On the Distribution of the Product and Ratio of Independent Central and Doubly Non-central Generalized Gamma Ratio random variables

Transcription

1 On the Dstbuton of the Poduct Rato of Independent Cental Doubly Non-cental Genealzed Gamma Rato om vaables Calos A. Coelho João T. Mexa Abstact Usng a decomposton of the chaactestc functon of the logathm of the poduct of ndependent Genealzed Gamma Rato om vaables we obtan explct expessons fo both the pobablty densty cumulatve dstbuton functons of the poduct of ndependent cental o non-cental om vaables wth genealzed F o Genealzed Gamma Rato dstbutons unde the fom of patcula mxtues of Paeto nveted Paeto dstbutons. The expessons obtaned do not nvolve any unsolved ntegals ae much adequate fo compute mplementaton the development of asymptotc nea-exact dstbutons. By consdeng not necessaly postve powe paametes we wee able to obtan as patcula cases not only the poduct of Beta pme, folded T, folded Cauchy F om vaables but also the denstes dstbutons fo the ato of two ndependent Genealzed Gamma Rato om vaables o two ndependent poducts of such vaables. Poducts of Genealzed Gamma Rato dstbutons may be appled n the study of multvaate lnea functonal models. As a by-poduct we also obtan closed fom epesentatons fo the dstbuton of the dffeence of two ndependent sums of a fnte numbe of Gamma om vaables wth dffeent ate paametes ntege shape paametes, unde the fom of fnte mxtues of Gamma dstbutons, as well as the dstbutons fo the poduct ato of genealzed Paeto dstbutons, unde the fom of fnte mxtues of Paeto nveted Paeto dstbutons. Key wods: patcula mxtues, Paeto nveted Paeto dstbutons, GIG dstbuton, sum of Exponentals, dffeence of Exponentals, folded T, folded Cauchy, Beta pme, Beta second knd Intoducton The poblem of obtanng an explct expesson, wthout nvolvng any unsolved ntegals, fo both the pobablty densty functon p.d.f. cumulatve dstbuton functon c.d.f. of the poduct of ndependent Genealzed Gamma Rato om vaables.v. s o Genealzed F.v. s, as they ae also called, s a challengng one, moeove snce the chaactestc functon s not eadly avalable fo such.v. s. In ths pape we pesent the dstbuton fo the poduct of ndependent cental doubly non-cental Genealzed Gamma Rato GGR om vaables unde the fom of patcula mxtues of Paeto nveted Paeto dstbutons. Expessons fo the p.d.f. c.d.f.? of the cental case of such a poduct wee obtaned by Shah Rathe 974 n tems of Fox s H functon. Yet, only fo the cental case, also Pham-Ga Tukkan 2002 obtaned expessons fo the p.d.f. of the poduct of only two ndependent genealzed F.v. s n tems of the Laucella hypegeometc D-functon of two vaables. Howeve, even nowadays when good softwaes fo symbolc numec computaton ae avalable the computaton of Fox s H functon the Laucella functon s not eadly avalable, beng usually computed n tems of the ntegals that defne them. Although Pham-Ga Tukkan 2002 developed an effcent compute code to compute the Laucella hypegeometc functon, the appoach s not extensble to the poduct of moe than two.v. s as these authos stengthen, when we consde the poduct of moe than two GGR.v. s t seems that fequently, howeve, no closed fom soluton fo these opeatons can be obtaned one has to esot to appoxmate appoaches, ncludng smulaton. The same authos, when efeng to the esults n Shah Rathe 974 coespondng autho: Calos A. Coelho cmac@fct.unl.pt s Assocate Pofesso João T. Mexa s Pofesso of Statstcs at the Depatment of Mathematcs of the Faculty of Scences Technology of the New Unvesty of Lsbon, Capaca, Potugal

