Concomitants of Dual Generalized Order Statistics from Farlie Gumbel Morgenstern Type Bivariate Inverse Rayleigh Distribution

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1 J Stat Appl Po 4, No, Jounal of Statitic Application & Pobability An Intenational Jounal Concomitant of Dual Genealized Ode Statitic fom Falie Gumbel ogenten Type Bivaiate Invee Rayleigh Ditibution Haeeb Atha and Nayabuddin Depatment of Statitic and Opeation Reeach, Aligah ulim Univeity, Aligah- India Received: 7 Jul 4, Revied: 3 Nov 4, Accepted: Dec 4 Publihed online: a 5 Abtact: Dual genealized ode tatitic contitute a unified model fo decending ode andom vaiable, like evee ode tatitic and lowe ecod value In thi pape, we have conideed concomitant of dual genealized ode tatitic fo the Falie-Gumbel-ogenten type bivaiate invee Rayleigh ditibution and ingle and joint ditibution of concomitant of dual genealized ode tatitic ae obtained Futhe, Single and poduct moment ae deived and ecuence elation between moment ae etablihed Alo eult ae deduced fo ode tatitic and lowe ecod value and ome computation wok ae caied out Keywod: Dual genealized ode tatitic; lowe ecod value; concomitant; ingle and poduct moment AS Subject Claification: 6E5, 6G3 Intoduction The mot impotant ue of concomitant aie in election pocedue when k <n individual ae choen on the bai of thei X-value Then the coeponding Y -value epeent pefomance on an aociated chaacteitic Fo example, X might be the coe of a candidate on a ceening tet and Y the coe on a late tet Statement ae made uch a if a tudent i good in mathematic then he will be poo in language If the aveage coe in language of tudent good in mathematic i lowe than aveage coe in language of All tudent, then the tatement may be jutified To tet hypothee of thi kind we need the ditibution of the concomitant of ode tatitic Thu to tudy a vaiable aociated with anothe, ditibution of concomitant of ode tatitic ae uually cucial The Falie Gumbel ogenten FG family of bivaiate ditibution ha found extenive ue in pactice Thi family i chaacteized by the pecified maginal ditibution function F X x and F Y y of andom vaiable X and Y epectively and a paamete α, eulting in the bivaiate ditibution functiond f i given by F X,Y x,y=f X xf Y y+α F X x F Y y], with the coeponding pobability denity functionpd f f X,Y x,y= f X x f Y y+αf X x F Y y] Hee, f X x and f Y y ae the maginal pd f of f X,Y x,y The paamete α i known a the aociation paamete; the two andom vaiable X and Y ae independent when α i zeo Such a model wa oiginally intoduced by ogenten 956 and invetigated by Gumbel 96 fo exponential maginal The geneal fom in i due to Falie 96 and Coeponding autho haeebatha@hotmailcom c 5 NSP Natual Science Publihing Co

2 54 Haeeb Atha, Nayabuddin: Concomitant of Dual Genealized Ode Statitic Johnon and Kotz 975 The admiible ange of aociation paamete α i α and the Peaon coelation coefficient ρ between X and Y can neve exceed /3 The conditional d f and pd f of Y given X, ae given by and c f Beg and Ahanullah, 7 F Y X y x=f Y y+α F X x F Y y] 3 f Y X y x= f Y y+α F X x F Y y] 4 In thi pape, we conide FG type bivaiate invee Rayleigh ditibution with pd f and coeponding d f The conditional pd f of Y given X i fx,y= 4θ θ θ x 3 y 3 e x e θ y +α e θ x e θ y ], <x,y<, α 5 Fx,y=e θ x e θ y +α e θ x e θ y ], <x,y<, α 6 fy x= θ The maginal pd f and d f of X ae fx= θ θ x 3 e y 3 e θ y +α e θ x e θ y ], <x,y<, α 7 x, <x<, θ >, 8 epectively Fx=e θ x, <x<, θ >, 9 Let X be a continuou andom vaiable with d f Fx and the pd f fx; x α,β Futhe, Let n N,k, m = m,m,,m n R n, = n j= m j uch that γ = k + n + fo all,,,n Then X d,n, m,k, =,,,n ae called dual genealized ode tatitic dgo if thei pd f i given by Bukchat et al 3] n n k γ j Fxi ] m Fxn i fx i ] k fxn j= on the cone F >x x x n > F If m i =, i =,,,n, k =, then X d,n,m,k educe to the n + th ode tatitic X n +:n fom ample X,X,,X n and when m=, then X d,n,m,k educe to th, k lowe ecod value The pd f of X d,n,m,k i and joint pd f of X d,n,m,k and X d,n,m,k, i < n i f Xd,n,m,k = C! Fx]γ fxg m Fx, f Xd,,n,m,kx,y= C!! Fx]m fx g m Fx whee h m Fy hm Fx ] Fy] γ fy, α x<y β, C = γ i, γ i = k+n im+ c 5 NSP Natual Science Publihing Co

