Distributions of Ratios: from Random Variables to Random Matrices*

Home , Beta distribution, Univariate distribution

American Open Journal of Statistics, 2011, 1, 93-104 doi:10.4236/ojs.2011.12011 Published Online July 2011 (http://www.SciRP.org/journal/ojs)

Distributions of Ratios: From Random Variables to Random Matrices*

Thu Pham-Gia#, Noyan Turkkan Department of Mathematics and Statistics, Universite de Moncton, Canada E-mail: #[email protected] Received March 25, 2011; revised April 17, 2011; accepted April 25, 2011

Abstract

The ratio R of two random quantities is frequently encountered in probability and statistics. But while for unidimensional statistical variables the distribution of R can be computed relatively easily, for symmetric positive definite random matrices, this ratio can take various forms and its distribution, and even its definition, can offer many challenges. However, for the distribution of its determinant, Meijer G-function often provides an effective analytic and computational tool, applicable at any division level, because of its repro- ductive property.

Keywords: Matrix Variate, Beta Distribution, Generalized-F Distribution, Ratios, Meijer G-Function, Wishart Distribution, Ratio

1. Introduction at the present time , we still know little about their numerical values, to be able to effectively compute the In statistical analysis several important concepts and power of some tests. methods rely on two types of ratios of two independent, The use of special functions, especially Meijer G-fun- or dependent, random quantities. In univariate statistics, ctions and Fox H-functions [4] has helped a great deal in for example, the F-test, well-utilized in Regression and the study of the densities of the determinants of products Analysis of variance, relies on the ratio of two indepen- and ratios of random variables, a domain not fully ex- dent chi-square variables, which are special cases of the plored yet, computationally. A large number of common gamma distribution. In multivariate analysis, similar pro- densities can be expressed as G-functions and since blems use the ratio of two random matrices, and also the products and ratios of G-functions distributions are again ratio of their determinants, so that some inference using a G-functions distributions, this process can be repeatedly statistic based on the latter ratio, can be carried out. La- applied. Several computer applicable forms of these tent roots of these ratios, considered either individually, functions have been presented by Springer [5]. But it is or collectively, are also used for this purpose. the recent availability of computer routines to deal with But, although most problems related to the above ra- them, in some commercial softwares like Maple or tios are usually well understood in univariate statistics, Mathematica, that made their use quite convenient and with the expressions of their densities often available in effective [6]. We should also mention here the increasing closed forms, there are still considerable gaps in multi- role that the G-function is taking in the above two soft- variate statistics. Here, few of the concerned distributions wares, in the numerical computation of integrals [7]. are known, less computed, tabulated, or available on a In Section 2 we recall the case of the gamma distribu- computer software. Results abound in terms of approxi- tion and various results related to the ratio of two gam- mations or asymptotic estimations, but, as pointed out by mas in univariate statistics. In Section 3, going into ma- Pillai ([1] and [2]), more than thirty years ago, asymp- trix variate distributions, we consider first the classical totic methods do not effectively contribute to the practi- case where both A and B are Wishart matrices, leading to cal use of the related methods. Computation for hyper- the two types of matrix variate beta distributions for their geometric functions of matrix arguments, or for zonal ratios. The two associated determinant distributions are polynomials, are only in a state of development [3], and, the two Wilks’s lambdas, with density expressed in terms *Research partially supported by NSERC grant A9249 (Canada). of G-functions. It is of interest to note that the variety of