2 say that these esults, although vey convenent notatonwse, ae dffcult to be pogammed on a compute hence ae dffcult to be used n applcatons. Yet the same authos say that they ae, howeve, essental when the numbe of vaables n the poduct, o quotent, s lage than 2. In ths pape ou am s to obtan explct smple expessons fo the p.d.f. c.d.f. of the poduct of ndependent Genealzed Gamma Rato.v. s whch may be eadly mplemented computatonally that, gven ts stuctue, may also gve us eady access to asymptotc nea-exact dstbutons. Gven the appoach followed, not only the dstbutons fo the non-cental case ae eadly at h but also the dstbuton fo the poduct of any non-null powe of GGR.v. s. As patcula mmedate cases we have the poduct of cental non-cental ndependent genealzed second knd Beta o Beta pme, folded T folded Cauchy.v. s yet, of couse, F.v. s. Gven the fact that the chaactestc functons fo the GGR om vaables ae not eadly avalable gven the fact that we ae dealng wth a poduct of om vaables, t s he to cay ou wok though the decomposton of the chaactestc functon of the logathm of the poduct of the GGR om vaables. Thus, ths was ou choce. Anothe novelty s that, although usually only postve powe paametes ae consdeed fo the GGR dstbutons, actually nothng foces those paametes to be postve, gven that the coect appoach s taken, beng the case that actually a negatve powe paamete n the GGR dstbuton only takes us to consde the ecpocal of that gven om vaable wth the symmetcal postve powe paamete. Gven the way he poblem s appoached, even negatve powe paametes may be easly consdeed n the GGR dstbutons also the dstbuton of the ato of two GGR om vaables o of the ato of two poducts of GGR om vaables ae patcula cases of the esults obtaned n the pape. Poducts of seveal ndependent GGR om vaables ae elated to a test statstc used n the multvaate lnea functonal model Povost, Some pelmnay esults 2. The Genealzed Gamma Rato GGR dstbuton In ode to establsh some of the notaton, nomenclatue a esult used ahead we wll stat to defne what we ntend by a Genealzed Gamma Rato GGR dstbuton. Let X Γ, λ X 2 Γ 2, λ 2 be to ndependent.v. s wth Gamma dstbutons wth shape paametes 2 ate paametes λ λ 2, that s, fo example, X has p.d.f. pobablty densty functon Let then f X x = λ Γ e λ x x,, λ > 0; x > 0. Y = X /β, Y 2 = X /β 2, β IR\{0} Z = Y /Y 2. We wll say that Y Y 2 have Genealzed Gamma dstbutons that Z has a GGR dstbuton. Usng stad methods we have the p.d.f ṣ of Y =, 2 Z gven by f Y y = β λ Γ e λ y β y β, y > 0, =, 2 f Z z = β k B, 2 + kz β 2 z β, z > 0 2

3 k = λ /λ 2 B, s the Beta functon. We wll denote the fact that Z has the GGR dstbuton wth paametes k,, 2 β by Z GGRk,, 2, β. The non-cental moments of Z ae easly deved as E Z h = k h Γ + h Γ Γ 2 h Γ 2, < h < 2. If β =, = m/2 2 = n/2, wth m, n IN, then Z has an F dstbuton wth m n degees of feedom. Ths s the eason why the dstbuton of Z s also called a Genealzed F dstbuton Shah Rathe, The Genealzed Intege Gamma GIG dstbuton In ths subsecton n the two followng ones we wll establsh some dstbutons that wll be used n the next secton. Let X Γ, λ =,..., p be p ndependent.v. s wth Gamma dstbutons wth shape paametes IN ate paametes λ > 0 =,..., p. We wll say that then the.v. p Y = has a GIG dstbuton of depth p, wth shape paametes ate paametes λ, =,..., p, we wll denote ths fact by X Y GIG, λ ; p =,..., p. The p.d.f. c.d.f. cumulatve dstbuton functon of Y ae, see Coelho 998, espectvely gven by p f Y y = K P y e λ y wth c, k = k K = c, p, = k = F Y y = K p P y = p P y e λ y λ, P y =! k +! k! k k! y k 2 y! λ k p λ λ, =,..., p, 3 = = [, 2,..., p ], R,, p,, λ = R,, p,, λ c, k, k =,..., =,..., p 4 p k λ λ k =,...,. 5 k 3

4 2.3 The dstbuton of the sum of om vaables wth Exponental dstbuton the dstbuton of the dffeence of two of these om vaables Let X Expλ =,..., p be p ndependent Exponental.v. s wth ate paametes λ =,..., p, let p Y = X. = The dstbuton of the.v. Y s a patcula case of the GIG dstbuton Coelho, 998 of depth p, wth all the shape paametes equal to, whose p.d.f. may be wtten as f Y y = K p c e λy K = p λ c = p k We wll denote the fact that Y has ths dstbuton by λ λ k =,..., p. Y SEλ, {,..., p}. Let then Y SEλ, {,..., p} Y 2 SEν, {,..., p} be two ndependent.v. s let Z = Y Y 2. The p.d.f. s of Y Y 2 may then be espectvely wtten as, fo =,..., p, f Y y = K p p c e λy f Y2 y 2 = K 2 d e νy2 c = maxz,0 K = p k p λ, K 2 = λ λ k, d = p ν 6 p k ν ν k 7 so that the p.d.f. of Z wll be gven by + p p f Z z = K K 2 c e λ y d e ν y z dy = K K 2 p p + e νkz c d k 4 maxz,0 e λ+ν ky dy

