The Exponentiated Gamma Distribution Based On Ordered Random Variable

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1 Applied Mthemtics E-Notes, , c ISSN Ailble free t mirror sites of men/ The Exponentited Gmm Distribtion Bsed On Ordered Rom Vrible Deendr Kmr Receied 15 October 2014 Abstrct In this pper, e ge some ne explicit expressions recrrence reltions for mrginl joint moment generting fnctions of dl generlized order sttistics from exponentited gmm distribtion. The reslts for order sttistics loer record les re dedced from the reltion deried. Frther, chrcterizing reslt of this distribtion on sing recrrence reltion for mrginl moment generting fnctions dl generlized order sttistics is discssed. 1 Introdction The concept of generlized order sttistics gos s introdced by Kmps [1] s generl frmeork for models of ordered rom ribles. Moreoer, mny other models of ordered rom ribles, sch s, order sttistics, k-th pper record les, pper record les, progressiely Type II censoring order sttistics, Pfeifer records seqentil order sttistics re seen to be prticlr cses of gos. These models cn be effectiely pplied, e.g., in relibility theory. Hoeer, rom ribles tht re decresingly ordered cnnot be integrted into this frmeork. Conseqently, this model is inpproprite to stdy, e.g. reersed ordered order sttistic loer record les models. Brkscht et l. [2] introdced the concept of dl generlized order sttistics dgos. The dgos models enble s to stdy decresingly ordered rom ribles like reersed order sttistics, loer k record les loer Pfeirfer records, throgh common pproch belo: Sppose X d 1, n, m, k,..., X d n, n, m, k, k 1, m is rel nmber, re n dgos from n bsoltely continos cmltie distribtion fnction cdf F x ith probbility density fnction pdf fx, if their joint pdf is of the form n 1 n 1 k [F x i ] m fx i [F x n ] k 1 fx n, 1 j1 γ j i1 for F 1 1 > x 1 x 2... x n > F 1 0, here γ j k + n jm + 1 > 0 for ll j, 1 j n, k is positie integer m 1. If m 0 k 1, then this model redces to the n r + 1-th order sttistic, from the smple X 1, X 2,..., X n 1 Mthemtics Sbject Clssifictions: 62G30, 62E10. Deprtment of Sttistics, Amity Institte of Applied Sciences Amity Uniersity, Noid , Indi. 105

2 106 The Exponentited Gmm Distribtion ill be the joint pdf of n order sttistics. If k 1 m 1, then 1 ill be the joint pdf of the first n record les of the identiclly independently distribted iid rom ribles ith cdf F x corresponding pdf fx. In ie of 1, the mrginl pdf of the r-th dgos, is gien by f Xd r,n,m,kx The joint pdf of r-th s-th dgos, is C r 1 r 1! [F x]γ r 1 fxg r 1 m F x. 2 f Xd r,n,m,k,x d s,n,m,kx, y C s 1 r 1!s r 1! [F x]m fxgm r 1 F x [h m F y h m F x] s r 1 [F y] γ s 1 fy, 3 here C r 1 r γ i, h m x i1 { 1 m+1 xm+1 for m 1, lnx for m 1, g m x h m x h m 1, for x [0, 1. Ahsnllh Rqb [3], Rqb Ahsnllh [4, 5] he estblished recrrence reltions for moment generting fnctions mgf of record les from Preto Gmble, poer fnction extreme le distribtions. Recrrence reltions for mrginl joint mgf of gos from poer fnction distribtion, Erlng-trncted exponentil distribtion extended type II generlized logistic distribtion re deried by Srn Singh [6], Klshresth et l. [7] Kmr [8] respectiely. Kmr [9, 10, 11] he estblished recrrence reltions for mrginl joint mgf of dgos from generlized logistic, Mrshll-Olkin extended logistic type I generlized logistic distribtion respectiely. Al-Hssini et l. [12, 13] he estblished recrrence reltions for conditionl joint mgf of gos bsed on mixed popltion. Kmr [14] he estblished explicit expressions some recrrence reltions for mgf of record les from generlized logistic distribtion. Recrrence reltions for single prodct moments of dgos from the inerse Weibll distribtion re deried by Pls Szynl [15]. Ahsnllh [16] Mbh Ahsnllh [17] chrcterized the niform poer fnction distribtions bsed on distribtionl properties of dgos respectiely. Chrcteriztions bsed on gos he been stdied by some thors, Keseling [18] chrcterized some continos distribtions bsed on conditionl distribtions of gos. Bieniek Szynl [19] chrcterized some distribtions i linerity of regression of gos. Crmer et l. [20] ge nifying pproch on chrcteriztion i liner regression of ordered rom ribles. Rest of the pper is orgnized s follos: In Section 2 exct expressions recrrence reltions for mrginl joint mgf of dgos from exponentited Gmm distribtion EGD re presented, hile in Section 3, the exct expressions recrrence