3 J Stat Appl Po 4, No, / wwwnatualpublihingcom/jounalap 55 and h m x= m+ xm+, m logx, m= g m x=h m x h m, x, Let X i,y i,i =,,,n, be n pai of independent andom vaiable fom ome bivaiate population with d f Fx,y If we aange the X vaiate in decending ode a X d,n,m,k X d,n,m,k X d n,n,m,k, then Y vaiate paied not neceaily in decending ode with thee dgo ae called the concomitant of dgo and ae denoted by Y,n,m,k],Y,n,m,k],,Y n,n,m,k] The pd f of Y,n,m,k], the th concomitant of dgo, i given a g,n,m,k] y= The joint pdf of Y,n,m,k] and Y,n,m,k] < n i g,,n,m,k] y,y = f Y X y x f Xd,n,m,kx dx 3 x f Y X y x f Y X y x f Xd,,n,m,kx,x dx dx, 4 whee f Xd,,n,m,kx i the joint pd f of X d,n,m,k and X d,n,m,k, < n An excellent eview on concomitant of ode tatitic i given by Bhattachaya 984 and David and Nagaaja 998 Balaubamanian and Beg 996, 997, 998 tudied the concomitant fo bivaiate exponential ditibution of ahall-olkin, ogenten type bivaiate exponential ditibution and Gumbel bivaiate exponential ditibution and gave the ecuence elation between ingle and poduct moment of ode tatitic Begum and Khan 997, 998, tudied the concomitant of ode tatitic fo Gumbel bivaiate Weibull ditibution, bivaiate Bu ditibution, ahall and Olkin bivaiate Weibull ditibution and etablihed expeion fo ingle and poduct moment Ahanullah and Beg 6 tudied the concomitant fo Gumbel bivaiate exponential ditibution and deived the ecuence elation between ingle and poduct moment of genealized ode tatitic Pobability Denity Function of Y,n,m,k] Fo the FG type bivaiate invee Rayleigh ditibution a given in 5, uing 7 and in 3, the pd f of th concomitant of dgo Y,n,m,k] fo m i given a: Setting x = t in, we get = = C 4θ θ g,n,m,k] y=!m+ y 3 C θ!m+ y 3 C θ!m+ y 3 = e θ y i= i i θ γ i x 3 e x +α e θ x e θ y ]dx θ e y i= θ e y i= k m+ + n +i+ α C θ!m+ y 3 θ e y i= i + α i γ i i i γ i k m+ + n +i k+ m+ + n +i k i B i m+ + n +i, e θ ] y γ i + e θ ] y 3 k k+ } +α B m+ + n +i, B m+ + n +i, e θ ] y 4 c 5 NSP Natual Science Publihing Co