5 o, Then f we take K K 2 f Z z = K K 2 H = p h= p H c e λ z z 0 p H 2 d e ν z d h λ + ν h H 2 = W = k e Z p h= z 0 c h λ h + ν. 8 we wll have K K 2 f W w = K K 2 p H c kw λ w p H 2 d kw ν w The c.d.f. s of Z W ae, espectvely, K K 2 F Z z = K K 2 p p w k 0 < w k. d c H 2 + H e λ z ν λ z 0 H 2 d ν e ν z z K K 2 F W w = K K 2 p p d c H 2 + H kw λ ν λ w k H 2 d ν kw ν 0 < w k. The dstbuton of Z s also the dstbuton of the sum of p ndependent.v. s wth the dstbuton of the dffeence of two ndependent.v. s wth exponental dstbuton ethe wth smla o dffeent paametes. We should note that although namely n 6 t may seem that t would not be easonable to take p, as a matte of fact n both 8 9 takng p wll yeld pope legtmate dstbutons p.d.f. s. If we take nto account that f the.v. X has an Exponental dstbuton wth ate paamete λ, wth p.d.f. f X x = λ e λx, λ > 0; x > 0, the.v. Y = k e X has a Paeto dstbuton wth ate paamete λ lowe bound paamete k, wth p.d.f. y λ f Y y = λ k y, λ > 0; y k. We wll also say that the.v. X = X wth p.d.f. f X x = λ e λx, λ > 0; x < 0 5

6 has a symmetcal Exponental dstbuton wth ate paamete λ that the.v. Y = /Y wth p.d.f. y λ f Y y = λ k y has an nveted Paeto dstbuton wth ate paamete λ lowe bound paamete k. We may then also note that whle the dstbuton of Z, fo z 0, may be seen as a patcula mxtue of Exponental dstbutons wth ate paametes λ =,..., p, wth weghts p = K K 2 H c λ, =,..., p, wth p p = P [Z 0] fo z 0 as a patcula mxtue of symmetcal Exponental dstbutons wth ate paametes ν, wth weghts wth s = K K 2 H 2 d ν, =,..., p, p s = P [Z 0], the dstbuton of W may, fo w k, be seen as a patcula mxtue of Paeto dstbutons wth ate paametes λ =,..., p lowe bound paametes k, wth weghts p =,..., p, wth p p = P [W k ], fo w k, t may be seen as a mxtue of nveted Paeto dstbutons wth ate paametes ν =,..., p lowe bound paametes k, wth weghts s =,..., p, wth p s = P [W k ]. 2.4 The dstbuton of the dffeence of two GIG dstbutons Let, fo =,..., p l =,..., p 2, be two ndependent.v. s let Y GIG, λ, p, Y 2 GIG 2l, ν l, p 2, Z = Y Y 2. Then, consdeng takng K defned n a smla manne to K n espectvely, K 2 defned n a coespondng manne, usng p 2 nstead of p, 2l nstead of, ν l nstead of λ, fo l =,..., p 2 =,..., p, the p.d.f. of Z s gven by f Z z = K K 2 + maxz,0 p y k e λ y p2 l= 2l y z h h= e ν ly z dy 6

7 = K K 2 + p = K K 2 p p 2 maxz,0 l= p 2 + 2l l= p p 2 = K K 2 l= maxz,0 h= e ν lz y k 2l h h= what, takng, fo m > 0 k IN 0, gves P z = + maxz,0 p 2 P 2 z = e my y k dy = 2l y z h e λ+ν ly e ν l z dy h= y k y z h e λ +ν l y e ν lz dy p K K 2 f Z z = p K K 2 2l l= h= h p 2 2l l= h= h e mz h + z h e λ+ν ly y k+ dy maxz,0 k k!! z m k + z 0 k! m k+ z 0 P z e λ z P 2 z e ν z k+ z h h h t=0 z 0 z 0 k +! t! z t λ + ν l k+ t 5 h k +! z. 6 λ + ν k+ It s howeve nteestng useful to obseve that, gven that the dstbuton of Y Y 2 Y 2 Y ae symmetcal, that we may n 2 ntegate n ode to y 2 nstead of y, we may obtan the p.d.f. of Z gven by a smla expesson to the one n 4, wth P 2 z = P z = p 2 p 2 2l l= h= moeove snce ndeed h k+ z h h t=0 2l l= h= k k +! t! k k k h+ z k t=0 z t k λ + ν l k+ t = k h +! z λ + ν l h+ 7 h +! t! k z t, 8 λ + ν l h+ t k h +! z λ + ν l h+. Ths way, n ode to obtan a smple expesson fo the p.d.f. of Z we may consde the p.d.f. n 4 wth P z gven by 7 P2 z gven by 6. 7