3 D. Kmr 107 reltions for joint mgf for dgos from EGD re discssed In Section 4, chrcteriztion of EGD is obtined by sing the recrrence reltion for mrginl mgf of dgos. Some finl comments in Section 5 conclde the pper. Gpt et l. [21] introdced the EGD. This model is flexible enogh to ccommodte both monotonic s ell s nonmonotonic filre rtes. The cdf pdf of EGD re gien, respectiely by Note tht for F GD, F x [1 e x x + 1] α, x > 0, α > 0, 4 fx αxe x [1 e x x + 1] α 1, x > 0, α > 0. 5 αxf x [e x x + 1]fx. 6 For α 1, the boe distribtion corresponds to the gmm distribtion G1, 2. 2 Reltions for Mrginl Moments Generting Fnctions In this Section the exct expressions recrrence reltions for mrginl mgf of dgos from EGD re considered. For the EGD hen m 1, M Xd r,n,m,kt e tx fxdx C r 1 r 1! e tx [F x] γ r 1 fxgm r 1 F xdx. On sing 4 5 in 6 simplifiction of the reslting eqtion e get M Xd r,n,m,kt for m 1 αc r 1 r 1 r 1!m + 1 r 1 αγr 1 p p q M Xd r,n, 1,kt αkr r 1! p0 0 0 r p 0 p0 q0 r 1++p p 1 +p r 1 7 Γq + 2, 8 p + 1 t q+2 1 φ p r 1 αk 1 Γ + 2, 9 r + + p t +2 here φ p r 1 is the coeffi cient of e r 1+px x + 1 r 1+p in the expnsion of see Blkrishnn Cohn [22]. r 1 e px x + 1 p, p p1

4 108 The Exponentited Gmm Distribtion Differentiting both sides of 8 9 ith respect to t, j times e get M j X d r,n,m,k t αc r 1 r 1!m + 1 r 1 αγr 1 p M j αkr X d r,n, 1,kt r 1! p0 0 r p r 1 0 p0 q0 p q r 1++p 0 p 1 +p r 1 Γj + q + 2 p + 1 t j+q φ p r 1 If α is positie integer, the reltions then gie r 1 M j X d r,n,m,k t αc r 1 r 1!m + 1 r 1 αγr 1 p p q M j αkr X d r,n, 1,kt r 1! αk 1 p0 0 0 r 1++p 0 r P αk 1 Γj r + + p t j++2 αγ r 1 p 1 +p r 1 p0 q0 Γj + q + 2 p + 1 t j+q φ p r 1 αk 1 Γj r + + p t j++2 By differentiting both sides of eqtion ith respect to t then setting t 0, e obtin the explicit expression for single moments of dgos k record les from EGD in the form r 1 αγ E[X j d r, n, m, k] αc r 1 r 1 p r 1!m + 1 r 1 1 +p r 1 0 p0 q0 αγr 1 p Γj + q + 2 p q p + 1 j+q+2 14 E[X j αkr d r, n, 1, k] r 1! Specil Cses: αk 1 p0 0 r 1++p 0 1 φ p r 1 r p Γj αk r + + p j++2