4 56 Haeeb Atha, Nayabuddin: Concomitant of Dual Genealized Ode Statitic Fo eal poitive p, c and a poitive intege b, we have b a= a b Ba+ p,c= Bp,c+b 5 a Thu uing 5 in 4, we get C θ g,n,m,k] y=!m+ y 3 which afte implification yield g,n,m,k] y= θ y 3 θ e y θ e y B k m+ + n, k k+ } +α B m+ + n, B m+ + n, e θ ] y, 6 +α e θ The expeion 7 may alo be epeented a y g,n,m,k] y= fy+α Fy fy }] 7 } 8 The above expeion fo g,n,m,k] y doe not depend on Fx at all Obeving that Fy fy i the pd f of Y, the econd ode tatitic of a andom ample of ize two of the Y vaiate we find that the ditibution of the th concomitant depend only on the maginal ditibution of Y and the ditibution of Y : Now fom 8, we have g,n,m,k] y= f : y α It may be veified that g,n,m,k]ydy= + ] ] f : y f : y 9 γ i Remak : Set m=,k=in 7, to get the pd f of th concomitant of ode tatitic fom FG type bivaiate invee Rayleigh ditibution a Replace n + by, we get g n +:n] y= θ g :n] y= θ θ y 3 e y θ y 3 e y +α e θ +α e θ y y n + n+ }] n+ Remak : At m = in 7, we get the pd f of th concomitant of k th lowe ecod value fom FG type bivaiate invee Rayleigh ditibution a g,n,,k] y= θ θ y 3 e y +α e θ y + k k+ }] }] 3 oment of Y,n,m,k] In thi ection,we deive the moment of Y,n,m,k] fo FG type bivaiate invee Rayleigh ditibution by uing the eult of the peviou ection In view of 7, the moment of Y,n,m,k] i given a E Y a,n,m,k] ]=θ y a θ y 3 e y +α e θ y }] dy 3 c 5 NSP Natual Science Publihing Co

5 J Stat Appl Po 4, No, / wwwnatualpublihingcom/jounalap 57 If we put y = z in 3, then we have = θ z a e θ z +α e θz }] dz 3 Note that x α e x dx= Γα 33 Uing 33 in 3, we get afte implification ] E Y a =θ,n,m,k] a a Γ +α a }] 34 Remak 3: At m =, k = in 34, we get the moment of concomitant of ode tatitic fom FG type bivaiate invee Rayleigh ditibution a ] E Y a =θ n +:n] a a Γ +α a n + n+ }] Replace n + by, we get E Y a :n] ] =θ a a Γ +α a }] n+ Remak 3: Setting m = in 34, we get the moment of concomitant of k th lowe ecod tatitic fom FG type bivaiate invee Rayleigh ditibution a ] E Y,n,,k] a =θ a a Γ +α a k }] k+ 4 Relation Between Single oment of Concomitant In thi ection we hall peent eveal ecuence elation between pd f, moment and mg f of concomitant The following elation between pd f can be een in view of expeion 9 g,n,m,k] y g,n,m,k] y=α g,n,m,k] y g,n,m,k] y=α g,n,m,k] y g,n,m,k] y=α Now on application of 4 to 43, we have the following Theoem: + ] ] f : y f : y 4 γ i + ] ] f : y f : y 4 γ i+ + ] ] f : y f : y 43 γ i+ Theoem 4: Let n N, m R, n Fo a bivaiate andom vaiable X,Y having pd f 5, the following ecuence elation between moment of concomitant ae valid:,n,m,k] µa,n,m,k] = α,n,m,k] µa,n,m,k] = α + ] ] γ :] µa 44 :] i + ] ] γ :] µa 45 :] i+ c 5 NSP Natual Science Publihing Co