8 Then, usng 3, the c.d.f. of Z may be wtten as p K K 2 F Z z = p K K 2 P z e λ z P2 z e ν z z 0 z 0 9 wth P z = P 2 z = p 2 p 2 2l l= h= 2l l= h= If we consde, fo k > 0, the.v. we have wth k h p K K 2 f W w = p K K 2 Q logkw = Q 2 logkw = Q logkw = p 2 p 2 k h h +! λ + ν l h+ k +! λ + ν k+ W = k e Z Q logkw kw λ Q 2 logkw kw ν 2l l= h= 2l l= h= p K K 2 F W w = p K K 2 p 2 2l l= h= k k h k h w k t=0 h w t=0 k! t! h! t! w k z t λ k t z t ν h t l < w k 22 k h +! logkw λ + ν l h+ 23 h k +! logkw. 24 λ + ν k+ Q logkw kw λ w k Q 2 logkw kw ν 0 < w k 25 k h +! λ + ν l h+ k t=0 k! t! logkw t λ k t 26 Q 2 logkw = p 2 2l l= h= h h k +! λ + ν k+ h t=0 h! t! logkw t. 27 ν h t l As we dd wth 8 9, also n 4 though 8 we may take both p p 2, stll holdng pope dstbutons. 8

9 3 The dstbuton of the poduct of m ndependent om vaables wth GGR dstbutons 3. The case wth all dstnct shape paametes fo the numeato denomnato Let X GGRk, 2,, β =,..., m be m ndependent.v. s. We want an explct concse expesson fo the p.d.f. c.d.f. of the.v. W = X. 28 Based on the esults n subsecton. we have E X h = k h/β Γ 2 + h/β Γ h/β Γ Γ 2 f we take then we have so that f we take we have then f we have, fo =, what, usng W = Y = k /β X E Y h Γ 2 + h/β Γ h/β =, Γ Γ 2 W = X = Φ Z t = = k /β Y } {{ } =K Z = log W = Y = K W m log Y Φ log Y t = Γz = e γz z E Y t Γ 2 + t/β Γ t/β Γ Γ 2 = + z ez/ γ s the Eule gamma constant, may, afte some smplfcatons, be wtten as 29 Φ Z t = β 2 + k β 2 + k + t β + k β + k t. k=0 9

10 Snce the β =,..., m ae not necessaly all postve we take β = β, s = { f β > 0 2 f β < 0 s 2 = { 2 f β > 0 f β < 0, so that we may wte Φ Z t = β s 2 + k β s 2 + k + t β s + k β s + k t, k=0 what shows that, f β s βk s k β s 2 βk s 2k, k,, k {,..., m}, the dstbuton of Z s the same as the dstbuton of a sum of nfntely many ndependent.v. s dstbuted as the dffeence of two ndependent Exponental dstbutons, wth paametes β s 2 + k β s + k =,..., m; h = 0,,.... Altenatvely, Z s dstbuted as the dffeence of two ndependent.v. s, each one wth the dstbuton of the sum of nfntely many ndependent Exponental dstbutons. Thus, snce W = K e Z ; K = k /β, 30 takng s h = β s + h s 2h = β s 2 + h usng 0 above as a bass, we get lm K K 2 m F W w = m lm K K 2 n h=0 n h=0 d h c h H 2h s + H h 2h s h w/k s h, w K d h H 2h s w/k s 2h, 0 < w K 2h K K 2 ae defned n a smla manne as above, that s, K = h=0, fo =,..., m n = 0,,..., c h = n η= ν=0 η ν h n β s + h, K 2 = β s + h β ηs η + ν, d h = h=0 η= ν=0 η ν h n β s 2 + h, n β s 2 + h β ηs 2η + ν, 3 H h = m n l=0 d kl β k s 2k + l β s + h, H 2h = m n l=0 c kl β k s k + l β s 2 + h. To lghten the wtng we may eplace the pa of ndexes k, by h = km +, settng s h = β s + k, fo =, 2; =,..., m; k = 0,..., n. 32 We may now defne K, K 2, c, d, H H 2 =,..., mn +, wth n n a way smla to the one used n subsecton.2, that s, K = mn+ s, K 2 = mn+ s 2, 0

