Concomitants of Dual Generalized Order Statistics from Bivariate Burr III Distribution

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1 Journal of Statstcal Theory and Applcatons, Vol. 4, No. 3 September 5, 4-56 Concomtants of Dual Generalzed Order Statstcs from Bvarate Burr III Dstrbuton Haseeb Athar, Nayabuddn and Zuber Akhter Department of Statstcs & Operatons Research Algarh Muslm Unversty, Algarh -, Inda haseebathar@hotmal.com Receved 3 Aprl 4; Accepted March 5 In ths paper probablty densty functon of sngle concomtant and ont probablty densty functon of two concomtants of dual generalzed order statstcs from bvarate Burr III dstrbuton are obtaned and expressons for sngle and product moments are derved. Further, results are deduced for the order statstcs and lower record values. Keywords: Dual generalzed order statstcs; order statstcs; lower record values; concomtants; bvarate Burr III dstrbuton. Mathematcs Subect Classfcaton: 6G3.. Introducton The probablty densty functon pd f of bvarate Burr III dstrbuton [Rodrguez, 98] s gven as f x,y c c + α k x k + α k y k + + α x k + α y k c+, x,y >, k, k,c, α, α >. and the correspondng dstrbuton functon d f s Fx,y + α x k + α y k c, x,y >, k, k,c, α, α >.. The condtonal pd f of Y gven X s f y x c + α k y k+ + α x k c+ + α x k + α y k, y >..3 c+ The margnal pd f of X s f x c α k x k + { + α x k, x >..4 } c+ The margnal d f of X s Correspondng author 4

2 Haseeb Athar, Nayabuddn and Zuber Akhter Fx + α x k c, x >..5 Order statstcs, record values and several other model of ordered random varables can be vewed as specal case of generalzed order statstcs gos[kamps, 995]. Burkschat 3 ntroduced the concept of dual generalzed order statstcs dgos to enable a common approach to descendng ordered rv s lke reverse order statstcs and lower record values. Let X be a contnuous random varables wth d f Fx and the pd f f x, x α,β. Further, Let n N, k, m m,m,...,m n _ R n, M r n r m such that γ r k + n r + M r for all r,,...,n. Then X d r,n, m,k r,,...,n are called dgos f ther pd f s gven by [Burkschat et al. 3] k n γ n on the cone F > x x... x n > F. [ Fx ] m [Fxn f x ] k f xn.6 If m,,,...,n, k, then X d r,n,m,k reduces to the n r + th order statstcs, X n r+:n from the sample X,X,...,X n and at m, X d r,n,m,k reduces to r th, k lower record values. For more detals on order statstcs and record values, one may refer to Davd and Nagaraa 3 and Ahsanullah 4 respectvely. In vew of.6, the pd f of X d r,n,m,k s f r,n,m,k x and ont pd f of X d r,n,m,k and X d s,n,m,k, r < s n s f r,s,n,m,k x,y C r r! [Fx]γ r f xg r m Fx.7 C s r!s r! [Fx]m f x g r m Fx [h m Fy hm Fx ] s r [Fy] γ s f y, x > y.8 where C r r γ h m x m + xm+ logx, m, m and g m x h m x h m, x,. Let X,Y,,,...,n, be the n pars of ndependent random varables from some bvarate populaton wth dstrbuton functon Fx, y. If we arrange the X varates n descendng order as 4

