A pathway to matrix-variate gamma and normal densities

Transcription

1 Linea Algeba and its Applications A pathway to matix-vaiate gamma and nomal densities A.M. Mathai Depatment of Mathematics and Statistics, McGill Univesity, 805 Shebooke Steet West, Monteal, Que., Canada H3A K6 Received 8 May 004; accepted 1 Septembe 004 Submitted by R.A. Bualdi Abstact A geneal eal matix-vaiate pobability model is intoduced hee, which coves almost all eal matix-vaiate densities used in multivaiate statistical analysis. Though the new density intoduced hee, a pathway is ceated to go fom matix-vaiate type-1 beta to matix-vaiate type- beta to matix-vaiate gamma to matix-vaiate Gaussian o nomal densities. Othe densities such as extended matix-vaiate Student t, F, Cauchy density will also come in as paticula cases. Connections to the distibutions of quadatic foms and genealized quadatic foms in the new matix ae established. The pesent day analysis of these poblems is mainly confined to Gaussian andom vaiables. Thus, though the new distibution, all these theoies ae extended. Connections to cetain geometical pobability poblems, such as the distibution of the volume of a andom paallelotope in Euclidean space, is also established. 004 Elsevie Inc. All ights eseved. AMS classification: 15A5; 6E15 Keywods: Random matices; Matix-vaiate statistical distibutions; Extended gamma; Beta; t; F; Cauchy distibutions; Quadatic foms Tel.: ; fax: addesses: mathai@math.mcgill.ca, mathai8@hotmail.com /$ - see font matte 004 Elsevie Inc. All ights eseved. doi: /j.laa

2 318 A.M. Mathai / Linea Algeba and its Applications Intoduction Let X = x ij, i = 1,...p, j = 1,...,, p, ofankp and of eal scala vaiables x ij s fo all i and j, subject to the condition that the ank of X is p, having the density fx,wheefxis a scala function of X given by fx= c A 1 XBX A 1 α I a1 qa 1 XBX A 1 1 q 1.1 fo A = A > 0andp p, B = B > 0and, a,,q scalas, a>0, >0, I a1 qa 1 XBX A 1 > 0, whee A and B ae fee of the elements in X and c is the nomalizing constant. Fo convenience let A 1 and B 1 denote the eal symmetic positive definite squae oots of A and B espectively. A pime denotes the tanspose, denotes the deteminant of, I is the identity matix, >0 means that the eal symmetic matix is positive definite. Also t will denote the tace of and R will denote the eal pat of. The nomalizing constant c can be evaluated by using the following tansfomations. Let Y = A 1 1 p XB dy = A B dx by using Theoem 1.18 of [3]. Let U = YY dy = π p Γ U p+1 du p by using Theoem.16 of [3], whee fo example, Γ p α = π pp 1 4 ΓαΓ α 1 Γ α p 1, α > p 1, 1. taking α as eal, and if complex the condition is Rα > p 1. Let V = a1 qu dv =[a1 q] pp+1 du by using Theoem 1.0 of [3]. Then c 1= fxdx = X A B p YY α I a1 qyy 1 q dy Y π p = Γ p A B p u At this stage we can conside thee possibilities: i q<1, ii q>1, iii q = 1. Let us conside these one by one. U α+ p+1 I a1 qu 1 q du. 1.3 Case i: q<1. Then a1 q > 0 and then by making the tansfomation V = a1 qu we have

3 A.M. Mathai / Linea Algeba and its Applications c 1 π p = Γ p A B p V α+ [a1 q] pα+ p+1 I V 1 q dv. V 1.4 Now, evaluating the integal in 1.4 by using a matix-vaiate type-1 beta, see Section of [3], we have c 1 π p Γ p α + Γp = Γ p A B p [a1 q] pα+ Γ p α + + fo α + > p 1. We will assume the paametes to be eal fo convenience. Case ii: q>1. In this case wite 1 q = q 1 so that q 1 > 0. Then in 1.3 I a1 qu 1 q 1.5 = I + aq 1U q and then make the tansfomation V = aq 1U. Then c 1 π p = Γ p A B p V α+ [aq 1] pα+ p+1 I + V q 1 dv. V Evaluating the integal by using a matix-vaiate type- beta integal, see Section of [3], we have the following: c 1 π p Γ p α + Γp q 1 α = Γ p A B p [aq 1] pα+ Γ p q fo α + > p 1, q 1 α > p 1. Case iii: q = 1. Iespective of whethe q appoaches 1 fom the left o fom the ight it can be shown that the deteminant containing q in 1.3 and 1.6 has the following fom, which will be stated as a lemma: Lemma 1.1 lim I a1 qu 1 q = e atu. q 1 This esult can be seen by obseving the following: Fo a eal symmetic positive definite matix U thee exists a matix Q such that