11 , fo =,..., c = H = mn+ k mn+ h= s s k d h s s 2h so that we may wte the p.d.f. of the.v. W as F W w = lm K K 2 mn+ mn+ lm K K 2 d H 2 s 2 d H 2 s 2, d =, H 2 = mn+ k mn+ h= s 2 s 2k c h s h s 2 c + H s w/k s, w K w/k s 2, 0 < w K The geneal case In the geneal case we wll admt that t s possble that some of the paametes s = β s, {,..., m}, wll be equal that also some of the paametes s 2 = β s 2, {,..., m}, wll be equal. Moe pecsely, wthout any loss of genealty, let us suppose that τ of the m paametes s ae equal to s, τ 2 ae equal to s 2, so on, that τ p ae equal to s p, wth p < m p τ = m, that, smlaly, η of the m paametes s 2 ae equal to s 2, η 2 ae equal to s 22, so on, that η p2 ae equal to s 2p2, wth p 2 < m p 2 η = m. Then the chaactestc functon n 29 may be wtten as p Φ Z t = s + β k τ s + β k t p 2 τ s2 + β k η s2 + β k t η k=0 = lm p n k=0 k=0 s + β k τ s + β k t p 2 τ k=0 n s2 + β k η s2 + β k t η that s the chaactestc functon of the dffeence of two ndependent.v. s wth GIG dstbutons, the fst one, that s, the one wth postve sgn, wth depth p n + wth n, wth ate paametes assocated shape paametes s + βk,..., s + β k k=0,...,n k=0,...,n τ,..., τ n+,..., τ,..., τ n+,..., s p + βp k k=0,...,n,..., τ p,..., τ p n+ the second one, that s, the one wth negatve sgn, wth depth p 2 n + wth n, wth shape paametes assocated ate paametes s 2 + βk,..., s 2 + β k k=0,...,n k=0,...,n η,..., η,..., η,..., η n+ n+,..., s 2p + βp 2 k k=0,...,n,..., η p2,..., η p2 n+.

12 Let us consde the vectos [ ] τ = τ,..., τ p, τ,..., τ p,..., τ,..., τ p, η = n+ tmes [ ] η,..., η p2, η,..., η p2,..., η,..., η p2, n+ tmes, fo =,..., p n + l =,..., p 2 n +, wth = kp + h l = kp 2 +, fo k = 0,..., n, h =,..., p =,..., p 2, τ = τ h, η l = η,, smlaly to the vectos s s 2 consdeed n the pevous subsecton, the vectos s = [ s,..., s p, s + β,..., s p + β p,..., s + β n,..., s p + β p n ] s 2 = [ ] s 2,..., s 2p, s 2 + β,..., s 2p + βp,..., s 2 + βn,..., s 2p + βp n, once agan, fo =,..., p n + l =,..., p 2 n +, wth, l, k, h defned as above, s = s h + β hk s 2l = s 2 + β k. The c.d.f. of W = K e Z, fo K defned as n 30, may then be deved fom 25, takng nto account the shape ate paametes mentoned above, as p n+ lm K K 2 Q logw/k w/k s w K F W w = 34 p 2n+ lm K K 2 Q 2 logw/k w/k s 2 0 < w K now K, K 2, Q Q 2 ae defned as n subsecton 2.4, wth p eplaced by p n+, p 2 eplaced by p 2 n +, λ eplaced by s, ν l eplaced by s 2, eplaced by τ 2l eplaced by η l. 3.3 The double non-cental case The followng Lemma s a useful esult fo some of the wok ahead ad ts poof s staghtfowadly obtaned fom the Theoem of global pobablty Robbns, 948; Robbns Ptman, 949. Lemma : Let Z Z 2 be two ndependent.v. s takng values on the non-negatve nteges, such that P Z = = u P Z 2 = = v,, = 0,,.... Let futhe X Y be two.v. s let X = X Z = Y = Y Z 2 =, fo, = 0,,..., let yet g, be a measuable functon. Then Φ X t = u Φ X t, Φ Y t = v Φ Y t f then Φ Z t = Z = gx, Y =0 =0 u v Φ gx,y t. A patcula case of the above Lemma s clealy the case the.v. s X Y ae mxtues. We wll use the above Lemma exactly n ths case. 2