3 Concomtants of Dual Generalzed Order Statstcs... X,n,m,k X,n,m,k... Xn,n,m,k then Y varates pared not necessarly n descendng order wth these dual generalzed ordered statstcs are called the concomtants of dual generalzed order statstcs and are denoted by Y [,n,m,k],y [,n,m,k],...,y [n,n,m,k]. The pd f of Y [r,n,m,k], the r th concomtant of dual generalzed order statstcs s gven as g [r,n,m,k] y and the ont pdf of Y [r,n,m,k] and Y [s,n,m,k], r < s n s g [r,s,n,m,k] y,y f y x f r,n,m,k x dx.9 x. Probablty Densty Functon of Y [r,n,m,k] f y x f y x f r,s,n,m,k x,x dx dx.. For the bvarate Burr III dstrbuton as gven n., the pd f of g [r,n,m,k] y n vew of.3,.4,.5,.7 and.9 s gven as r α k C r g [r,n,m,k] y r!m + r c c + y +k r α k x +k + α x k + α y k c+ + α x k dx.. cγ r Let t α x k, then the R.H.S. of. reduces to r α k C r r!m + r c c + y +k r Snce x v a + x µ x + y ρ dx where +t + α y k c+ +t cγr dt.. a, b F ; x a p b p x p c p c p p! s the Gauss hypergeometrc seres µ, v Γv Γµ v + ρ Γµ + ρ a µ y v ρ F ; y a.3 µ + ρ [ arg a < Π, Re v >, arg y < Π, Re ρ > Re v µ], Γλ + n λ n, λ,,,..., Γλ [Prudnkov et. al,986].5 s the Pochhammer symbol..4 4

4 Haseeb Athar, Nayabuddn and Zuber Akhter Therefore., becomes r α k C r g [r,n,m,k] y r!m + r c c + y +k r + α y k c+ cγ r + c γ r c, F ; α y k..6 c γ r + 3. Moment of Y [r,n,m,k] For the bvarate Burr III dstrbuton as gven n., the a th moment of Y [r,n,m,k] s gven as, E Y a [r,n,m,k] y a g [r,n,m,k] y dy 3. r α k C r r!m + r c c + r cγ r + c γ r c, y a y +k + α y k c+ F ; α y k dy 3. c γ r + r α k C r r!m + r c c + r cγ r + cγ r c p p y a y +k + α y k c+ α y k p dy. 3.3 p cγ r + p p! Now lettng t + α y k n 3.3, we get k α a r C r r r!m + r c c + cγ r + Snce p cγ r c p p p cγ r + p p! t c+ t p a k dt. 3.4 y x λ x y µ dx Γλ µ Γµ Γλ y µ λ, < Re µ < Re < λ 3.5 [c.f. Erdély et al., 954]. 43

5 Concomtants of Dual Generalzed Order Statstcs... Therefore R.H.S. of 3.4 becomes k α a r C r r!m + r c c + r cγ r + p cγ r c p p p cγ r + p p! Γc + a k p Γp + a k. 3.6 Γc + Now on applcaton of.5 and usng λ n n λ n, n,,3,..., λ, ±, ±,... [Srvastava and Karlsson,985] 3.7 n 3.6 and smplfyng, we get E Y a [r,n,m,k] k α a r C r r!m + r c c + r cγ r + p cγ r c p p a k p cγ r + p c a k p p! Γc + a k Γ a k. 3.8 Γc + We have the generalzed Gauss seres or the generalzed hypergeometrc seres. α... α p pf q,z β... β q k α k α k...α p k z k β k β k...β q k k!. 3.9 Thus applyng 3.9 n 3.8, we have E Y a [r,n,m,k] α a k C r c c + r!m + r r r cγ r + Γc + a k Γ a k Γc + cγ r c,, a k 3 F,. 3. cγ r +, c a k Notng that [prudnkov et al, 986] 44

6 Haseeb Athar, Nayabuddn and Zuber Akhter N,, a 3F, b, a m, m,, b b a m b + N a 3 F,. 3. b N, a, Now usng 3. n 3., we can wrte α k a c C r r!m + r r r Γc + a k Γ a k Γc + c + a k a k c,, cγ r 3 F,. 3. c, + a k, Usng relaton 3.9 n 3., we get α k a c C r r!m + r r r Γc + a k Γ a k Γc + c + a k a k, cγ r F ; a k Notng that [Prudnkov et al., 986]. a b F ; Γ c Γ c a b, Re c a b >, c, ±, ±,..., 3.4 Γ c a Γ c b c Applyng 3.4 n 3.3, we get 45