4 30 A.M. Mathai / Linea Algeba and its Applications QQ = I, Q Q = I, Q UQ = diagλ 1,...,λ p, λ j > 0, j = 1,...,p 1.8 whee diagλ 1,...,λ p denotes a diagonal matix with the diagonal elements λ 1,..., λ p. Then I a1 qu = I a1 qqq UQQ = I a1 qq UQ = I a1 qdiagλ 1,...,λ p p = 1 a1 qλ j. But lim 1 a1 qλ j 1 q = e aλ j. q 1 Then p lim I a1 qu 1 q = e a λ j = e atu q 1 which establishes the esult. Hence in case iii c 1 π p = Γ p A B p U α+ p+1 e atu du U π p = Γ p A B p Γ p α + a pα+, α + > p by using Section 5.1.1of[3].. A geneal density Fo X, A, B, a,, q as defined in 1.1 let fx= c A 1 XBX A 1 α I a1 qa 1 XBX A 1 1 q.1 fo q /= 1, and fo q = 1 = c A 1 XBX A 1 α e at [ A 1 XBX A 1 ]. whee c in.1 isgivenby1.5 foq<1 and by 1.7 foq>1. Fom 1.9 we have the c in.. In.1 a necessay condition to be met is that I a1 qa 1 XBX A 1 > 0. Note that when q moves fom to 1, that is, <q<1 then.1 maintains a matix-vaiate type-1 beta fom and when q becomes geate than 1 then the type-1 beta fom switches to a type- beta fom. That is, to the left of 1foq a type-1 beta fom is available and to the ight of 1 fo q a type- beta fom

5 A.M. Mathai / Linea Algeba and its Applications is available. Both these type-1 and type- beta foms go to a matix-vaiate gamma fom at q = 1. Thus the pathway fo q descibes a wide ange of statistical densities coveing type-1 and type- beta foms and gamma foms. It may also be noted fom 1.1 that one need not go fo the symmetic squae oots A 1 and B 1 of A and B, one needs to obtain only a epesentation A = A 1 A 1 and B = B 1B 1. Then one A 1 could be eplaced by A 1 and one B 1 by B Abitay moments Abitay hth moment fo the deteminant A 1 XBX A 1 o that fo XBX can be obtained fom c 1 in 1.5, 1.7, 1.9 fo the cases q<1, q>1, q = 1 espectively, by changing α to α + h and then taking the atio of the nomalizing constants. Thus we have the following, whee E denotes the expected value. Theoem.1 E A 1 XBX A 1 h 1 Γ p α + h + = [a1 q] ph Γ p α + Γ p α + + Γ p α + h + + fo q<1,α+ h + > p Γ p α + h + Γ p = q 1 α h [aq 1] ph Γ p α +.4 Γ p q 1 α fo q>1, q 1 α h > p 1,α+ h + > p 1 1 Γ p α + h + = a ph Γ p α + fo q = 1, α + h + > p 1..5 One may wonde whethe.3 and.4 goto.5 whenq 1 fom the left and ight espectively. This can be seen fom an asymptotic expansion fo gamma functions o fom Stiling s appoximation. These will be stated as lemmas. Lemma.1. Fo z and a a bounded quantity, Γz + a πz z+a 1 e z,.6 whee means appoximately equal to. Then by applying lemma.1 and witing Γ p in explicit foms one has the following esults.

6 3 A.M. Mathai / Linea Algeba and its Applications Lemma. 1 lim q 1 [a1 q] ph Γ p α + + Γ p α + h + + This can be seen by obseving the following: Γ p α + + p Γ p α + h + = + Γ α + h + + When q goes to 1 fom the left p Γ α + h + + Hence, = p π 1 q 1 q = 1..7 [a] ph Γ α + + j 1. Then, fo example, 1 q + p + 1 j 1 α+h+ + j 1 1 e 1 q = p α+h+ π p + 1 q + pp+1 4 e p 1 q. 1 q 1 [a1 q] ph p Γ Γ α + + α + h + + j 1 j 1 j 1 = 1 a ph.. This establishes that.3 goes to.5 whenq 1 fom the left. In a simila way one can see that.4 also goes to.5. Thus q is a pathway fom moments in.3 and.4togotothemomentsin.5. One can make some inteesting obsevations fom.3.5. Fom.3 we have, E a1 qa 1 XBX A 1 h p Γ α + + h j 1 Γ p α + + j 1 = Γ α + j 1 Γ α h j 1 p = E xj h,.8 whee x j is a eal scala type-1 beta andom vaiable with the paametes α + j 1, 1 q + p + 1, j = 1,...,p.