13 We wll say that the.v. Y has a non-cental Genealzed Gamma dstbuton wth shape paamete, ate paamete λ, powe paamete β non-centalty paamete δ f ts p.d.f. may be wtten as a mxtue wth Posson weghts wth ate δ/2 of Genealzed Gamma p.d.f. s wth shape paametes + = 0,,..., ate paamete λ powe paamete β, that s, f f Y y = β λ + β p Γ + e λy y β++, p = δ/2 e δ/2 = 0,,... 35! clealy wth p =. We wll denote the fact that the.v. Y has a non-cental Genealzed Gamma dstbuton wth the above paametes by Y Γ, λ, β; δ, 36 If the.v. Y has the non-cental Genealzed Gamma dstbuton n 36, then t s staghtfowad to show that the.v. X = Y β, β IR\{0}, has a non-cental Gamma dstbuton wth shape paamete, ate paamete λ non-centalty paamete δ. Let us suppose that ae two ndependent.v. s let Y Γ, λ, β; δ Y 2 Γ 2, λ 2, β; δ 2 Z = Y /Y 2. Then, the.v. Z wll have what we call a double non-cental Genealzed Gamma Rato o double non-cental Genealzed F dstbuton. Usng Lemma above, the p.d.f. of Z s, fo k = λ /λ 2, f Z z = =0 p ν β k + B +, k z β 2 z β + k = λ /λ 2 p = δ /2! e δ /2 ν = δ 2/2 e δ2/2.! We wll denote the fact that the.v. Z has ths dstbuton by Z GGR, 2, k, β; δ, δ 2. In ths secton we wll be nteested n obtanng the dstbuton of W = n Z 37 Z GGR, 2, k, β ; δ, δ 2. Usng Lemma expesson 3, n subsecton 3., fo the c.d.f. of W n the cental case, we obtan, fo the case all shape paametes n the numeato ae dffeent all shape paametes n the denomnato ae also dffeent, that s the case, β s β ks k β s 2 β ks 2k fo all k wth, k {,..., m}, 3

14 defnng s k s 2k as n 3, we have fo s kh = β k s k + h + s 2kh = β k s 2k + h +, F W w = =0 =0 lm K K 2 lm K K 2 n h=0 n h=0 m p k ν k d hk H 2kh s 2kh c hk +H kh s kh m d hk p k ν k H 2kh w/k s 2kh s 2kh w/k s kh w K 0 < w K K = n βηs η + + h K 2 = η= h=0 n βηs 2η + + h, η= h=0 H kh = m n η= l=0 d ηl β ηs 2η ++l β k s k++h, H 2kh = m n η= l=0 c ηl β ηs η ++l β k s 2k++h, wth c hk = n η= ν=0 η k ν h β k s k++h β ηs η ++ν, d hk = n η= ν=0 η k ν h β k s 2k++h β ηs 2η ++ν yet, fo, = 0,,... k =,..., m, p k = δ k/2! e δ k/2 ν k = δ 2k/2 e δ2k/2. 38! In case that all the non-centalty paametes n the numeato of W ae the same, say δ k = δ, k {,..., m}, all non-centalty paametes n the denomnato ae also the same, wth say δ 2k = δ 2, k {,..., m}, the only dffeence n the dstbuton of W would be that the weghts p k ν k would be no moe a functon of k the c.d.f. of W could be wtten as F W w = =0 =0 p ν p ν lm K K 2 lm K K 2 n m h=0 n m h=0 d hk H 2kh s 2kh +H kh c hk s kh d hk H 2kh s 2kh w/k s kh w K w/k s 2kh 0 < w K Let k m epesent the emande of the ntege ato of k by m let k m = + k m. Then, an altenatve epesentaton fo the c.d.f. of W n ths case may be deved fom the c.d.f. n 33 n subsecton 3., obtaned fo the cental case of W, we have, the c.d.f. of W n 37 s, fo k = s k + β k p 2k = s 2k + β k p2, 4