7 Concomtants of Dual Generalzed Order Statstcs... α k a C r r!m + r r r c + a k Γc + a k Γ a k c Γc γ r + a 3.5 c k α k a C r r!m + r r r c + a k Γc + a k Γ a k Γc + k a B + n r + + m + c k m +,. 3.6 Snce, b a a b Ba + k, c Bk, c + b 3.7 a where Ba,b s the complete beta functon. Therefore, E y a [r,n,m,k] α k a C r Γc + + a m + r k Γ a k c Γc Γ k+n rm++ a c k m+ Γ k+nm++ a c k m+ 3.8 E y a k [r,n,m,k] α a Γc + + a k Γ a k c Γc r + a c k γ. 3.9 Remark 3. : If m, k n 3., we get sngle moment of concomtants of order statstcs from bvarate Burr III dstrbuton as; E y a [n r+:n] n! n r! α k a Γc + + a k Γ a k c Γc Γn r + + a c k Γn + + a c k. If we replace n r + by r, then E y a [r:n] n! r! α a Γc + + a k k Γ a k c Γc Γr + a c k Γn + + a c k and at r n, we get 46

8 Haseeb Athar, Nayabuddn and Zuber Akhter E y a n [n:n] n c + a k α k a Γc + + a k Γ a k. 3. Γc It may be noted that for the order statstcs, the k th moments of Y [r:n] s µ [r:n] k n n y r r n r µ [:] k y. 3. Therefore, E y a [r:n] n r n r r c + a k α k a Γc + + a k Γ a k Γc 3. as obtaned by Begum and Khan,998. Remark 3. : Set m n 3., to get moment of k th lower record value from bvarate Burr III dstrbuton as; E y a [r,n,,k] α a k Γc + + a k Γ a k c Γc + a c k k r Jont Probablty Densty Functon of Y [r,n,m,k] and Y [s,n,m,k] For the bvarate Burr III dstrbuton as gven n., the ont pd f of Y [r,n,m,k] and Y [s,n,m,k] n vew of.3,.4,.5,.8 and. s gven as g [r,s,n,m,k] y,y C s r!s r!m + s c c + α k α k y +k y +k r s r + r s r where, x +k + α x k cs r+ m+ c + α x k + α y k Ix,y dx 4. c+ x Ix,y α k x +k + α x k cγ s Settng t + α x k, then the R.H.S. of 4. reduces to + α x k + α y k dx. 4. c+ 47

9 Concomtants of Dual Generalzed Order Statstcs... Ix,y λ β +α x k t α + t λ β dt 4.3 where α cγ s c, β c + and λ α y k Note that [Prudnkov et al., 986]. + z a p p a p z p. 4.4 p! Now n vew of 4.4, 4.3 after smplfcaton yelds Ix,y λ β l l β l λ l + α x k α l α l. 4.5 Now puttng the value of Ix,y from 4.5 n 4., we obtan g [r,s,n,m,k] y,y c c + α k α k C s r!s r!m + s y +k y +k r s r + r s r λ β l l α l β l λ l x +k + α x k cs r+ m+ c+cγ s c l Settng t + α x k n 4.6, and after smplfcaton we get + α x k + α y k dx. 4.6 c+ g [r,s,n,m,k] y,y C s r!s r!m + s c c + α k y +k y +k r s r + r s r λ β δ β l β l λ l α + l p β p δ p θ α + l + p p! 4.7 where δ α y k and θ cs r + m + c. 48

10 Haseeb Athar, Nayabuddn and Zuber Akhter By settng d α and g θ α n 4.7, we get c c + α k C s r!s r!m + s r s r + r s r y +k y +k λ β δ β l β l λ l d + l p β p δ p g + p + l p!. 4.8 Now after puttng the value of λ and δ n 4.8, we have c c + α k C s r!s r!m + s r s r + r s r y +k y +k α y k β α y k β l Notng that [Srvastava and Karlsson, 985]. l α y k β l d + l p β p p α y k g + p + l p! 4.9 λ + m λ λ + m λ m 4. λ + m + n λ λ + m+n λ m+n. 4. Usng relaton 4. and 4. n 4.9, t becomes c c + C s r!s r!m + s r s r + r s r dg α k y +k α k y α y k β α y k +k β l p p α y k g p+l β l d l β p g + p+l d + l We have the Kampede Feret s seres [Srvastava and Karlsson, 985] F p:q;k l:m;n a p : b q ; c k ; x,y α l β m γ n p! l α y k. 4. r s Therefore after usng 4.3, we fnally get p a r+s q b r k c s x r l α r+s m β r n γ s r! y s s!

