ET AL. x fx x x e x, () where α > ad β >. May authors have developed structural odels wth the uderlyg gaa dstrbuto. Cosder aother rado varable Y such that the dstrbuto of the rado varable Y gve a realzato of X at x s a gaa wth the paraeters a ad x. That s the desty of Y X s gve by x f y x y e y a a xy YX, a () where a > ad x > are the shape ad rate paraeters respectvely. So the jot desty of the rado varables defed above s gve by the expresso below f x, y = f y X = x * f ( x) X, Y Y X = x X a a x y x xy = e e, x > ad y >, ( a) ( ) (3) wth paraeters α >, β > ad a >. Equato (3) tegrates to, so ths s a legtate dstrbuto. Fgure shows the plot of the jot dstrbuto defed Equato (3) for dfferet values of α, β ad a. Thus the cuulatve dstrbuto of the rado varable X ad Y s XY,, x a, xy a F x, y, x ad y. (4) 7

5 A BIVARIATE DISTRIBUTION WITH CONDITIONAL GAMMA a) α =.5, β =.3, a = 3. b) α =.5, β = 3.3, a =. Fgure. Jot Probablty Desty Fucto of (X,Y) Propertes The a propertes of the dstrbuto as defed (3), such as the argal destes, ther oets, ther product products ad covarace, are derved here. Margal Desty ad Moets: The argal desty of X s gve by (). Margal desty of Y s gve by the theore below. Theore : If the jot desty of (X,Y) s gve (3), the the argal desty of Y s gve by a y f y y, y, a ad a B a, y (5) 7

6 SEN ET AL. Proof. The argal desty of Y s gve by y a a xxy f y y x e dx a a a z y z e dz, wth z x a y a y a a y a a y Probablty desty fucto of Y s a specal for of Geeralzed Beta pre desty wth shape paraeter ad scale paraeter β. Fgure descrbes ts pdf for dfferet values of α. Probablty desty of geeralzed beta pre dstrbuto wth scale p ad shape q s gve by f y;,, p, q p p x x p q q (6) qb, Let T be a rado varable such that T ~ Beta(a, α). The gve by (5). Y T, has desty T Theore : Let Y be a rado varable wth a pdf gve (5). The th oet of the rado varable Y exsts oly f α >. Proof: Fro the prevous theore t ca be see that f T: Beta(a, α) ad t Y, the the desty of Y wll be sae as defed (5). Ad the th oet t of Y s 73

7 A BIVARIATE DISTRIBUTION WITH CONDITIONAL GAMMA 74,,,, provded, a a t E Y E t t t t Beta a t t t Beta a Beta a Beta a wth, a Beta a a The choce of α > s ade so that E[Y] ad Var[Y] wll both exst. Fgure. The Probablty Desty Fucto of Y as defed (5)

8 SEN ET AL. Product Moets Theore 3: The product oet of the rado varables (X,Y) assocated wth the pdf defed (3) ca be expressed as a a E X Y for a,, ad Proof: For > ad > oe ca wrte a a xxy E X Y x y dxdy a x a a xy y e dydx e x x y e a a x x e a a a x a x e x a a a a provded the tegrals exst. Now for the th product oet by choosg = the above expresso, oe ca wrte the product oet as a E X Y a Note that the product oet depeds oly o a. dx 75

9 A BIVARIATE DISTRIBUTION WITH CONDITIONAL GAMMA Covarace Matrx Wth the desty of X ad Y as gve Equatos () ad (5), respectvely, the varace-covarace atrx of X ad Y s gve by the followg theore Theore 4: the Deote the Varace-Covarace atrx of X ad Y by Cov(X,Y), Cov X, Y a a a a (7) Proof: Usg Theore, the varace ad expectato of the rado varable Y ca be coputed Var Y E Y E Y a a a a a (8) Equato () ples that the dstrbuto of X s a Gaa dstrbuto wth shape α ad rate β. So varace of X s gve by Var X (9) Now the covarace betwee X ad Y ca be wrtte as 76