15 wth s k s 2k defned by 32, lm K K 2 =0 F W w = =0 lm K K 2 mn+ K = k, K 2 = wth yet c k = mn+ mn+ h= h k p k = δ k/2! mn+ mn+ p,k m ν,k m H 2k +H k c k k p,k m ν,k m H 2k 2k, H k = h s k e δ k/2 mn+ h= d k = d k 2k w/k k s w K d k w/k s 2k 2k d h h s 2k mn+ h= h k mn+, H 2k = 2h s 2k ν k = δ 2k/2 e δ2k/2.! h= 0 < w K c h 2h s k In case all the non-centalty paametes n the numeato of W ae the same, say δ k = δ, k {,..., m}, all non-centalty paametes n the denomnato ae also the same, wth say δ 2k = δ 2, k {,..., m}, the only dffeence n the dstbuton of W would be that the weghts p k ν k would be no moe a functon of k the c.d.f. of W could be wtten as F W w = =0 =0 p ν p ν lm K K 2 mn+ mn+ lm K K 2 H 2k d k 2k +H k c k k H 2k w/k k s w K d k w/k s 2k 0 < w K. 2k Fo the double non-cental case coespondng to the geneal case studed n subsecton 3.2 above, some of the paametes s = β s, {,..., m}, wll be equal also some of the paametes s 2 = β s 2, {,..., m}, wll be equal, we have, fom 34, wth F W w = lm K K 2 =0 lm K K 2 =0 p n+ p n+ p n+ k = s k + βk p, 2k = s 2k + βk p2, K = Q k logw/k w/k s k Q 2k logw/k w/k s 2k 5 k τ k, K 2 = w K w K p 2n+ η s k 2k,

16 Q k logw/k = Q 2k logw/k = τ k g= τ k g= p 2 n+ c kg l= p 2n+ c kg l= η l h= η l h= g u=0 h u=0 g u h u h u! k + s 2l h+u g u! k + s 2l g+u g u t=0 g u t=0 g u! t! h u! t! logw/k t k g u t logw/k t h u t 2l, fo k =,..., p fo l =,..., p 2, c k,τ k, = p τk! l= l k wth, fo l =,..., τ k k =,..., p, c k,τ k l, = l l k τ l, d l,η l, = l h=, fo m =,..., η l l =,..., p 2, d l,η l m, = m m h=, fo h = 0,..., τ k m = 0,..., η l, p2 ηl! k l s 2k η k 2l τ k l + h! τ k l! R h, k, p, τ, c k,τ k l h, η l m + h! η l m! R 2 h, l, p 2, η, 2 d l,η l m h, p R h, k, p, τ, = τk k l h, l= l k p2 R 2 m, l, p 2, η, 2 = n= n l η l s 2l 2n m. 4 Conclusons Fnal Remaks We should stengthen that the esults obtaned may be easly dectly genealzed to the case we ae nteested n the dstbuton of the.v. Z = n γ Y α γ IR +, α IR\{0} Y ae ndependent.v. s wth GGR dstbutons, snce t s staghtfowad to show that f then γ Y α Y GGR, 2, k, β ; δ, δ 2 39 GGR, 2, k γ β/α, β α ; δ, δ 2 If the.v. X has a stad Beta dstbuton, the dstbuton of ethe X/X o X/ X s then usually called a stad Beta pme o Beta second knd dstbuton. Howeve we should note that the dstbuton of ethe X/X o X/ X s actually only a patcula GGR dstbuton. Actually t s easy to show that f Y GGR, 2, k, β ; 0, 0. 6