11 Concomtants of Dual Generalzed Order Statstcs... g [r,s,n,m,k] y,y c c + C s r!s r!m + s r s r + r s r dg α k y +k α k y α y k β +k α y k F :; β :; g + ; d + ; g; β; d; β; ; α y k, α y k Product Moments of two concomtants Y [r,n,m,k] and Y [s,n,m,k] For Burr type III dstrbuton, the product Moments of two concomtants Y [r,n,m,k] and Y [s,n,m,k] s gven as E Y a [r,n,m,k] Y b [s,n,m,k] In vew of 4.4 and 5., we have y a y b g [r,s,n,m,k] y,y dy dy. 5. E Y a [r,n,m,k] Y b [s,n,m,k] A y a y b α k y +k α k y α y k β α y k +k β F :; :; g ; β ; d; β; ; g + ; d + ; α y k, α y k dy dy 5. where, Thus, A c c + C s r!s r!m + s r s r + r s r dg. 5.3 E Y a [r,n,m,k] Y b [s,n,m,k] A y a y b α k y +k α k y α y k β α y k +k β l p p α y k g p+l β l d l β p g + p+l d + l p! l α y k dy dy. 5.4 By Srvastava and Karlsson 985, we have 5

12 Haseeb Athar, Nayabuddn and Zuber Akhter λ m+n λ m λ + m n. 5.5 On applyng 5.5 n 5.4, we get A y b α k y +k α y k β l g l g + l β l d l l α y k d + l { y a α k y +k β p g + l p α y k β p A y b α k y +k α y k β l α y k g + + l p g l g + l β l d l p! p d + l y a α k y +k β; g + l α y k F β g + l + Now lettng α y k z n 5.7, we have dy } dy 5.6 l α y k ; α y k dy dy. 5.7 A α a k y b α k y +k α y k β l g l g + l β l d l l α y k d + l z β+ k a F β; g + l g + l + ; z dz dy 5.8 Snce a, b a, c b F ; x x a F c c ; x x. 5.9 Therefore on usng relaton 5.9 n 5., we have A α a k y b α k y +k α y k β l g l g + l β l d l l α y k d + l p β p { p g + l + p } z k a z β+p dz dy. 5. 5

13 Concomtants of Dual Generalzed Order Statstcs... Now usng relaton 3.5 n 5., we get A α a k g g + l Γβ + a k Γ a k Γβ y b α k y +k α y k β l l α y k β l d l d + l Now n vew of 3.4, 5. becomes β + a k ; F ; dy. 5. g + l + k α a g A Γβ + a k Γ a k { y b α k y +k } Γβ α y k dy β β l d l l l α y k d + l Usng relaton 4. n 5., we get g + β a k + l. 5. α a k A g g + β a k Γβ + a k Γ a k { Γβ y b α k y +k } α y k dy β β l d l g + β a k l l d + l α y k g + β a k l l. 5.3 Further settng z α y k n 5.3, t becomes α a+b k A g g + β a k Γβ + a k Γ a k Γβ z β+ b Set z t n 5.4, to get k 3 F d; g + β a k ; β d + ; g + β a k ; z dz. 5.4 α a+b k A g g + β a k Γβ + a k Γ a k Γβ 5

14 Haseeb Athar, Nayabuddn and Zuber Akhter d; g + β a t β+ k b k ; β 3 F ; t dz. 5.5 d + ; g + β a k Note that [Prudnkov et al., 986] a, a, a 3 x s 3F ; x dx b, b Γb Γb Γs Γa s Γa s Γa 3 s Γa Γa Γa 3 Γb s Γb s Now usng relatons 5.6 n 5.5, we get 5.6 α a+b k g A g + β a k b k d + β b k Γβ + a k Γ a k Γβ Γβ + b k Γ b k. 5.7 Γβ Fnally puttng the values of A, d, g and β n 5.7, we have E Y a [r,n,m,k] Y b [s,n,m,k] α a+b c c + C k s Γβ + a k Γ a k r! s r!m + s Γβ Γβ + b k Γ b k Γβ r s r r + s r [c{k + n s + m + } + b k ] [c{k + n r + m + } + a k + b k ] 5.8 α a+b k c c + C s Γβ + a k Γ a k r! s r!m + s Γβ Γβ + b k Γ b k r Γβ s r s r k b B + n s + + m + r k B + n r + + a m + + b ck m +, ck m +,