special mathematical functions used in the previous sec- the same ratios TXXX1112  and TXX21 2. tion can all be expressed as G-functions, making the lat- T2 has the Generalized-F distribution, TGF2 ~,,  , ter the only tool really required. with density : In Section 4, extensions of the results are made in sev- t 1 eral directions, to several Wishart matrices and to the ft;,,    , matrix variate Gamma distribution. Several ratios have Bt,1  their determinant distribution established here, for exam- where   122,,   1 and we have ple the ratio of a matrix variate beta distribution to a FT  2, 2, . 12,212 1 2 Wishart matrix, generalizing the ratio of a beta to a chi I Naturally, when 12  , Tbet1~a1, 2 and square. Two numerical examples are given and in Sec- II Tbeta21  , 2 . tion 5 an example and an application are presented. Fi- Starting now from the standard beta distribution de- nally, although ratios are treated here, products, which fined on (0,1), Pham-Gia and Turkkan [9] give the ex- are usually simpler to deal with, are also sometines pression of the density of R, and also of WXXY  , studied. using Gauss hypergeometric function 21F . . For the general beta defined on a finite interval (a,b), Pham-Gia 2. Ratio of Two Univariate Random and Turkkan [10] gives the density of R, using Appell’s Variables function FD . , a generalization of 21F . . In [11], several cases are considered for the ratio X/Y, with We recall here some results related to the ratios of two YGa~,  . In particular, the Hermite and Tricomi independent r.v.’s so that the reader can have a compara- functions, H . and  . , are used. The General- tive view with those related to random matrices in the ized-F variable being the ratio of two independent next section. gamma variables, its ratio to an independent Gamma is 2 First, for XN~,XX , independent of also given there, using  . . Finally, the ratio of two 2   YN~,YY, RXY has density established nu- independent Generalized-F variables is given in [11], merically by Springer ([5], p. 148), using Mellin trans- using Appell function FD . again. All these operations form. Pham-Gia, Turkkan and Marchand ([8]) estab- will be generalized to random matrices in the following lished a closed form expression for the more general case section. 22 of the bivariate normal XY,~ NXYXY ,;,; , using Kummer hypergeometric function 11F . . Sprin- 3. Distribution of the Ratio of Two Matrix ger ([5], p.156) obtained another expression but only for Variates the case  XY 0 . The gamma distribution, XGa~,  , with density 3.1. Three Types of Distributions 1  gx;, x exp x   ,x0 has as spe- Under their full generality, rectangular (p × q) random cial case Ga n 2,2called the the Chi-square with n 2 matrices can be considered, but to avoid several difficul- degrees of freedom  , with density : n ties in matrix operations and definitions, we consider n 1 gxn;exp222, x2  xn/2  n x0. only symmetric positive definite matrices. Also, here, we  will be concerned only with the non-singular matrix, With independent Chi-square variables X ~  2 and with its null or central distribution and the exact, 1 n1 X ~  2 , we can form the two ratios non-asymptotic expression of the latter. 2 n2 WXXX1112 and WXX212 which have, re- For a random (p × q) symmetric matrix, p  1 , there spectively, the standard beta distribution on [0,1] (or beta are three associated distributions. We essentially distin- I of the first kind beta 12, , with density defined on guish between: 0,1 by: 1) the distribution of its elements (i.e. its pp 12  1  2 1 1 independent elements eij in the case of a symmetric fx x 1, x B12 , 12,0  . matrix), called here its elements distribution (with matrix and standard betaprime distribution ( beta of the second input), for convenience. This is a mathematical expres- kind betaII  , ) defined on 0, by 12   sion relating the components of the matrix, but usually, it  fy y1 1  B,1 y12 . is too complex to be expressed as an equation (or several  12   equations) on the elements eij themselves and hence, The Fisher-Snedecor variable F is just a multiple most often, it is expressed as an equation on its determi- 12, of the univariate beta prime W2 . nant.

For independent XGaiii~,,1 i ,2, we can form 2) the univariate distribution of its determinant (de-

qp1 nq noted here by X instead of det X ) (with positive  22 real input), called determinant distribution, and they obtained gKGGIG, i.e.