5 D. Kmr 109 i Ptting m 0, k 1 in 12 14, reltions for order sttistics cn be obtined s M j X r:n t αc r:n n r 0 αr+ 1 p0 αr + 1 p p 1 +p n r q0 p Γj + q + 2 q p + 1 t j+q+2 here n r E[Xr;n] j αc r:n 0 αr+ 1 p0 αr + 1 p C r:n n! r 1!n r!. p 1 +p n r q0 p Γj + q + 2 q p + 1 j+q+2, ii Ptting k 1 in 13 15, reltions for record les cn be obtined s M j X Lr t α r α 1 r 1++p r 1! p0 0 r p 0 1 φ p r 1 Γj r + + p t j++2 α 1 E[X j Lr ] α r α 1 r 1! p0 0 r P r 1++p 0 1 φ p r 1 Γj r + + p j++2. α 1 A recrrence reltion for mgf of dgos from cdf 4 cn be obtined in the folloing theorem. THEOREM 1. For 2 r n n 2 k 1, 2,..., 1 t M j αγ X d r,n,m,k t r M j X d r 1,n,m,k t + 1 αγ r j αγ r M j 1 X d r,n,m,k t { te[φx d r, n, m, k] + E[ψX d r, n, m, k], 16

6 110 The Exponentited Gmm Distribtion here φx x j 1 e t+1x e tx ψx x j 2 e t+1x e tx. here PROOF. Integrting by prts of 7 sing 6, e get M Xd r,n,m,kt M Xd r 1,n,m,kt + j αγ r { MXd r,n,m,kt E[hX d r, n, m, k], 17 e t+1x hx x etx. x Differentiting both the sides of 17 j times ith respect to t, e get the reslt gien in 16. By differentiting both sides of eqtion 17 ith respect to t then setting t 0, e obtin the recrrence reltions for moments of dgos from EGD in the form E[X j d r, n, m, k] j E[Xj d r 1, n, m, k] + E[X j 1 d r, n, m, k] αγ r + j { E[X j 2 d r, n, m, k] E[ξX d r, n, m, k], 18 αγ r here ξx x j 2 e x. REMARK 2.1. Ptting m 0, k 1 in 16 18, reltions for order sttistics cn be obtined s M j X r:n t t 1 M j j X αr 1 r 1:n t αr 1 M j 1 X r 1:n t 1 + αr 1 {te[φx r 1:n] + je[ψx r 1:n ] EX j r:n EX j r 1:n j { EX j 1 r 1:n αr 1 + EXj 2 r 1:n EξX r 1;n. REMARK 2.2. Ptting k 1 in 16 18, reltions for k record les cn be obtined s 1 t M j X αk Lr t M j X Lr 1 t + j αk M j 1 X Lr t 1 { te[φxlr ] + je[ψx Lr ] αk E[X j Lr ] E[Xj Lr 1 ] + j { E[X j 1 αk Lr ] + E[Xj 2 Lr ] E[ξX Lr].

7 D. Kmr Reltions for Joint Moment Generting Fnctions In this Section exct moments recrrence reltions for joint mgf of dgos from EGD re considered. For the EGD hen m 1, M Xd r,n,m,k,x d s,n,m,kt 1, t 2 x C s 1 r 1!s r 1! e t1x+t2y f Xd r,n,m,kx d s,n,m,kx, ydxdy. x e t1x+t2y [F x] m fxgm r 1 F x [h m F y h m F x] s r 1 [F y] γ s 1 fydydx. 19 On sing 4 5 in 19 simplifiction of the reslting eqtion e get M Xd r,n,m,k,x d s,n,m,kt 1, t 2 α 2 C s 1 r 1!s r 1!m + 1 s 2 s r 1 0 l0 0 p0 s r 1 r 1 0 b0 0 l l++ αγs 1 r 1 b αs r + m l l Γ + 2Γp p!l + 1 t 2 +2 p l + 2 t 1 t 2 p++2 For m 1, M Xd r,n, 1,k,X d s,n, 1,kt 1, t 2 αk s s r 1 r 1!s r 1! c+1 0 +q+ d0 p0 q0 0 0 s r 1 b0 s+b+p 2 c0 1 s r 1++ φ p s 2φ q s + b + p 2 c αk 1 + q + Γc + 2Γ + d + 2 d!s + b + p 1 t 2 c d s + b + p + q + t 1 t 2 +d+2