7 A.M. Mathai / Linea Algeba and its Applications Thus, stuctually, a1 qa 1 XBX A 1, foq<1, is a poduct p of statistically independently distibuted eal type-1 beta andom vaiables with the paametes as mentioned above. Similaly fo q>1, aq 1A 1 XBX A 1 is a poduct of p statistically independently distibuted type- eal scala beta andom vaiables, and fom.5, aa 1 XBX A 1 is a poduct of p independently distibuted gamma andom vaiables. These poducts of independent eal scala type-1 beta and type- beta andom vaiables go to a poduct of independent eal scala gamma andom vaiables when q 1. Thus, though q a pathway is achieved to go to poduct of independent gamma vaiables fom poducts of independent type-1 beta and type- beta vaiables... Some special cases In.whenα = 0 one has the famous matix-vaiate Gaussian o nomal density. Fo q = 1 and α eplaced by the degees of feedom and with appopiate change in A = 1 V 1, V = V > 0andB = I and expected value of X null, we have the extended Wishat density. The standad Wishat density is the cental density in multivaiate statistical analysis. We have extended type-1 beta, extended type- beta, F, Student t, Cauchy and othe distibutions coming as special cases. Note that all these ae defined on ectangula matices and hence we call them the extended vesions. The following is a list of some paticula cases and the tansfomations ae listed to go fom the extended vesions to the egula cases. If a location matix is to be intoduced then one may eplace X by X M whee M is a p constant matix. q<1, a1 q = 1 Extended type-1 beta density q<1, a1 q = 1, Y = XBX Non-standad type-1 beta density q<1, a1 q = 1, Y = A 1 XBX A 1 Standad type-1 beta density q<1, a1 q = 1, α = 0, = 0 Extended unifom density q<1, a1 q = 1, Y = XBX Non-standad unifom density q<1, a1 q = 1, α + = p+1, = 0, Y = A 1 XBX A 1 Standad unifom density q<1, a1 q = 1, α = 0, 1 q = 1 m p 1 Inveted T density of Dickey q<1, a = 1, α = 0, = 1 Aq-binomial density q>1, aq 1 = 1 Extended type- beta density q>1, aq 1 = 1, Y = XBX Non-standad type- beta density q>1, aq 1 = 1, Y = A 1 XBX A 1 Standad type- beta density q>1, aq 1 = 1, q 1 = m, α = 0 T density of Dickey q>1, aq 1 = n 1, q 1 = n+1, Y = A 1 XBX A 1 Standad T density q>1, aq 1 = 1, α + = p+1, q 1 = 1, Y = A 1 XBX A 1 Standad Cauchy density q>1, aq 1 = m n, α + = m, q 1 = m+n, Y = BXB Non-standad F density q>1, aq 1 = m n, α + = m, q 1 = m+n, Y = A 1 BXB A 1 Standad F density q = 1, a = 1, = 1 Extended gamma density q = 1, a = 1, = 1, Y = XBX Non-standad gamma density q = 1, a = 1, = 1, Y = A 1 XBX A 1 Standad gamma density q = 1, a = 1, = 1, α = 0 Gaussian density q = 1, a = 1, = 1, α + = n, Y = XBX, A = 1 V 1 Wishat density

8 34 A.M. Mathai / Linea Algeba and its Applications Fo p = 1, = 1, that is, in the scala case, I a1 qz 1 q = 1 a1 qz 1 q. Let us see what we obtain if we expand this by Taylo seies. [ ] 1 + aq 1z q 1 = 1 az az + [ + q 1] 1!! [ + q 1][ + q 1] az3 3! [ + q 1][ + q 1][ + 3q 1] az4 4!.9 This is a type of q-binomial seies. Hence one can also look upon.1 foα = 0as a matix-vaiate analogue of a q-binomial seies..3. Special cases as quadatic foms One inteesting special case is when p = 1, >p. Then the constant matix A is a scala and without any loss of geneality we may take it as 1. x 1 A 1 XBX A 1 = x1,...,x B. = u say.10 x which is a eal positive definite quadatic fom in the fist ow of X, denoted by x 1,...,x. The density of this quadatic fom is available fom 1.3 fo the case q<1and fom 1.6 fo the case q>1. Denoting the density of u by gu we have the following: Theoem.. The density of u in.10 is given by gu = c 1 u α+ 1 [1 a1 qu] 1 q.11 with 1 a1 qu > 0, whee, fo q<1 [ ] α+ a1 q Γ α q + 1 c 1 = Γ α +, α + > 0,.1 Γ 1 q + 1 fo q>1 c 1 = Γ α + and fo q = 1 [aq 1] α+ Γ q 1 Γ q 1 α, α + > 0, q 1 α > 0,.13