3) the distribution of its latent roots (with p-vector in- II qn G  Beta , . put), called latent roots distribution. 22 These three distributions evidently become a single Similarly, for Type I ratio, we can also have 5 types of ra- one for a positive univariate variable, then called its den- 1/2 1 2T tios, and we will consider UXYXXYs   , sity. The literature is mostly concerned with the first dis- the symmetric form of the ratio XX Y1 for ele- tribution [12] than with the other two. The focal point of ments distribution. this article is in the determinant distribution. 2) In general, the density of the determinant of a random matrix A , can be obtained from its elements dis- 3.2. General Approach tribution f . of the previous section, in some simple cases, by using following relation for differentials: When dealing with a random matrix, the distribution of dAAAA tr  1 d ([15], p. 150). In practice, fre- the elements within that random matrix constitutes the quently, we have to manipulate A directly, often us- first step in its study, together with the computation of its ing an orthogonal transformation, to arrive at a product moments, characteristic function and other properties. of independent diagonal and off-diagonal elements. Let X and Y be two symmetric, positive definite inde- The determinants of the above different ratios pendent (p × q) random matrices with densities (or ele- Zi ,1,,5i   , have univariate distributions that are iden- ments distributions) fX X and fY Y . As in uni- tical, however, and they are of much interest since they variate statistics, we first define two types of ratios, type will determine the null distribution useful in some statis- I of the form X XY, and type II, of the form: X Y . tical inference procedures. However, for matrices, there are several ratios of type II 3) Latent roots distributions for ratios, and the distri- that can be formed: ZX 1Y, ZY 1 X and T 1 2 butions of some associated statistics, remain very com- 12 12 12 ZYXY3  , where Y is the symmetric plicated, and in this article, we just mention some of their 12 2 square root of Y, i.e. YY = Y, beside the two basic properties. For the density of the latent roots formed with the Cholesky decomposition of Y, 1 1 1 1 ll1,, p  , we have, using the elements density f . : ZUX T U and ZV T XV, where 4   5   2 U is upper triangular, with UU T  Y , and V, lower tri- p π 2 p angular, with VV T = Y . hl,, l l l f HLHT d H , 1 pij   pOpp 2 ij 1) For elements distribution, we only consider Z3 ,  which is positive and symmetric, but there are applica- ll1  p tions of Z4 and Z5 in the statistical literature. We can determine the matrix variate distribution of Z3 from where Op  is the orthogonal group, H is an or- those of X and Y. In general, for two matrix variates A thogonal  p p matrix, dH  is the Haar invariant and B, with joint density f AB, A, Bthe density of measure on Op  , and Ll diag1,, lp  ([16], p. 105). GA 12B A12 is obtained by a change of variables: p 12 hfGA  AG, A,dGAA  , with dA associated 3.3. Two Kinds of Beta, of Wilks’s Statistic, and with all elements of A. When A and B are independent, of Latent Roots Distributions we have fffGA, GA,  BAGA A. Gupta and Kabe [13], for example, compute the den- We examine here the elements, determinant and latent sity of G from the joint density g A, B, using the ap- roots distributions of a random matrix called the beta proach adopted by Phillips [14]. matrix variate, the homologous of the standard univariate

p beta.   First, the Wishart distribution, the matrix generalization gKGF   11,dF 22 F22 , F 11 of the chi-square distribution, played a critical part in the 11 22 FGFG11 22 development of multivariate statistics. VC~,Wnp   , is AA11 12 called a Wishart Matrix with parameters n and C if its where in the block division of A, A  , we density is: AA21 22 np1 have: AApp, qq while  F , F is the 1/2np n/2 11   22   21 22  fetrVCVVC222  n2(1)  p  joint characteristic function of A21 et A22 . tr  AB+  np11qp V >0,np . It is a particular case of the Gamma matrix For gKAB,exp A22 B , 2 variate W~,Gap C , with matrix density:

p1 p 2 1  CpnCpn,, 2  π  12  fetrWCWWC p  (2) fl  i pCpnn2, p 12 with WC0,pp 0 and  p 12 p nn12 pn1    1 2 Hence, If SW~,n , then SG~2,an 2.  llii1  ll ij p p iij1 THEOREM 1: Let AW~,p mA  et BW~,p mB  , We have also: lf1 f. with A and B independent  pp  positive definite ii i PROOF: The proofs of part 1) and part 3) are found symmetric matrices, and mA and mB integers, with in most textbooks in multivariate analysis. For part 2, see mmAB,  p. Then, for the three above-mentioned distributions, we have: [6]. QED. The following explicative notes provide more details 3.3.1. Elements Distribution on the above results. 12 12 1) The ratio UABAAB  has the ma- 3.4. Explicative Notes I trix beta distribution betap  mAB2, m 2, with density given by (3). 1) On Elements Distributions: The positive definite 2) Similarly, the ratio VBAB 12 12 has a symmetric random matrix U has a beta distribution of the I II 12 first kind, U ~,betap  a b if its density is of the form: betap  mAB2, m 2 distribution if  = I p and B is symmetric. Its density is given by (4). 1 1 ap1 bp1 2 2 UI p -U f (,UUI)= 0<<, (3) 3.3.2. Determinant Distribution p β p ab, 1) For U ~,betap  mAB m , its determinant U has Wilks’s distribution of the first type, denoted by It has a beta of the second kind distribution, denoted I by V ~,betaII a b if its density is of the form: U ~2mmpAB ,,2mB (to follow the notation p   p1 in Kshirsagar [17], expressed as a product of independent n betaI , and its density, is given by (5). ||V 2 II f ()VV ab ,0< (4) 2) For V ~,betap  mAB m , its determinant V has β pp(,)|ab IV | Wilks’s distribution of the second type, denoted by II where ab,1 p  2, are positive real numbers, and V ~2mmpmAB,,2B, expressed as a product II p of p independent univariate beta primes beta , and its β p ab,  is the beta function in R , i.e. density is given by (6). ab   β ab,  pp, with p  p ab 3.3.3. Latent Roots Distribution pp1 p 1) The latent roots of U: The null-density of the latent 4 i 1 12 12 p aaπ  . roots f1,, f p of UABAAB  or i1 2 1 AA B is: I  The transformations from U ~betap  a, b to p 2 II π CpnCpn ,,12   V ~,beta a b and vice-versa are simple ones, Also, fl  p i similarly to the univariate case, where the beta prime is p pCpnn2,12 also called the gamma-gamma distribution, frequently np1 p 1 np2 1 II  encountered in Bayesian Statistics [11], V ~,beta a b 2 2 p   fii1 fff  ij  iij1 can also be obtained as the continuous mixture of two Wishart densities, in the sense of V ~ Wn,,Ω with defined in the sector ff f0 , with p  12 p Ω ~ Wn,Ω . Also, for U and V above, 1 p  0 0  pni I 1II 2 ni IUp  ~,beta b a and V ~beta b, a . Cpni,2p , 2 2) For the general case where    I p , V is not nec-  II essarily betap , as pointed out by Olkin and Rubin [18]. p j 1 Several other reasons, such as its dependen cy on  and where p vppπ  14    v ,(If np1  j1 2 on (A + B), m akes V difficult to use, and Perlman [19] suggested using but np2  , we can make the changes VABBAAB*= + 12 1I+12 ~betaI m 2, m 2 pn,,12 n n1,, pn 1 n 2 p to obtain the right ex-   pA B pression). which does not have these weaknesses. We will use this definition as the matrix ratio type II of A and B when 2) The latent roots of V: For GB 12 AB 12 or considering its elements distribution. 1 AB , the roots ll1,, p have density: 2) On Determinant Distributions: a) For