10 SEN ET AL., Cov X Y E XY E X E Y E XE Y X E X E Y a E X E X E Y Y X Gaa a x X a a a as :, () Usg the Equatos (8), (9) ad () the result follows. Note that the covarace betwee X ad Y exsts oly whe α, ad s postve whe α <. Varace of Y oly exsts whe α >. Multvarate Exteso Case Cosder the ultvarate case of the odel: take + rado varables as follows: X : Gaa, X X : Gaa a, b x X X : Gaa a, b x X X : Gaa a, b x where X X ad X j X are depedet copoets for j ad (, j) {,,, }. The usg the sae arguet as Propertes, the jot depedet copoet odel s bult ad the argal desty fucto for each rado varable X s derved. I geeral, the desty fucto of X s gve by a a a b x a bx f x,for,,..., a () Usg the depedece assupto of the above odel, the jot desty of X, X,, X s the derved. The derved jot desty wll be of the for 77

11 A BIVARIATE DISTRIBUTION WITH CONDITIONAL GAMMA,,..., f x x x f x f x x The desty of the jot dstrbuto (X, X,, X ) ad ts varace covarace expresso are derved ext. For the desty of (X, X,, X ), the tegrato of the jot desty wth respect to the varable X s eeded. Desty Fucto To derve the desty fucto, the tegral below s coputed,..., f x x f x f x x dx () Ad solvg the tegral (), the jot desty s as follows f x,..., x a a a b x a bx a (3) where x >, a > for all =,,,, ad α >, β >. I the dstrbuto obtaed fro (3), f the choces of β = ad b = for all =,,, are ade, the the verted Drchlet dstrbuto s obtaed. The applcato of ths dstrbuto ca be foud ay places the lterature. Tao ad Cutta (965) troduced ths type of dstrbuto ad dscussed about ther applcatos. Covarace The covarace betwee X ad X j for j s derved Theore 5. Theore 5: If the rado varables X, X,, X have the desty fucto defed (3), the the covarace betwee X ad X j for j s gve by the expresso below 78

12 SEN ET AL. aa j Cov X, X for j, b ca be equal to b bb j j j Proof: Usg the sae arguets Theore, the th oets of X are derved. Based o the desty of X defed by () E X a a Fro (3) ths useful detty s obtaed b wth (4) a x a a a b a bx (5) Usg the detty (5), the (,,, ) th xed oet s gve as E X... X a b a (6) provded > =. I partcular, the covaraces betwee X ad Xj, for =,,,, s as Cov X, X j = bb j aa j Note that the covarace betwee X ad X for =,,, s also derved as 79

13 A BIVARIATE DISTRIBUTION WITH CONDITIONAL GAMMA, E X E X X E X E X Cov X X E X X E X E X a E X E X E X X X Gaa a b x Xb a X b b a b a as :, a (7) Bvarate cases wll reduce to Equato (). Lkelhood ad Estato for Bvarate Case I ths secto, the axu lkelhood estato process ad Fsher forato atrx for the bvarate odel are troduced. Statstcal aalyss software (SAS) s used to geerate data ad R s used to get the axu lkelhood estates (MLEs). Log lkelhood Let (x, y ), for =,,,, be a saple of sze fro the bvarate gaa dstrbuto as defed Equato (3). The, the log lkelhood fucto s, ;,, log log log L x y a x x log log a x y x y a (8) The frst order dervatves of the log lkelhood wth respect to the three paraeters are Log(,, a) = log ( ) log ( x ) ( ) (9) = 8

