Ž. Ž. Ž. 2 QUADRATIC AND INVERSE REGRESSIONS FOR WISHART DISTRIBUTIONS 1

Transcription

1 The Annals of Statistics 1998, Vol. 6, No., QUADRATIC AND INVERSE REGRESSIONS FOR WISHART DISTRIBUTIONS 1 BY GERARD LETAC AND HELENE ` MASSAM Universite Paul Sabatier and York University If U and V are indeendent random variables which are gamma distributed with the same scale arameter, then there exist a and b in such that U U V. au V. and U U V b U V. This, in fact, is characteristic of gamma distributions. Our aer extends this roerty to the Wishart distributions in a suitable way, by relacing the real number U by a air of quadratic functions of the symmetric matrix U. This leads to a new characterization of the Wishart distributions, and to a shorter roof of the 196 characterization given by Olkin and Rubin. Similarly, if U. exists, there exists c in such that U U V c U V. Wesołowski has roved that this also is characteristic of gamma distributions. We extend it to the Wishart distributions. Finally, things are exlained in the modern framework of symmetric cones and simle Euclidean Jordan algebras. 1. Introduction. Let and be two ositive numbers. We denote by. 1.1, du ex u u 10,. u du the gamma distribution on 0,. with scale arameter and shae arameter. Let U and V be two indeendent, ositive, non-dirac random variables, and consider the following five roerties: 1. U U V. and U V are indeendent.. There exist two real numbers a and b such that 1.. U U V. au V., 1.3 U U V b U V.. There exist two real numbers a and c such that U U V. au V., 1.4. U U V. c U V.. Received August 1996; revised July Suorted by NATO Grant and NSERC Grant A8947. AMS 1991 subject classifications. Primary 6H05; secondary 60E10. Key words and hrases. Natural exonential families, Wishart distributions, Jordan algebras, conditional moments. 573

2 574 G. LETAC AND H. MASSAM 3. There exist, q and such that the distributions of U and V are, resectively, and q,. 3. Same as 3., lus 1. It is known that 1., and 3. are equivalent, and that and 3. are equivalent, too. Let us first concentrate on the first grou of equivalences; if either or 3. holds, the following formulas are true: a, b. q q 1. q. The imlication is the easiest to rove, since if t q and if, one has t t. q t. q 1.6. ž UV. ex U V../ 1... q. The roof of has been given by Lukacs However, it has been noted by several authors e.g., Wang that 1. is quite simle and that, this together with 3., is a little bit easier than the direct roof of At the beginning of Section 4, we will give a roof of 3. using the variance function of the natural exonential family ; 04,. The most natural generalization of 1.1. is the Wishart distribution on the sace E of r, r. symmetric matrices. Throughout this aer, we will use the same notation as Casalis and Letac We denote by E and E the cones of ositive and ositive definite matrices, resectively, and we write , 1, 3,..., r 1. 4 r 1.,. For in, there exists a ositive measure on E such that, if belongs to E, then the measure on E :. 1.8, du ex trace u det du is a robability distribution. Clearly, if we relace by. in 1.8., we get that, for. in E, H, r 1.9. ex trace u.. du. det I.. We call, the Wishart distribution with scale arameter and shae arameter. It should be ointed out that, in order to avoid tedious constants, we differ slightly from the traditional notation W m,. r for Wishart distributions, as found, for instance, in Muirhead The corresondence between the two notations is simly given by W m,. m, r. A natural question is now the following. Does there exist a generalization, to the Wishart distributions, of the equivalence between 1., and 3.? For the equivalence between 1. and 3., the answer is yes and has been rovided