I p U ~,betap  mAB m its determinant U has Wilks’s 1 distribution of the first type, denoted by with A   . I j1 nq jj11 q  U ~2mmAB,,2pm B (to follow the notation in   [17]), expressed as a product of p univariate betas of the 22 22 first kind, and this expression has been treated fully in In the general case npq,, can have non-integral [6]. This result is obtained via a transformation and by values and we can also have the case qp . considering the elements on the diagonal. PROOF: See (6). PROPOSITION 1: For integers np, and q , with REMARK: The cdf of Y is expressible in closed form, nq p, using the hypergeometric function 21F of matrix argu- a) The density of the ratio U ~,qI np,, is: (see (5)). ment. ([12], p. 166) and the moments are

ni1 h ppah  bh p  EY  , for 2   ab where K   . pp  i1 nq1 i  ap 12  hbp    12 2 3) On Latent Roots distributions: II 1 b) For V ~,betap  mAB m , the latent roots of AB , The distribution of the latent roots f12,,,ff p of 1 12 12 I BA, and BAB are the same, and so are the X ~betap  a, b was made almost at the same time by three determinants, and V can be expressed as a five distinguished statisticians, as we all know. It is product of p independen t univariate beta primes, which, sometimes referred to as the generalized beta, but the in turn, can be expressed as Meijer’s G-functions, i.e. marginal distributions of some roots might not be uni- II V ~2Λ mmpmAB ,,2 B, or the Wilks’s distribu- variate beta. Although they are difficult to handle, par- tion of the second type . Hence, for the above ratio V tially due to their domain of definition as a sector in R p , p their associations with the Selberg integral, as presented VVTj ,0 , with j1 in [20], has permitted to derive several important results. A similar expression applies for the latent roots of mj11mj II AB   Y ~,betaII a b . Tbetaj ~,, p   22 where Xbeta~,II   if it has as univariate density 4. Extensions and Generalizations  1 1 x From the basic results above, extensions can be made fx  B, 1 x into several directions and various applications can be (6) found. We will consider here elements and determinant 1 11(,1)   H x distributions only. Let us recall that for univariate distri- 11 (1,1)    butions there are several relations between the beta and PROPOSITION 2: Wilks’s statistic of the second the Dirichlet, and these relations can also be established kind, II npq,, , has as density (see (7)). for the matrix variate distribution.

 nn1 np1   1, 1, ,  1   pp 22 2  KuG pp  nq nq1 nq p 1 ,0u  hu  1, 1, ,  1  (5)  22 2       0,u  0

 qq11 qp   ,  , ...,      222  22  fx AG pp  x  (7) pp   nq nq31 nq p  1, , ...,     22222