6 58 Haeeb Atha, Nayabuddin: Concomitant of Dual Genealized Ode Statitic,n,m,k] µa,n,m,k] = α + ] ] γ :] µa :] 46 i+ Remak 4: At m=,k= in Theoem 4, we get ecuence elation between the moment of concomitant of ode tatitic fom FG type bivaiate invee Rayleigh ditibution a n +: n] α ] µa n +: n] = n+ :] µa :] 47 n +: n] µa n +: n ] n +: n] µa n +: n ] Now afte eplacing n + by, we get αn + ] = nn+ :] µa 48 :] α ] = nn+ :] µa 49 :] : n] α ] µa : n] = n+ :] µa :] 4 : n] α ] µa = : n ] nn+ :] µa :] 4 : n] µa : n ] αn ] = nn+ :] µa 4 :] Remak 4: At m = in Theoem 4, we get ecuence elation between the moment of concomitant of ecod tatitic fom FG type bivaiate invee Rayleigh ditibution a k k ] ],n,,k] µa,n,,k] = α k+ k+ :] µa 43 :] Theoem 4: Let n N, m R, n Fo a bivaiate andom vaiable X,Y having pd f 5, the following ecuence elation between mg f of concomitant ae valid,n,m,k] t,n,m,k] t=α,n,m,k] t,n,m,k] t=α,n,m,k] t,n,m,k] t=α + ] ] γ :] t :] t 44 i :] t :] t + γ i+ ] :] t :] t ] 45 + γ i+ ] ] 46 Remak 43: At m =,k = in Theoem 4, we get ecuence elation between the mg f of concomitant of ode tatitic fom FG type bivaiate invee Rayleigh ditibution a n +: n] t n +: n] t= α ] n+ :] t :] t 47 n +: n] t n +: n ] t= αn + ] nn+ :] t :] t 48 c 5 NSP Natual Science Publihing Co

7 J Stat Appl Po 4, No, / wwwnatualpublihingcom/jounalap 59 n +: n] t n +: n ] t= α ] nn+ :] t :] t 49 If we eplace n + by, we get : n] t : n] t= α ] n+ :] t :] t 4 : n] t : n ] t= α ] nn+ :] t :] t 4 : n] t : n ] t= αn ] nn+ :] t :] t 4 Remak 44: At m= in Theoem 4, we get ecuence elation between the mg f of concomitant of ecod tatitic fom FG type bivaiate invee Rayleigh ditibution a k k ],n,,k],n,,k] = α ] k+ k+ :] :] 43 Table 4 : ean of the concomitant of ode tatiticremak 3 n α =, θ = α = 5, θ = 5 α = 5, θ = c 5 NSP Natual Science Publihing Co

8 6 Haeeb Atha, Nayabuddin: Concomitant of Dual Genealized Ode Statitic Table 4 : ean of the concomitant of lowe ecod valueremak 3 α = 5, θ = 5 α =, θ = α = 5, θ = Joint Pobability Denity Function of Y,n,m,k] and Y,n,m,k] Fo the FG type bivaiate invee Rayleigh ditibution a given in 5, uing 7 and in 4, the joint pd f of th and th concomitant of dgo Y,n,m,k] and Y,n,m,k] fo m i whee, g,,n,m,k] y,y = C!!m+ θ x Ix,y = i= By etting x = t in 5, we get afte implification y 3 θ y 3 i+ j j= i j e θ y e θ y θ x 3 e θ +i jm+ x +α e θ x e θ y ]Ix,y dx 5 θ γ j e x Ix,y = γ j Subtituting the value of Ix,y in 5,we get θ x 3 e θγ j x +α e θ x e θ y ]dx 5 + α e θ θ γ j y e x γ j θ C θ e θ y y g,,n,m,k] y,y = 3 y 3 e θ y!!m+ +α θ x 3 e θγ i x e θ Let x = t in 54, we get afte implification g,,n,m,k] y,y = θ y 3 θ y 3 e θ y y e θ y i= e θ γ j + x ] 53 γ j + i+ j j= i θ + α e y γ j e θ x γ j γ j + e θ ] x dx 54 +α e θ y e θ y ] j c 5 NSP Natual Science Publihing Co