3 WISHART DISTRIBUTIONS 575 by Olkin and Rubin This aer is difficult to read; Casalis and Letac 1996., using many of Olkin and Rubin s ideas, rovides a simler and clearer roof of or, more accurately, of the corresonding statements for the general Wishart distribution given by 1.8.; see Section 5. The statement for the general Wishart distribution is not as easy as 1.6. and is also roved in Casalis and Letac The first roblem considered in the resent aer is the equivalence between and 3. for the Wishart distributions Surrisingly enough, as in the one-dimensional case, 3. will be much simler than Since, as we will see in Proosition 5.1, the imlication 1. is also easy to rove, we shall end u with a new and shorter roof of Olkin and Rubin s theorem in Section 5. Now, finding the correct generalization of 1. to the Wishart distribution was one of the numerous bright ideas of Olkin and Rubin. For., the generalization of 1.. is clear, and we just have to write a trivial extension of a result by Rao to obtain it. However, the roer generalization of 1.3. is not that obvious, but can be found in the toolbox of the roof of given in Casalis and Letac Relacing the real number U by the square of the symmetric matrix U is not the right idea see the remarks following Corollary.3 for the exlicit exression of U U V.. What we use is a air of quadratic functions of U valued in the linear sace L E. s of symmetric endomorhisms of the Euclidean sace E of symmetric real r, r. matrices. The Euclidean structure on E is given by the scalar roduct ² A, B: trace AB. Thus dim E r r 1. and dim L E. r s r. 1 r r. 8. These two quadratic functions are: U U, defined by E E, H U U. H. U trace UH., and U, defined by E E, H U. H. UHU. For examle, if r, and if we take a b c U, c a b then U belongs to E. if and only if a b c 1. In this case, as an orthogonal basis of E we can take the three matrices: I, J, K Thus, if H xi yj zk, we have U U. H. ax by cz. U and U. H. a b. x y. c x y. a b. cz acx bcy a b c. z acx bcy a b c. z a b. x y. c x y. a b. cz ;

4 576 G. LETAC AND H. MASSAM 4 that is, the matrices reresentative of U U and U in the basis I, J, K are, resectively, and a ab ac ba b bc ca cb c a b c ab ac. ab a b c bc ac bc a b c Let us show that it is natural to relace 1.3. for real symmetric r, r. matrices by the following: There exists a,. real matrix b b ij. i, j, such that 1.1. U U U V. b1 U V. U V. b1 U V., U U V. b1 U V. U V. b U V.. The main idea leading to the consideration of the air can be found in Casalis and Letac 1996., where quadratic mas Q: E L E. s with a certain invariance roerty were considered. This roerty can be resented here as follows: For any A in the linear grou GL r,. of invertible real r, r matrices, and for any U, H in E, one has t t t 1.13 Q AU A H AQ U AHA A a reformulation of 6.1. in Casalis and Letac for symmetric matrices. It is easily seen that the functions Q defined by or fulfill condition 1.13.; we shall see in Proosition.1 that the linear sace of ossible Q has dimension, and that form a basis of this sace. Therefore it is natural to consider this air of quadratic functions of U. Of course, to justify 1.1., we also have to rove that these two identities are satisfied when U and V are two indeendent and Wishart-distributed random variables with the same scale arameter ; that is, we have to rove the imlication 3... In other words, we have to find the values of a, b 11, b, b and b corresonding to a and b in We shall actually rove in Section 3 that if and q are the shae arameters of U and V, and if A., then the matrix b in 1.1. is A.A q... Such a formula for Wishart distributions has an ancestor in Casalis 199., Theorem 1. The result in 1.1. can be resented differently if we diagonalize the matrix b. One

5 WISHART DISTRIBUTIONS 577 obtains a air of identities. Let 1 be an element of, let 1 X. U U U.., 1. ž / 1 1 Y. U U U. 1. and let X q. and Y q. be defined in an analogous way by relacing U,. by U V, q.. The identities have a martingale flavor and read X. U V. X q., Y. U V Y q.. Note that, for 1, U is concentrated on the cone of matrices of rank 1 and thus U U U. Let us now consider and 3.. Their equivalence has been observed rather recently in Wesołowski 1990., where q c. 1 Our second aim in the resent aer is to generalize the equivalence between and 3. to the Wishart distributions. Here, things are a little bit simler than in the quadratic case, and the real number U can safely be relaced by the inverse matrix U. However, the analogue of has to be sulemented by the consideration of U U, the element of the linear sace L E. of endomorhisms of the Euclidean sace E of symmetric real r, r. matrices defined by E E, H U U. H. U trace UH., which necessarily occurs when one considers the differential of ² :. U ex, U. For examle, if r and if a b c U, c a b as above, then the reresentative matrix of U U in the basis I, J, K 4 will be a ab ac ba b bc. a b c ca cb c Thus, for symmetric real r, r. matrix, 1.4. will be relaced by the following: There exist three real numbers c, c1 and c such that 1.16 U U V c U V, 1 E U U U V cid c U V U V.