8 112 The Exponentited Gmm Distribtion Differentiting both side of i times ith respect to t 1 then j times ith respect to t 2, e get M i,j X d r,n,m,k,x d s,n,m,k t 1, t 2 α 2 C s 1 r 1!s r 1!m + 1 s 2 s r 1 0 l0 0 s r 1 r 1 0 b0 0 l i l++ αγs 1 r 1 b p0 αs r + m l l Γi + + 2Γj + p p!l + 1 t 2 i++2 p + l + 2 t 1 t 2 j+p M i,j X d r,n, 1,k,X d s,n, 1,k t 1, t 2 αk s s r 1 r 1!s r 1! i+c+1 0 +q+ d0 p0 q0 0 0 s r 1 b0 s+b+p 2 c0 1 s r 1++ φ p s 2φ q s + b + p 2 c αk 1 + q + Γi + c + 2Γj + + d + 2 d!s + b + p 1 t 2 i+c d s + b + p + q + t 1 t 2 j++d+2 If α is positie integer, the reltions then gie M i,j X d r,n,m,k,x d s,n,m,k t 1, t 2 α 2 αγ s 1 C s 1 r 1!s r 1!m + 1 s 2 s r 1 0 r 1 αs r +m+1 1 l0 s r 1 0 r 1 b0 0 l i l++ αγs 1 b 0 p0 αs r + m l l Γi + + 2Γj + p p!l + 1 t 2 i++2 p + l + 2 t 1 t 2 j+p++2 24

9 D. Kmr 113 M i,j X d r,n, 1,k,X d s,n, 1,k t 1, t 2 αk s s r 1 r 1!s r 1! i+c+1 αk 1 d0 p0 q0 0 s r 1 0 +q+ 0 b0 s+b+p 2 c0 1 s r 1++ φ p s 2φ q s + b + p 2 c αk 1 + q + Γi + c + 2Γj + + d + 2 d!s + b + p 1 t 2 i+c d s + b + p + q + t 1 t 2 j++d+2 By differentiting both sides of eqtion ith respect to t 1, t 2 then setting t 1 t 2 0, e obtin the explicit expression for prodct moments of dgos k record les from EGD in the form E[Xdr, i n, m, k, X j d s, n, m, k] α 2 αγ s 1 C s 1 r 1 r 1!s r 1!m + 1 s 2 s r 1 0 r 1 αs r +m+1 1 l0 s r 1 0 b0 0 l i l++ αγs 1 b 0 p0 αs r + m l l Γi + + 2Γj + p p!l + 1 i++2 p + l + 2 j+p E[Xdr, i n, 1, k, X j d s, n, 1, k] αk s s r 1 r 1!s r 1! i+c+1 αk 1 d0 p0 q0 0 s r 1 0 +q+ 0 b0 s+b+p 2 c0 1 s r 1++ φ p s 2φ q s + b + p 2 c αk 1 + q + Γi + c + 2Γj + + d + 2 d!s + b + p 1 i+c d s + b + p + q + j++d+2 Specil Cses:

10 114 The Exponentited Gmm Distribtion i Ptting m 0, k 1 in 24 26, reltions for order sttistics cn be obtined s M i,j X r:n,x s:n t 1, t 2 αr+ 1 α 2 C r,s:n 0 n s b0 0 s r 1 0 αs r + 1 l0 l i l++ αr + 1 b 0 p0 n s s r 1 αs r + 1 l Γi + + 2Γj + p p!l + 1 t 2 i++2 p + l + 2 t 1 t 2 j+p++2 l αr+ 1 E[Xr:n, i Xs:n] j α 2 C r,s:n 0 b0 s r 1 0 αs r + 1 l0 i++1 p0 n s 0 l 1 +l++ αr + 1 n s b 0 s r 1 αs r + 1 l l Γi + + 2Γj + p p!l + 1 i++2 p + l + 2 j+p++2, here C r,s;n n! r 1!s r 1!n s!. ii Ptting k 1 in 25 27, reltions for record les in the form M i,j X Lr,X Ls t 1, t 2 α s s r 1 r 1!s r 1! i+c+1 d0 α 1 p0 q0 0 s r 1 0 +q+ 0 b0 s+b+p 2 c0 1 s r 1++ φ p s 2φ q s + b + p 2 c α 1 + q + Γi + c + 2Γj + + d + 2 d!s + b + p 1 t 2 i+c d+2 s + b + p + q + t 1 t 2 j++d+2

11 D. Kmr 115 E[X i Lr, Xj Ls ] α s s r 1 r 1!s r 1! i+c+1 d0 α 1 p0 q0 0 s r 1 0 +q+ 0 b0 s+b+p 2 c0 1 s r 1++ φ p s 2φ q s + b + p 2 c α 1 Γi + c + 2Γj + + d + 2 d!s + b + p 1 i+c d+2 s + b + p + q + j++d+2. + q + Mking se of 6, e cn derie recrrence reltions for joint mgf of dgos. THEOREM 2. For 1 r < s n n 2 k 1, 2,..., 1 t 2 M i,j αγ X d r,n,m,kx d s,n,m,k t 1, t 2 s M i,j X d r,n,m,kx d s 1,n,m,k t 1, t 2 + j M i,j 1 αγ X d r,n,m,kx d s,n,m,k t 1, t 2 s { t 2 E[φX d r, n, m, kx d s, n, m, k] +je[ψx d r, n, m, kx d s, n, m, k], 28 1 αγ s here φx, y x i y j 1 e t1x+t2+1y e t1x+t2y ψx x i y j 2 e t1x+t2+1y e t1x+t2y. PROOF. Integrting by prts of 19 sing 6, e get M Xd r,n,m,kx d s,n,m,kt 1, t 2 M Xd r,n,m,kx d s 1,n,m,kt 1, t 2 + t { 2 M Xd r,n,m,kx αγ d s,n,m,kt 1, t 2 s E[hX d r, n, m, kx d s, n, m, k] 29 e t 1x+t 2+1y hx, y y et1x+t2y. y

12 116 The Exponentited Gmm Distribtion Differentiting both the sides of boe eqtion i times ith respect to t 1 then j times ith respect to t 2 simplifying the reslting expression, e get the reslt gien in 28. By differentiting both sides of eqtion 28 ith respect to t 1, t 2 then setting t 1 t 2 0, e obtin the recrrence reltions for prodct moments of dgos from EGD in the form E[Xdr, i n, m, kx j d s, n, m, k] E[Xdr, i n, m, kx j d s 1, n, m, k] + j { E[X i αγ dr, n, m, kx j 1 d s, n, m, k] + E[Xdr, i n, m, kx j 2 d s, n, m, k] s j αγ s E[ξX d r, n, m, kx d s, n, m, k], 30 here ξx, y x i y j 2 e y. REMARK 3.1. Ptting m 0, k 1 in 28 30, e obtin reltions for order sttistics M i,j X r:nx s:n t 1, t 2 1 t t1 αr αr 1 M i,j X r 1:nX s:n t 1, t 2 i αr 1 M i,j { t 1 E[φX r 1:n X s:n ] + je[ψx r 1:n X s:n ] X r 1:nX s:n t 1, t 2 EXr:nX i s:n j { EXr 1:nX i s:n j i EXr 1:n i 1 αr 1 s:n + EXr 1:n i 2 s:n EφX r 1;n X s;n. REMARK 3.2. Ptting m 1 k 1in 28 30, e obtin reltions for k record les in the form 1 t 2 M i,j αγ X d r,n, 1,kX d s,n, 1,k t 1, t 2 s M i,j X d r,n, 1,kX d s 1,n, 1,k t 1, t 2 + j M i,j 1 αγ X d r,n, 1,kX d s,n, 1,k t 1, t 2 s { t 2 E[φX d r, n, 1, kx d s, n, 1, k] +je[ψx d r, n, 1, kx d s, n, 1, k] 1 αγ s