9 J Stat Appl Po 4, No, / wwwnatualpublihingcom/jounalap 6 +α e θ y It may be veified that g,,n,m,k]y,y dy dy = e θ } y + γ i } }] 55 Remak 5: By etting m =, k = in 55, we get the joint pd f of two concomitant of ode tatitic and at m =, we have the joint pd f of two concomitant of k th lowe ecod value fo FG type bivaiate invee Rayleigh ditibution 6 Poduct oment of Two Concomitant Y,n,m,k] and Y,n,m,k] Poduct moment of two concomitant Y,n,m,k] and Y,n,m,k] i given a E Y a,n,m,k] Yb = y a,n,m,k] yb g,,n,m,k]y,y dy dy 6 In view of 55 and 6, we have E Y a,n,m,k] Yb = y b 3,n,m,k] e θ y Let y = z in 6and implify, we have = Γ a θ a +α e θ y +α e θ y y a 3 e θ y + γ i γ i y b 3 e θ y + +α + e a θ } y e θ } y +α a + 4 e θ e θ y + γ i } }]dy } dy 6 y + γ i } }] dy 63 Set y = z in 63, to get E Y = a,n,m,k] Yb Γ a,n,m,k] Γ b θ a+b +α a b +4 + γ i } +α + a b }] } 64 Remak 6: Set m =, k = in 64, to get poduct moment of concomitant ode tatitic fom FG type bivaiate invee Rayleigh ditibution a E Y a n +:n] Yb n +:n] = Γ a Γ b θ a+b +α a b c 5 NSP Natual Science Publihing Co

10 6 Haeeb Atha, Nayabuddin: Concomitant of Dual Genealized Ode Statitic n +n + n + n + } +4 n+n+ n+ n+ } +α a b + By eplacing n + by and n + by, we have E Y a :n] Yb :n] n + n+ }] = Γ a Γ b θ a+b +α a b +4 + n+n+ } n+ n+ } +α a b }] + n+ Remak 6: At m = in 64, we get the poduct moment of concomitant of k th lowe ecod tatitic fom FG type bivaiate invee Rayleigh ditibution a E Y a,n,,k] Yb,n,,k] = Γ a Γ b θ a+b k k k +4 k+ k+ } +α a b k + + α a b k+ k+ } }] 7 Relation Between Poduct oment of Two Concomitant In thi ection we hall peent ecuence elation between joint pd f, poduct moment and mg f of two concomitant Fom 55, we have g,,n,m,k] y,y = fy fy α f : y f : y ] fy α f : y f : y ] fy + α f : y f : y ] f : y f : y ] Now uing Fy= Fy Fy in 7, we get ] + 4 g,,n,m,k] y,y = fy fy + α f :y f : y ] fy α f :y f : y ] fy + α 4 f :y f : y ] f : y f : y ] ] ] γ i } 7 ] + γ i } 7 c 5 NSP Natual Science Publihing Co

11 J Stat Appl Po 4, No, / wwwnatualpublihingcom/jounalap 63 g,,n,m,k] y,y g,,n,m,k] y,y = α } f : y f : y ] f : y f : y ] 73 Uing 73, the ecuence elation fo joint moment geneating function of Y,n,m,k] and Y,n,m,k] i given a,,n,m,k] t,t,,n,m,k] t,t = α } : t : t ] : t : t ] 74 Remak 7: Set m =, k = in 74, to get the ecuence elation fo joint mg f of two concomitant of ode tatitic fom FG type bivaiate invee Rayleigh ditibution n +,n +:n] t,t n +,n +:n] t,t = α Replacing n + by and n + by, we get,:n] t,t, :] t,t n+ :t : t ] : t : t ] = α n+ :t : t ] : t : t ] Remak 7: By etting m = in 74, we get the ecuence elation fo joint mg f of two concomitant of k th lowe ecod tatitic fom FG type bivaiate invee Rayleigh ditibution a,,n,,k] t,t,,n,,k] t,t = α k k } : t : t ] : t : t ] k+ k+ Now uing 73, the ecuence elation fo poduct moment of Y,n,m,k] and Y,n,m,k] i given a µ a,b,,n,m,k] µa,b,,n,m,k] = α + } γ :] µa ] µb :] :] µb ] 75 :] i Remak 73: Set m =, k = in 75, to get the ecuence elation fo poduct moment of two concomitant of ode tatitic fom FG type bivaiate invee Rayleigh ditibution a µ a,b n +,n +:n] µa,b = α n +,n +:n] n+ µa :] µa ] µb :] :] µb :] ] Replace n + by and n + by, we have µ a,b,:n] µa,b, :n] = α n+ µa :] µa :] ] µb :] µb :] ] Remak 74: Set m = in 75, to get the ecuence elation fo poduct moment of two concomitant of k th lowe ecod tatitic fom FG type bivaiate invee Rayleigh ditibution a µ a,b,,n,,k] µa,b,,n,,k] = α k k+ k } k+ :] µa :] ] µb :] µb :] ] c 5 NSP Natual Science Publihing Co