12 12 4.1. Extensions TWWWjj 00 1  jk , then

1) the matrix vector WW1,, k  has the matrix 4.1.1. To Several Matrices variate Dirichlet of type I distribution, i.e. I There is a wealth of relationships between the matrix WW1,, k ~Dir nj, n0 , with density: np0 1 variate Dirichlet distributions and its components [12], np1 k j k 2 but because of space limitation, only a few can be pre- 2 fcWW1,, kj  WI  Wjj, W 0, sented here. The proof of the following results can be j1 j1 found in [18] and [19], where the question of independ- k Cpn, ence between individual ratios and the sum of all matri- k  i k i0 ces is discussed. I  W j  0 , where c  , with nn  j j1 Cpn, j0 1) In Matrix-variate statistics, if S01,,,SS k  are 1 (k + 1) independent (p × p) random Wishart matrices,  pn p  2 pp12 nj1 S ~,Wn Σ , then, the matrix variates U k , de- and Cp,2n  . ipi i i1  j1 2  fined “cumulatively” by: I 12 12 and W jp~2,beta ni n ni 2 . USSSSS101101=+ + ,,  2) the matrix vector (TT1,, k ) has the ma trix variate 12 12 Dirichlet of type II distribution , i.e. USjj=++00 S SSjj ++ S ,, II TT10,, kj ~Dir n , n  , with density: US=++ S12 SS ++S12 , kk00kk npj 1 k 2 US=+SS+ T 001 k   j j1 fcTT1,, k n , Tj 0 are mutually independent, with U j ,1,,jk  , having a I k k 2 matrix variate beta distrib. betapjon, n n j1 .  I  Tj 2) Similarly, as suggested by Perlman [19], defining, j1 j1 12 1 12 VSSSSSSS=++ ++ ++ and T ~22betaII n, n  n . jjjjoo-1oj ipi  i   THEOREM 3: Under the same hypothesis as Theo- then V ~,betaII n n n and are mutually jj21oj rem 2 the densities of the determinants of independent. Concerning their determinants, we have: R WW,ij  and of R*  TT, can be ob tained THEOREM 2: Let S ,,,SS be (k + 1) indep ij ij 01 k in closed form in terms of G-functions . (p × p) random Wishart matrices and U j and V j I PROOF: We have W j nnpni00,, and j II Tji nnpn00,,  , 1 jk and the conclusion defined as above. Then the two products Ω j  Ui i1 is immediate from Section 2 of Theorem 1. j QED. and Ω j Vi 1 jk, as well as any ratio i1 4.1.2. To the Matrix Variate Gamma Distribution UU, jk and VV, jk , have the densities of jk jk In the preceding sections we started with the Wishart their determinants expressed in closed forms in terms of distribution. However, it is a more ge neral to consider Meijer G-functions. the Gamma matrix variate distribution, WC~,Ga  . jj1  I W p PROOF: We have U j ~2 nk ,p,2 nk, kk00 Here, we know that ~  X i , with independent C i1 jj1 1 jk V ~2II np,,2 n , and the result is i 1 j kk XGa~,1  . Hence, the density of W is : kk00 i  2 immediate from Sect ion 2 of Theorem 1 and the expression of the G-function for products and ratios of inde- p 0 3p 1 fw C KG0 p  w1, , , , w0 pendent G-function distributions given in the Appendix. 22 Exact expressions are not given here to save space but (8) are available upon request. 1  p i 1  QED. with K   . k  i1 2  Also, considering the sum TS  j , if we take: Although for two independent Matrix Gammas, W j0 1 12 12 12 12 and W2 , their symmetric ratio of the second type, WTST00 , WTSTjj and 12 12 WWW212, suffers from the same definition diffi-

Copyright © 2011 SciRes. OJS T. PHAM-GIA ET AL. 99 culty as with the Wishart, the same recommendations p θ    2 , made by Perlman [19] can be implemented to obtain the 11 ni  i1 well-defined elements distribution o f the Generalized F j1j1 Matrix variate W ~,GFp 12 . The matrix variate II θ2 ~2ab ,pb,2~ betaa, b  , GFp  a, b -distribution, a scaled form of the 22 betaII ,  , is encountered in multivariate regression, p  II II jj11 just like its univariate counterpart. But we have, for the θ3 ~2abp ,,~2 b  betaa , b , 22 ratio of the two determinants p p i 1 θ ~,Cga  1. X 4    i p i1 2 i1 WWW12 p Ti , with independent i1 The expressions of the densities of θi ,1 i  4 , in Yi i1 terms of G-functions are given in previous section s, ex- cept for θ1 the density of which is given by (12) below. II ii11 I II Tbetai ~,12 (9)  and  here are generalized Wilks’s variables, 22 with a and b positive constants, instead of being integers. Hence, the distribution of W , the determinant of THEOREM 4: For any of the above matrix variates W ~,GFp 12 has density: θ j 1 11p   a) The density of the determinant of any product  ij, ,,,   2 222  = θθij, and ratio  ij, = θθij, 0,ij 4, in the pp 22  fzAG x above list, can be expressed in terms of Meijer G-func- pp  31p  tions. 1211, , , 1  22 b) Furthermore, subject to the independence of all the (10) factors involved, any product and ratio of different 1 2 1 θ ,1 i  4 , and of different  and  , p i ij, ij, jj11 0, ij 4, can also have their d ensities ex pressed in with A  1 2 , which is j1 22 terms of G-functions. of the same form as (4). PROOF: The proof is again based on the reproduc tive The determinants of the product and ratio of the two property of the Meijer G-functions when product and ratio operati ons are performed, with the complex expres- GFp variates, can now be computed, using the method given in the Appendix, extending the univariate case sions for the se operations presented in [6], and repro- established for the generalized-F [1 1]. duced in the Appendix. Computation details can be pro- vided by the authors upon request.