17 wth k = β = then the.v. s /+Y Y /+Y have stad Beta dstbutons wth paametes 2 o 2, espectvely, whle fo geneal k > 0 geneal β IR\{0} the.v. X = Y / + Y has what we call a genealzed Beta dstbuton wth p.d.f. f X x = β k B, 2 + k x x 2 z β z β whch clealy educes to the stad Beta p.d.f. fo k = β =, whle the.v. X = / + Y has of couse a smla p.d.f. wth 2 swapped. Fo ths eason the dstbuton of Y s also called, fo k = β = a Beta pme dstbuton fo geneal k β a genealzed Beta pme dstbuton. Thus, fo the non-cental dstbuton n 39 we may say that Y has also a non-cental genealzed Beta pme dstbuton thus, the dstbutons obtaned n secton 3 ae also the dstbutons of the poduct of ndependent cental non-cental genealzed Beta pme.v. s. Clealy f n 39 we have = m/2, 2 = n/2, wth m, n IN k = β = we have the.v. s Y wth ethe cental o non-cental F dstbutons, accodng to the case that δ = 0 o δ 0, wth m n degees of feedom. The esults n secton 3 may then be eadly appled to both the cental non-cental cases. Also, f n 39 we have = /2, 2 = n /2, k = /n β = 2, wth n IN, we have Y wth the so-called folded T dstbuton, that s the dstbuton of a.v. Y = T, T s a.v. wth a Student T dstbuton wth n degees of feedom. If nstead we take 2 = /2 k = we wll have Y wth the so-called folded Cauchy dstbuton, that s the dstbuton of the absolute value of a.v. wth a stad Cauchy dstbuton, o a Student T dstbuton wth only degee of feedom. In both cases, once agan, the esults n secton 3 may be eadly appled to both the cental non-cental cases. Besdes, gven the way ou appoach was conducted, even the doubly non-cental case, each of the GGR.v. s s the ato of two non-cental Gamma.v. s, was then eadly at h. Also, snce, fom the begnnng, n subsecton 2., opposte to what s commonly done, we consdeed the powe paametes as eal only non-null, allowng them to be negatve, the esults obtaned may be dectly extended to the dstbuton of the ato of two ndependent GGR om vaables o the dstbuton of the ato of two poducts of ndependent GGR om vaables. In ode to obtan the dstbuton of the ato of two ndependent GGR om vaables one smply has to consde m = 2 n 28, takng then fo the om vaable n the denomnato the symmetc of ts powe paamete. The dstbuton fo the ato of two poducts of ndependent GGR om vaables may then be obtaned by takng the dstbuton of the poduct of the whole set of om vaables, takng the symmetc of the powe paametes fo the om vaables n the denomnato. As a by-poduct, n subsectons we also obtan closed fom epesentatons, not nvolvng any nfnte sees o unsolved ntegals, fo the dstbuton of the dffeence of two ndependent sums of a fnte numbe of Exponental om vaables wth all dffeent ate paametes o Gamma om vaables wth all dffeent ate paametes ntege shape paametes, unde the fom of patcula mxtues of ethe Exponental o Gamma dstbutons, accodng to the case. Also, f we consde the exponental of a Gamma om vaable wth ntege shape paamete as a genealzed Paeto dstbuton, then the dstbuton of the om vaable W n subsecton 2.3 s the dstbuton of the ato of two ndependent poducts of Paeto dstbutons, whle the dstbuton of the same om vaable n subsecton 2.4 s the dstbuton of the ato of two ndependent poducts of genealzed Paeto dstbutons, expessed as a patcula mxtue of Paeto nveted Paeto dstbutons. Gven the fom of the exact dstbutons obtaned fo the poduct of ethe cental noncental GGR.v. s asymptotc nea-exact dstbutons ae eadly at h, beng not teated n ths pape due to length lmtatons. They ae ntended to be publshed n a sepaate pape. We may thnk of asymptotc dstbutons by smple tuncaton of the sees obtaned what would ndeed gve manly unsatsfactoy esults, manly n tems of c.d.f. quantles, owng to the fact that then the weghts would not add up to the ght values, peventng ths way the c.d.f ṣ fom eachng the value. Much bette esults may be obtaned f we consde nea-exact dstbutons, based on the concept of keepng a good pat of the exact chaactestc functon unchanged appoachng the emanng, desably, much smalle pat, by an asymptotc esult Coelho, 2003, 2004 what, gven 7

18 the fom obtaned fo the exact dstbutons, would lead us to consde fo example the tuncaton of the nfnte sees obtaned, coupled wth one o two moe tems that would both make the weghts add up to the ght value the fst two, thee o fou moments to match the fst exact ones. REFERENCES Coelho, C. A The Genealzed Intege Gamma dstbuton a bass fo dstbutons n Multvaate Statstcs, J. Multv. Analyss, 64, Coelho, C. A A Genealzed Intege Gamma dstbuton as an asymptotc eplacement fo a Logbeta dstbuton Applcatons, Ame. J. Math. Manag. Scences, 23, Coelho, C. A The Genealzed Nea-Intege Gamma dstbuton: a bass fo nea-exact appoxmatons to the dstbuton of statstcs whch ae the poduct of an odd numbe of ndependent Beta om vaables, J. Multv. Analyss, 89, Pham-Ga, T. Tukkan, N Opeatons on the genealzed-f vaables applcatons. Statstcs, 36, Povost, S. B On the dstbuton of some test statstcs connected wth the multvaate lnea functonal elatonshp model. Commun. Statst. - Theoy Meth., 5, Robbns, H Mxtue of dstbuton. Ann. Math. Stat., 20, Robbns, H., Ptman, E. J. G Applcaton of the method of mxtues to quadatc foms n nomal vaates. Ann. Math. Stat., 20, Shah, M. C. Rathe, P. N Exact dstbuton of poduct of genealzed F -vaates. The Canadan Jounal of Statstcs, 2,