13 D. Kmr 117 E[Xdr, i n, 1, kx j d s, n, 1, k] E[Xdr, i n, 1, kx j d s 1, n, 1, k] + j { E[X i αγ dr, n, 1, kx j 1 d s, n, 1, k] + E[Xdr, i n, 1, kx j 2 d s, n, 1, k] s j αγ s E[ξX d r, n, 1, kx d s, n, 1, k]. 4 Chrcteriztion This Section contins chrcteriztion of EGD by sing the recrrence reltion for mgf of dgos. Let L, b st for the spce of ll integrble fnctions on, b. A seqence f n L, b is clled complete on L, b if for ll fnctions g L, b the condition b gxf n xdx 0, n N, implies gx 0.e. on, b. We strt ith the folloing reslt of Lin [23]. PROPOSITION 1. Let n 0 be ny fixed non-negtie integer, < b gx 0 n bsoltely continos fnction ith g x 0.e. on, b. Then the seqence of fnctions {gx n e gx, n n 0 is complete in L, b iff gx is strictly monotone on, b. Using the boe Proposition e get stronger ersion of Theorem 1. THEOREM 3. A necessry sffi cient conditions for rom rible X to be distribted ith pdf gien by 5 is tht M j X d r,n,m,k t M j X d r 1,n,m,k t + j { M j 1 αγ X d r,n,m,k t E[hX dr, n, m, k]. 31 r PROOF. The necessry prt follos immeditely from eqtion 17. On the other h if the recrrence reltion in eqtion 31 is stisfied, then on sing eqtion 2, e he C r 1 r 1! 0 C r 1 γ r r 2! tc r 1 0 αγ r r 1! + tc r 1 αγ r r 1! e tx [F x] γ r 1 fxgm r 1 F xdx e tx [F x] γ r +m fxgm r 2 F xdx e t+1x 0 0 x [F x] γ r 1 fxgm r 1 F xdx x j 1 [F x] γ r 1 fxgm r 1 F xdx

14 118 The Exponentited Gmm Distribtion tc 1 r 1 e tx αγ r r 1! 0 x [F x]γ r 1 fxgm r 1 F xdx. 32 Integrting the first integrl on the right-h side of the boe eqtion by prts simplifying the reslting expression, e get tc { r 1 e tx [F x] γ r 1 gm r 1 F x F x etx x + 1 fx dx r 1! αx 0 It no follos from Proposition 1, tht hich proes tht fx hs the form 4. 5 Conclding Remrks αxf x [e tx x + 1]fx i In this pper, e proposed ne explicit expressions recrrence reltions for mrginl joint moment generting fnctions of dgos from EGD. Frther, chrcteriztion of this distribtion hs lso been obtined on sing recrrence reltion for mrginl moment generting fnctions of dgos. Specil cses re lso dedced. ii The recrrence reltions for moments of ordered rom ribles re importnt becse they redce the mont of direct compttions for moments, elte the higher moments in terms of the loer moments they cn be sed to chrcterize distribtions. iii The recrrence reltions of higher joint moments enble s to derie single, prodct, triple qdrple moments hich cn be sed in Edgeorth pproximte inference. Acknoledgments. The thor is grtefl to nonymos referees the Editor for ery sefl comments sggestions. References [1] U. Kmps, A Concept of Generlized Order Sttistics. Tebner Skripten zr Mthemtischen Stochstik. [Tebner Texts on Mthemticl Stochstics] B. G. Tebner, Stttgrt, [2] M. Brkscht, E. Crmer U. Kmps, Dl generlized order sttistics, Metron, , [3] M. Ahsnllh M. Z. Rqb, Recrrence reltions for the moment generting fnctions of record les from Preto Gmble distribtions, Stoch. Model. Appl., 21999,