4.2. Further Matrix Ratios QED. REMARK: Mathai [23] considered several types o f

integral equations associated with Wilks’ work and pro- There are various types of ratios encountered in the sta- vide solutions to these equations in a general theoretical tistical literature, extending the univariate results of context. The method presented here can be used to give a Pham-Gia and Turkkan [21] on divisions by the univari- G-function or H-function form to these solutions, that ate gamma variable. We consider the following four ma- can then be used for exact numerical computations. trix variates, which include all cases considered previously ( θ is a special case of θ ): 1 4 5. Example and Application Let θ1 ~,Wishart Wpi n  , with integer n, θ  Beta type I ( betaI a, b ), θ  Beta type II or 2 p  3 We provide here an example using Theorem 4, with two GF (a,b) ( BetaII a, b or GF a, b ), ab,12 p , p  p    graphs and also an engineering application. θ  4 Gamma ,,C (Ga p ,,CC 0,0). The elements distributions of various products 5.1. Example θθθ12 12, and ratios θθθ12 12 and j ij jij 12 12 θθ θθθ , for independent θθ, , ij iij ij II nnii Let θi~betap,,,1,2 p i  . The densities of Y = 1,ij 4, can be carried out, but will usually lead to 22 quite complex results. Some results when both θi and θ θ are θ -matrices are obtained by Bekker, Roux and θθ and R = 1 can be obtained in closed form as j 2 12 θ Pham-Gia [22]. 2 However, for their determinants, we have: follows:

 qq1 qp1qq 1 qp1  11,,, 11 2,,, 2  22  222 22 222 22      pp fyAG pp  y  (11)   nqnq33 nq  p11nq nq  nq p  111, 11,, 111 2 21,2 2  , , 2 2  2  22222 2 2222

Pi 1 for y  0 with pp12 p, A  AA12 and Ai  , i  1, 2 . j1 nq jj11 q  ii i  2222 Similarly, the ratio R has density:

 q1 q 111 q1 p 111  nq22 nq 22 nq22  p 2  ,,,   ,  ,,  222 22  2 22 2 2 pp  fr AG  r  (12) pp   nq11 nq 113n 11  q p 111 q2 q 23 q2 p 2   1, ,,  1, ,,    22222 22222  for r  0 . and products matrix, and the sample covariance matrix PROOF: Using (7) above, we can derive (11) and (12) SA n 1 . We have: by applying the approach presented in the Appendix. 22 QED. A   nnp 1 (13) Figure 1 and Figure 2 give, respectively, these two where the  2 variables, with j degrees of freedom, densities, for pp124, q 1  9, q 2  12, n 1  13 and j np 1 jn are independent. n2  18, using the MAPLE software. S This is also the distribution of n p . 5.2. Application to Multivariate Process Control 

Now let S1 and S2 be two independent sample co- 1) Ratios of Variances: The generalized variance is variance matrices of sizes n and n respectively. treated in [24]. We consider X ~ N μ, Σ and a ran- 1 2 p  From (1), we ha ve SW~1,nn 1. dom sample XX,, of X. We define ip i ii 1 N a) The samp le generalized va riance N T Yn1 pi S  has density: Α XXXX, the sample sum of square s ii ii  1

  p1 np1  11p 0 y nn23 G i ,,, ,0y  p  0 pip  hyi  2 j0 nji  2 22 2  (14)  2   0, yii 0, 1,2 