14 SEN ET AL. Log(,, a) = x () Log(,, a) = x log ( x ) ( a ) () a = d where ( x) l( ( x)) s the Dgaa fucto. dx Solvg above Equatos (9-) sultaeously, the MLEs of the paraeters ca be forulated. As the MLEs are ot a closed for, a R code s developed to get the estates. Fsher Iforato Matrx The Fsher forato atrx g s gve by the expectato of the covarace of partal dervatves of the log lkelhood fucto. Let (θ, θ, θ 3 ) = (α, β, a); the the copoets of the Fsher forato atrx are gve by g j log f ( x, y,,, a) log( ) Hece, g log( ) j () Ivertg the fsher forato atrx, the asyptotc stadard errors of the axu-lkelhood estates ca be obtaed. Exaple Usg Sulated Data A uber of sulatos are perfored to evaluate the statstcal propertes ad the estato are coputed usg axu lkelhood ethod. Because of the coplexty of the target desty ad of the lkelhood, there s o closed for of the estators. Effectve saple szes wll be drectly pactg the estates. R 8

15 A BIVARIATE DISTRIBUTION WITH CONDITIONAL GAMMA progra s used to do the optzato, but SAS 9.3 verso s used to sulate data wth saples of szes = ad 5. Accordgly, for each set of paraeters ad saple sze, X s sulated fro a gaa dstrbuto wth paraeters α ad β. The, for each X, geerate X based o X accordg to Equato (3) wth the sae value of a. The sulato results preseted uder the table gve the estates of the paraeters. Fgure 3 gves the plot of log lkelhood ad shows the uqueess of the soluto estate for each paraeter at saple sze. The results show that the larger the saple sze, the ore accurate the estates are. A plot of the estates versus saple sze s gve Fgure 4. Sulato results Table. Estato of paraeters for dfferet saple szes Estates (SE) Actual Values = = 5 α =.5 ˆ =.36 (.89) ˆ β =.3 ˆ =.6 (.474) =. (.75) ˆ =.6 (.78) a = 3. â = 3.9 (.44) â = 3.75 (.77) α = 6.3 β =. ˆ = 6.86 (.53) ˆ =.5 (.848) ˆ = 5.87 (3.43) ˆ =.757 (.3) a =. â =.69 (.3) â =.8 (.45) 8

16 SEN ET AL. Fgure 3. MLE estates of paraeters for a saple of Fgure 4. Paraeter estato for creasg saple sze 83

17 A BIVARIATE DISTRIBUTION WITH CONDITIONAL GAMMA Cocluso I ths paper, a bvarate codtoal gaa ad ts ultvarate for are proposed. Ther assocated propertes are preseted ad the sulato studes have show sgfcat proveet the paraeter estatos, takg to accout the tracorrelato ad depedece aog the observed xg rado varables. Whle our proposed odel process s guded by a foral ft crtera, Bayesa approach s aother opto to detere the paraeters. However, the proposed approach has the advatage of gvg a sple pleetato for xed outcoe data. Refereces Balakrsha, N. & La, C. D. (9). Cotuous bvarate dstrbutos (d ed.). New York: Sprger. Des, J. E. & Schabel, R. B. (983). Nuercal ethods for ucostraed optzato ad olear equatos. Upper Saddle Rver, NJ: Pretce Hall. Dawara, N. (8). Multvarate gaa dstrbuto based o lear relatoshp. Joural of Appled Statstcal Scece, 7(4), -. Joe, H. (997). Multvarate odels ad depedece cocepts. Lodo: Chapa ad Hall. Johso, N., Kotz, S. & Balakrsha, N. (997). Cotuous ultvarate dstrbutos. New York: Joh Wley. Kbble, W. F. (94). A two-varate gaa type dstrbuto, Sakhya, 5, Kotz, S., Balakrsha, N. & Johso, N. L. (), Cotuous Multvarate Dstrbutos Volue : Models & Applcatos. New York: Joh Wley. Nadarajah, S. (7). The bvarate gaa expoetal dstrbuto wth applcato to drought data. Joural of Appled Matheatcs ad Coputg, 4(-), -3. Nadarajah, S. (9). A bvarate dstrbuto wth gaa ad beta argals wth applcato to drought data. Joural of Appled Statstcs, 36(3), Tao, G. G. & Cutta, I. (965). The verted Drchlet dstrbuto wth applcatos. Joural of the Aerca Statstcal Assocato, 6(3),