15 D. Kmr 119 [4] M. Z. Rqb M. Ahsnllh, Reltions for mrginl joint moment generting fnctions of record les from poer fnction distribtion, J. Appl. Sttist. Sci., , [5] M. Z. Rqb M. Ahsnllh, On moment generting fnction of records from extreme le distribtion, Pkistn J. Sttist., , [6] J. Srn A. Singh, Recrrence reltions for mrginl joint moment generting fnctions of generlized order sttistics from poer fnction distribtion, Metron, 2003, [7] A. Klshresth, R. U. Khn D. Kmr, On moment generting fnction of generlized order sttistics from Erlng-trncted exponentil distribtion, Open J. Stt., 22012, [8] D. Kmr, On moment generting fnctions of generlized order sttistics from extended type II generlized logistic distribtion, J. Sttist. Theory Appl., , [9] D. Kmr, Moment generting fnctions of loer generlized order sttistics from generlized logistic distribtion its chrcteriztion, Pcific Jornl of Applied Mthemtics, 52013, [10] D. Kmr, Reltions for mrginl joint moment generting fnctions of Mrshll-Olkin extended logistic distribtion bsed on loer generlized order sttistics chrcteriztion, Americn Jornl of Mthemticl Mngement Sciences, , [11] D. Kmr, Reltions for mrginl Joint moments generting fnctions of extended type I generlized logistic distribtion bsed on loer generlized order sttistics chrcteriztion, Tmsi Oxford Jornl of Mthemticl Science, , [12] E. K. Al-Hssini, A. A. Ahmd M. A. Al-Kshif, Recrrence reltions for joint moment generting fnctions of generlized order sttistics bsed on mixed popltion, J. Sttist. Theory Appl., 62005, [13] E. K. Al-Hssini, A. A. Ahmd M. A. Al-Kshif, Recrrence reltions for moment conditionl moment generting fnctions of generlized order sttistics, Metrik, , [14] D. Kmr, Recrrence reltions for mrginl joint moment generting fnctions of generlized logistic distribtion bsed on loer record les its chrcteriztion, ProbStt Form, 52012, [15] P. Pls D. Szynl, Recrrence reltions for single prodct moments of loer generlized order sttistics from the inerse Weibll distribtion, Demonstrtio Mthemtic, 22001,

16 120 The Exponentited Gmm Distribtion [16] M. Ahsnllh, A chrcteriztion of the niform distribtion by dl generlized order sttistics, Comm. Sttist. Theory Methods, , [17] A. K. Mbh M. Ahsnllh, Some chrcteriztion of the poer fnction distribtion bsed on loer generlized order sttistics, Pkistn J. Sttist., , [18] C. Keseling, Conditionl distribtions of generlized order sttistics some chrcteriztions, Metrik, , [19] M. Bieniek D. Szynl, Chrcteriztions of distribtions i linerity of regression of generlized order sttistics, Metrik, , [20] E. Crmer, U. Kmps C. Keseling, Chrcteriztion i liner regression of ordered rom ribles: nifying pproch, Comm. Sttist. Theory Methods, , [21] R. C. Gpt, R. D. Gpt P. L. Gpt, Modelling filre time dt by Lehmn lternties. Comm. Sttist. Theory Methods, , [22] N. Blkrishnn A. C. Cohn, Order Sttistics Inference: Estimtion Methods, Acdemic Press, Sn Diego, [23] G. D. Lin, On moment problem, Tohok Mth. Jornl, ,