9 A.M. Mathai / Linea Algeba and its Applications c 1 = aα+ Γ α +, α + > Distibutions of quadatic foms in eal Gaussian andom vaiables ae discussed in [6] and the distibutions of genealized quadatic foms with Gaussian vecto andom vaiables ae consideed in [7]. But if the p, p eal andom matix X has a matix-vaiate distibution as in 1.1, which coves ectangula matix-vaiate type- 1beta, type- beta, gamma type and Gaussian type distibutions, then the density of the genealized quadatic fom follows tivially fom 1.1. This will be given as the next theoem. Theoem.3. When the p, p eal andom matix X has the matix-vaiate distibution as given in 1.1 then the genealized quadatic fom Y = A 1 BXB A 1 has the following density, denoted by f 1 Y = c Y α+ p+1 I a1 qy 1 q,.15 whee, fo q>1 c = fo q>1 c = [a1 q] pα+ Γ p α + + Γ p α + Γp [aq 1] pα+ Γ p q 1 Γ p α + Γp q 1 α, α + > p 1,, α + > p 1,.16 q 1 α > p 1.17 and fo q = 1 c = apα+ Γ p α +, α + > p Connection to geometical pobability poblems While consideing the distibutional aspects of the volume content of a -paallelotope geneated by the convex hull of linealy independent andom points in Euclidean n-space many authos had consideed the poblem when the points ae

10 36 A.M. Mathai / Linea Algeba and its Applications isotopic and ae distibuted accoding to a beta type-1, type- and Gaussian situations, see fo example [8,9,10]. The distibutions of the andom points that they consideed wee paticula cases of.11 with B = I. Moe geneal situations in this categoy of poblems ae consideed in [4]. Since the deteminant of the type A 1 XBX A 1, appeaingin1.1, can be consideed to be volume of an appopiately defined paallelotope a moe geneal model in this categoy of poblems is available fom 1.1. Note that the p matix X of full ank can also be looked upon as p linealy independent points in a -dimensional Euclidean space. Then XX is the squae of the volume of the paallelotope geneated by the convex hull of these p points in -space, p. Hence a1 qa 1 XBX A 1 is the squae of the volume of the paallelotope geneated by p points in a tansfomed space. Also fom.3.5 it is seen that A 1 XBX A 1 is stuctually a poduct of p independent type-1 beta, type- beta and gamma andom vaiables coesponding to q<1, q>1 and q = 1 espectively. The same stuctue is appeaing in geometical pobability poblems also. Fo such stuctues, appoximations and asymptotic esults ae deivedin[]andinchapte4of[5]. Hence appoximations and asymptotic esults will not be discussed hee. 4. Remaks In 1.1 we dealt with a geneal model when the elements in X ae eal scala andom vaiables. If a elocation paamete matix is to be intoduced then we may eplace X by X M whee M is a location paamete matix. If the model in 1.1 is to be extended to the complex domain then the Jacobians and the integals will be slightly diffeent. The necessay tools ae available in [3] and the pocedue is paallel to the eal case. Hence we will not deal with the case hee when X is in the complex domain. When p = 1, = 1, A = I, B = I, = 1andα = 0 we obtain Tsallis statistics as a special case fom 1.1. The q-binomial function [1 1 qt] 1 q 1 is also a solution of the powe law dy = y q 4.1 dt which is associated with the genealized entopy k f 1 q tdt 1, 4. q 1 whee ftis a density function and k is a constant. When q 1, 4. goes to Shannon s entopy. These consideations ae vey elevant in physics poblems. Nowadays Tsallis statistics is a hot subject, applicable in a wide ange of poblems in astophysics, extending the theoies in vaious topics in astophysics aeas. Fo a window into the vast aeas of eseach activities one may stat with [11].