2) Now, the two types of ratios X ~,N p μ00  , using the variations of the ratio of two 12 12 12 12 V = S212S S and U = SS12 SSS 112, random sample covariance matrices taken from that en- can be determined. vironment, against the fluctuations of its control envi- I We have U ~12,Beta n12 n 12 and ronment Y ~,N p μ00  , represented by a similar ratio. I U ~2nn12,, pn 2 1. V has a density which is For clarity, we will proceed in several steps: II not necessarily Beta n1212, n 12, but we a) Let θX  S12S , with S1 and S2 being two II have: V ~2nn12,, pn 2 1. random samples of sizes n1 and n2 from

2) Application: In an industrial environment we wish N p  μ00,  and, similarly, θY  S34S , with S3 to monitor the variations of a normal process and S4 being two random samples of sizes n3 and n4

60 and similarly for the numerator. c) Applying the ratio rule in the Appendix for these 50 two independent G-function distributions, we have the

40 density of  . f(y) QED. 30 6. Conclusions 20

10 As shown in this article, for several types of random matrices, the moments of which can be expressed in terms 0 of gamma functions, Meijer G-function provides a pow- 0.00 0.01 0.02 0.03 0.04 0.05 erful tool to derive, and numerically compute, the densi- y ties of the determinants of products and ratios of these

Figure 1. Density of X1X 2, with matrices. Multivariate hypothesis testing based on de- II II terminants can now be accurately carried out since the X12~,,,~,,1349 X 18412 . expressions of null distributions are, now, not based on

asymptotic considerations. 40

7. References

30 [1] K. C. S. Pillai, “Distributions of Characteristic Roots in f(r) Multivariate Analysis, Part I: Null Distributions,” Com-

20 munications in Statistics—Theory and Methods, Vol. 4, No. 2, 1976, pp. 157-184. [2] K. C. S. Pillai, “Distributions of Characteristic Roots in

10 Multivar iate Analysis, Part II: Non-Null Distributions,” Commun ica tions in Statistics—Theory and Methods, Vol. 5, No. 21, 1977, pp. 1-62.

0 [3] P. Koev and A. Edelman, “The Effective Evaluation of 0.0 0.1 0.2 0.3 0.4 the Hypergeometric Function of a Matrix Argument,” r Mathematics of Computation, Vol. 75, 2006, pp. 833-846. doi:10.1090/S0025-5718-06-01824-2 Figure 2. Density of [4] A. M. Mathai and R. K. Saxena, “Generalized Hyper- XXX,~II 13 ,,,~ 4 9 X II 18 ,, 4 12 . 1212    geometric Functions with Applications in Statistics and Physical Sciences,” Lecture Notes in Mathematics, Vol. 348, Springer-Verlag, New York, 1973. from N p μΣ,  . Hence, [5] M. Springer, “The Algebra of Random Variables,” Wiley, SS II nn2, pn ,  1 θY 34  34 4 New York, 1984.   II θX SS12nn122, pn , 2  1[6] T. Pham-Gia, “Exact Distribution of the Generalized p Wilks’s Distribution and Applications,” Journal of Muo- II n3 1 ii11n4 1  beta , tivariate Analysis, 2008, 1999, pp. 1698-1716. i1 2222  p [7] V. Adamchik, “The Evaluation of Integrals of Bessel II nn1211ii11Functions via G-Function Identities,”Journal of Compu- beta , i1 2222tational and Applied Mathematics, Vol. 64, No. 3, 1995, pp. 283-290. doi:10.1016/0377-0427(95)00153-0 b) For the denominator, its density is given by (7): [8] T. Pham-Gia, T. N. Turkkan and E. Marchand, “Distribu- p 1 tion of the Ratio of Normal Variables,” Communica- gx  tions in Statistics—Theory and Methods, Vol. 35, 2006, nj nj j1 12 pp. 1569-1591. 22  [9] T. Pham-Gia and N. Turkkan, “Distributions of the Ratios of Independent Beta Variables and Applications,” Com- n 1 nnp222  2 ,,,  munications in Statistics—Theory and Methods, Vol. 29, pp 22 2No. 12, 2000, pp. 2693-2715. G pp r nn34np2 doi:10.1080/03610920008832632 11,,, 1 22 2 [10] T. Pham-Gia and N. Turkkan, “The Product and Quotient