15 Concomtants of Dual Generalzed Order Statstcs... On usng relaton 3.7 n 5.9, we get α a+b k c + C s Γβ + a k Γ a k r! s r!m + s Γβ Γβ + b k Γ b k Γβ k B + n r + a + b m + ck m +, r k b B + n s + m + ck m +, s r, 5. whch after smplfcaton yelds E Y a [r,n,m,k] Y b [s,n,m,k] α a+b k c + Γβ + a k Γ a k Γβ Γβ + b k Γ b k Γβ r a+b + c k γ s r+ + b c k γ. 5. Remark 5. : By settng m, k n 5.8, we get product moments of concomtant of order statstcs from bvarate Burr III dstrbuton as E Y a [n s+:n] Y b [n r+:n] α a+b c c + n! k r! s r!n s! Γβ + b k Γ b k Γβ r Γβ + a k Γ a k. Γβ s r r + s r [cn s b k ] [cn r a k + b k ] 5. Replace n s + by r and n r + by s n 5., we get E Y a [r:n] Y b [s:n] α a+b c c + n! Γβ + a k k Γ a k r! s r!n s! Γβ Γβ + b k Γ b k Γβ n s s r n s + s r [cr + c + b k ] [cs + c + a k + b k ] 54

16 Haseeb Athar, Nayabuddn and Zuber Akhter E Y a [r:n] Y b [s:n] n! r! α a+b Γc + + a k k Γ a k Γc + + b k Γ b k Γc + Γr + b ck Γs + b ck as obtaned by Begum and Khan, 998. Γs + a+b ck Γn + + a+b ck Remark 5. : If m n 5., we get product moment of concomtants of k th lower record values from bvarate Burr III dstrbuton as E Y a [r,n,,k] Y b [s,n,,k] α a+b k c + Γβ + a k Γ a k Γβ Γβ + b k Γ b k Γβ + a+b c k k r + b s r. 5.3 c k k Puttng the value of β n 5., we get E Y a [r,n,,k] Y b [s,n,,k] α a+b Γc + + a k k Γ a k Γc + Γc + + b k Γ b k Γc + + a+b c k k r + b c k k s r 5.4 Acknowledgement The authors are thankful to Professor M. Ahsanullah, Rder Unversty, Lawrencevlle, NJ, USA for hs frutful suggestons. References [] Ahsanullah, M. 4: Record Values-Theory and Applcatons. Unversty Press of Amerca, Lanham, Maryland. [] Begum, A.A. and Khan, A.H. 998: Concomtants of order statstcs from bvarates Burr III dstrbuton. Journal of Statstcal Researh, 3, 5 5. [3] Burkschat, M.; Cramer, E. and Kamps, U. 3: Dual generalzed order statstcs. Metron, LXI, 3-6. [4] Davd, H. A. and Nagaraa, H. N. 3: Order Statstcs, John Wley, New York. [5] Erdély, A., Magnus, W., Oberhetnger, F. and Trcom, F. G.954: Tables of Integral Transforms. McGraw-Hll, New York. [6] Kamps, U. 995: A concept of generalzed order statstcs. B.G. Teubner Stuttgart, Germany. [7] Prudnkov, A. P., Brychkov, Yu. A. and Marchev, O. I. 986: Integral and seres: More Specal Functons Vol.3. Gordon and Breach Scence Publsher, New York. 55

17 Concomtants of Dual Generalzed Order Statstcs... [8] Rodrguez, R. N. 98: Burr dstrbutons: Encyclopeda of Statstcal Scence. Vol., Kotz and Johnson, ed., John Wley and Sons, New York. [9] Srvastava, H. M. and Karlsson 985: Multple Gaussan Hypergeometrc seres. John Wley & Sons, New York. 56