11 A.M. Mathai / Linea Algeba and its Applications Extensions of the ideas in 1.1 to densities involving many matix vaiables in the eal o complex domain ae also staightfowad. As an example let us look into the matix-vaiate Diichlet type family of distibutions. Fo a discussion of matixvaiate Diichlet and Liouville distibutions see [1]. Let X j, p j, j p, j = 1,...,kbe eal matix andom vaiables having a joint density of the following type: k fx 1,...,X k = C k A 1 j X j M j B j X j M j A 1 α j j k I 1 q A 1 j X j M j B j X j M j A 1 j, 4.3 whee C k is the nomalizing constant, A j, p p, B j, j j, j = 1,...,k ae eal symmetic positive definite constant matices and M j, p j, j = 1,...,k ae constant matices. The nomalizing constant C k can be evaluated by using the steps descibed in this pape. Many inteesting popeties can be seen fom the model in 4.3. Fo q<1 and the last facto in 4.3 emaining positive, the density is an extended Diichlet type-1 type, then when q>1the model switches to an extended type- type. But when q = 1 the andom matices ae independently distibuted and of the extended gamma types. This statistical independence popety is a supising esult. Thee ae vaious genealizations of the Diichlet model available in the liteatue. Such genealizations can also be extended to the ectangula matix-vaiate cases, eal o complex, and those can then be extended to thei q-vesions by using the pocedue discussed in this pape. Anothe obsevation that one can make is the following: In the eal scala case ou model in 1.1 becomes fx= cy α [1 1 qy] 1 q, y = xx = x 4.4 with 1 1 qy > 0, taking A = I, B = I. In this case, we can eplace y by z δ, δ>0. Then when q 1 one can go to genealized gamma, Weibull and othe distibutions. But in the matix case, powes such as δ ae not feasible, even though we ae dealing with eal symmetic positive definite o hemitian positive definite matices, because when tansfomations ae needed the Jacobians do not go into nice foms. Even fo δ = see the complicated fom of the Jacobian fom [3]. The special cases available fom 4.4, which itself is a special case of 1.1, ae the following: q = 1, α = 0, a = 1 Gaussian o nomal density fo <x< q = 1, α = 3 4, a = 1 Maxwell Boltzmann density in physics q = 1, α = 1, a = 1 Rayleigh density q = 1, α = n 1, a = 1 Hemet density q = 0, α = 0, = 1 U-shaped density q =, a = v 1, = v+1, α = 0 Student-t fo v degees of feedom, <x<

12 38 A.M. Mathai / Linea Algeba and its Applications q =, a = 1, = 1 Cauchy density fo <x< q<1, a1 q = 1, x = y Standad type-1 beta density q>1, aq 1 = 1, x = y Standad type- beta density α = 1, = 1, a = 1, x = y Tsallis statistics in astophysics, powe law, q- binomial density α = 1, q = 0, = 1, x = y Tiangula density q =, α + 1 = m, a = m n, = m+1, x = y F-density q = 1, α = 1 mg, a = 1, = KT, x = y Helley s density in physics q = 1, a = 1, x = y Gamma density q = 1, a = 1, = 1, α + 1 = v, x = y Chisquae density fo v degees of feedom q = 1, a = 1, α = 1, x = y Exponential density Laplace density with y= z, <z< q = 1, a = 1, x = z δ, δ>0 Genealized gamma density q = 1, a = 1, α = 1, x = z δ, δ>0 Weibull density q =, a = 1, =, α = 1, x = e y Logistic density fo <y< q =, a = e δ, = 1, α = 1, x = e γy, γ>0 Femi Diac density in physics Acknowledgment The autho would like to thank the Natual Sciences and the Engineeing Reseach Council of Canada fo financial suppot fo caying out this eseach. Refeences [1] R.D. Gupta, D.St.P. Richads, Multivaiate Liouville distibutions, J. Multivaiate Anal [] A.M. Mathai, On a conjectue in geometic pobability egading asymptotic nomality of a andom simplex, Ann. Pobab [3] A.M. Mathai, Jacobians of Matix Tansfomations and Functions of Matix Agument, Wold Scientific Publishing, New Yok, [4] A.M. Mathai, Random p-content of a p-paallelotope in Euclidean n-space, Adv. Appl. Pobab [5] A.M. Mathai, An Intoduction to Geometical Pobability: Distibutional Aspects with Applications, Godon and Beach, New Yok, [6] A.M. Mathai, S.B. Povost, Quadatic Foms in Random Vaiables: Theoy and Applications, Macel Dekke, New Yok, 199. [7] A.M. Mathai, S.B. Povost, T. Hayakawa, Bilinea foms and zonal polynomials, Lectue Notes in Statistics, 10, Spinge-Velag, New Yok, [8] R.E. Miles, Isotopic andom simplices, Adv. Appl. Pobab [9] H. Ruben, The volume of an isotopic andom paallelotope, J. Appl. Pobab [10] H. Ruben, R.E. Miles, A canonical decomposition of the pobability measue of sets of isotopic andom points in R n, J. Multivaiate Anal [11] T. Yanano, Some popeties of q-logaithm and q-exponential functions in Tsallis statistics, Physica A