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Appendix  aa,,, ,  j,1 j ,1  jp,,jj jp  mrjj   fxjj k jH pq  cx jj  , The H-function H  x is defined as follows: jj  bbjj,1, ,1 ,..., jqjq,, ,     jj  aa11,,,,  pp mr  H pq x x  0 bb,,,, j 11 qq (6) mr i.e. it is s bsjj 1 as jj 1 The product of these variables, PX , is also a  jj11 xsds  j  qp j1 2πi L 1 bsjj   as jj  H-Function Random variable, with density jm11 jr  the integral along the complex contour L of a ratio of s fy kHMR products of gamma functions. H-function has a relation- pjPQ  j1 ship with the generalized hypergeometric function:

S  pqF 1 . . An equivalent form is obtained by replacing s us.. a1,1 , 1,1 , , asp , , sp ,  cyss,0 y by (-s), which shows that the integrant is, in fact, a Mel-  j J 1 ls. . b , ,..., b , lin-transform. Numerically, H-functions can be com- 1,1 1,1sq ,s sq , s puted by using the residue theorem in complex analysis, together with Jordan’s lemma, but because of the com- (7) plexity of this operation, it is only quite recently that it is ssss where M  mRj ,, rPjj pQ ,  qj, and available on softwares, although, in the past, several au- jjjj1111 thors have suggested their own versions in their pub- the two parameter sequences, (u.s.) and (l.s.), are as lished works. follows: In this Appendix, the densities of the product and ratio a) The upper sequence of parameters (u.s.), of total of two independent random variables, whose densities length P, consists of s consecutive subsequences of the are expressed in H-functions, is obtained. For G-func- type aa,, , followed by s consecutive  1,1 1,1  1,rr11 1,  tions, we have ij1,ij , , and since these values are not affected by the operations the following results subsequences of the type aa1, rr1,, 1, 1 1,PP 1, , remain valid.  11  11 i.e. we have: 1) Product: Let X j ,j 1, , s, be s independent H-function random variables , each with pdf :

aaaa,,,, ;,,,, ;;,,,, a  a  :: 1,1 1,1 1,r 1, r  2,1 2,1  2,r 2, r   s,1 s ,1   sr,, sr  11 2 2 ss (8) aaaaaa,,,,;, ,,,;,,,, 1,(rr11 1) 1,( 1) 1, ppr 112 1, 2,( 1) 2,( r 2  1) 2, ppsrsrspsp 22 2, ,(ss 1) ,(  1) ,, ss

b) Similarly, the lower parameter sequence (l.s.), of secutive subsequences of the type: total length Q, consists of s consecutive subsequences of bb,,,, , i.e. we have:  1,(mm11 1) 1,( 1)   1, q 1 1,q1 the type bb, ,, ,, followed by s con- 1,1 1,1 1,mm11 1, 

bbbb, ,, , ; , ,, , ;; b ,  ,, b ,  ::  1,1 1,1   1,m 1, m   2,1 2,1   2,m 2, m   s,1 s ,1   sm,, sm  11 2 2 ss (9) bbbbbb,,,,;, ,,,;,,,, 1,(m11 1) 1,( m 1) 1,q 112 1, q 2,( m 1) 2,( m 2  1) 2,q 22 2, q sm,(ss 1) sm ,(  1) sq,, s sqs

2) Ratio: For the ratio WXX 12, its density is:

us.. a , , , , ,1b 2 , c  1,1 1,1   2,qqq222 2, 2,  fw() AHmrm12 21 r1 w ,w 0 (10) W pqq1212 p c2 ls.. b , , , , ,1a 2 , 1,1 1,1 2,qqq222 2, 2,

kk mrq12,, 1 m 1 and pr22 : where A  12 . 2 b, ,, b, c2  1,1 1,1   1,mm111,  a) The upper sequence of parameters (u.s.), of total 12aa,, ,12,   , (12)  2,1 2,1 2,1   2,rr22 2, 2,r2 length pq1 2, consists of the four consecutive subsequences: bb,,, ,, and  1,mm11 1 1, 1   1,q1 1,q1 aa,, , of length r 11 11 1rr11 1  1 12aa,,,12, . 12,,b ,1b 2 , of length  2,rrr222 1 2, 1 2, 1 2, pp 22 2, 2, p2 2,1 2,1 2,1 2,mm222, 2,m2 PROOF: The proofs of the above results, based on the m (11) 2 Mellin transform of a function f x defined on R , aa,,,,  of length pr , and 1,rr11 1 1, 1 1, pp 1 1, 1 11 and its inverse Mellin transform , as defined previously, 12,b ,,1, b2 are quite involved, but can be found in Springer (1984, p. 2,mmm222 1 2, 1 2,, 1 2, qq 22 2, 2,q2 214). of length qm . 22 QED. b) The lower sequence (l.s.), of total length pq21 , also has 4 subsequences of respective lengths