12 64 Haeeb Atha, Nayabuddin: Concomitant of Dual Genealized Ode Statitic Table 7: Poduct moment between the concomitant of ode tatitic Remak 6 α = 5 θ = 5 n \ c 5 NSP Natual Science Publihing Co

13 J Stat Appl Po 4, No, / wwwnatualpublihingcom/jounalap 65 Table 7: Poduct moment between the concomitant of lowe ecod tatitic Remak 6 n α =, θ = α = 5, θ = 5 α = 5, θ = Acknowledgement The autho acknowledge with thank to the Refeee and Edito, JSAP fo thei fuitful uggetion Refeence ] Ahanullah, and Beg, I Concomitant of genealized ode tatitic fom Gumbel bivaiate exponential ditibution, J Statit Theoy and Application 6, 8-3, 6 ] Beg, I and Ahanullah, Concomitant of genealized ode tatitic fom Falie Gumbel ogenten type bivaiate Gumbel ditibution, Statitical ethodology, 7 3] Begum, A A and Khan, A H Concomitant of ode tatitic fom Gumbel bivaiate Weibull ditibution, Cal Statit Aoc Bull 47, 3-4, 997 4] Begum, A A and Khan, A H Concomitant of ode tatitic fom bivaiate Bu ditibution, J Appl Statit Sci 7 4, 55-65, 998 5] Begum, A A and Khan, A H Concomitant of ode tatitic fom ahall and Olkin bivaiate Weibull ditibution, Cal Statit Aoc Bull 5, 65-7, 6] Balaubamnian, K and Beg, I Concomitant of ode tatitic in bivaiate exponential ditibution of ahall and Olkin, Cal Statit Aoc Bull 46, 9-5, 996 7] Balaubamnian, K and Beg, I Concomitant of ode tatitic in ogenten type bivaiate exponential ditibution, J App Statit Sci 54 4, 33-45, 997 8] Balaubamnian, K and Beg, I Concomitant of ode tatitic in Gumbel bivaiate exponential ditibution, Sankhya B, 6, , 998 9] Bhattachaya, PK Induced ode tatitic: Theoy and Application In: Kihnaiah, PR and Sen, PK Ed, Handbook of Statitic Elevie Science 4, , 984 ] Bukchat,, Came, E and Kamp, U Dual genealized ode tatitic eton, LXI, 3-6, 3 ] David, HA and Nagaaja HN Concomitant of ode tatitic In: N Balakihnan and CR Rao ed, Handbook of Statitic, 6, , 998 ] David, HA and Nagaaja HN Ode Statitic John Wiley, New Yok, 3 3] Falie, DJG The pefomance of ome coelation coefficient fo a geneal bivaiate ditibution, Biometika, 47, 37 33, 96 4] Gumbel, EJ Bivaiate exponential ditibution, J Ame Statit Aoc 55, , 96 5] Johnon, NL and Kotz, S On ome genealized Falie-Gumbel-ogenten ditibution, Commun tatit Theo eth 4, 45 47, 975 6] Kamp, U A concept of genealized ode tatitic BG Teubne Stuttgat, Gemany, 995 7] ogenten, D Einfache Beipiele Zweidimenionale Veteilungen, itteilungblatt fu athematiche Statitik 8, 34 35, 956 c 5 NSP Natual Science Publihing Co