6 578 G. LETAC AND H. MASSAM Statement 3. is, of course, relaced by a statement with Wishart distributions and r 1.. To justify and 1.17., we have to rove that these identities are actually satisfied for some c, c 1, c, when U and V are indeendent and have Wishart distributions with the same scale arameter i.e., we have to find the analogue of We shall rove in Section 6 that if and q are the shae arameters of U and V, then the constants in and are q r 1. q c, c1, r 1. r 1. q. q r 1. c. r 1. q. This leads to the rather unexected consequence, that if U and V are Wishart distributed with the same scale arameter, and with shae arameters r 1. and q in as defined in 1.7., then the conditional covariance of U and U, given U V, is q r 1.. q. see Corollary.6.. Let us now list the contents of the aer. In Section, we state the results which were roughly sketched above. We think that no new results on Wishart distributions should nowadays be resented outside the framework of the Wishart distributions on symmetric cones and simle Euclidean Jordan algebras. Following this olicy, as was done, for instance, in Casalis and Letac or in Massam 1994., we give our results on symmetric cones. It still remains, of course, that our results are new for the classical Wishart distributions. The rank and the Peirce constant of the simle Euclidean Jordan algebra are denoted, resectively, by r and d. The reader who is interested only in symmetric real matrices of order r should set d 1 in the formulas and follow the instructions given in Casalis and Letac 1996., Section 3. Section also contains a roof of the extension of Rao s aer, as well as a roof of the fact that quadratic mas E L E. s with the invari- ance roerty form a linear sace with dimension. Section 3 is comutational; we essentially rove the imlication 3... In Section 4, we give the roof of 3. and, as in Casalis and Letac 1996., the main tool is the variance function of the natural exonential family of the Wishart distributions with shae arameter. In Section 5, we rove the imlication 1. for Wishart distributions. Section 6 is arallel to Section 3; it is comutational and we essentially rove 3... In Section 7, we rove 3.; the basic tool in that roof is the derivation of Schwartz s distributions. In Section 8, we comment on an unsolved roblem raised by a referee. id E

7 WISHART DISTRIBUTIONS 579. The characterization of Wishart distributions by quadratic and inverse regressions. As exlained before, we kee the notation of Casalis and Letac Let E be a simle Euclidean Jordan algebra, with roduct E E Ea, b. a b, and scalar roduct ² a, b: trace a b.. We denote by LE and L E. s, resectively, the saces of endomorhisms and symmet- ric endomorhisms of the Euclidean sace E. The identity element of E is e. The closed cone of squares a a; a E4 is denoted by E, and E is its interior; r is the rank of E, d is its Peirce constant and n r dr r 1. is the dimension of E. To avoid trivialities, we always assume r 1. We denote by G the connected comonent, containing the identity, of the grou of linear automorhisms of the linear sace E which reserve E ; we write K for the intersection of G with the orthogonal grou of E. The determinant in E is written det: E. We write 4 1. d, d,3d,..., r 1. d r 1. d,. It is known Casalis and Letac 1996, Section 3 that if belongs to, then there exists a ositive measure on E such that for all in E one has H. ex ², u:. du. det.. For in E and in,, as defined in 1.8.,, is called the Wishart distribution with shae arameter and scale arameter. Definitions for objects in Jordan algebras are taken from Faraut and Koranyi 1994., with the excetion of the very definition of the Wishart distribution. What they call Wishart distributions are articular cases of our, distributions and are obtained as images of Gaussian distributions on a Euclidean sace F by reresentations of F into E. Thus they choose to generalize distributions only, not gamma distributions as we do. For instance, this imlies that, in their restricted sense, there exists no Wishart distribution on the excetional Jordan algebra of dimension 7 corresonding to r 3 and d 8, while they do exist in our sense. We now introduce two quadratic mas from E to the linear sace L E. s of symmetric endomorhisms of E defined as follows: 1. For x in E, x x in L E. s is defined by h x x. h. x² x, h :, E E.. For x in E, x in L E is defined by s 3. h x. h. x x h. x x. h, E E. The ma x x., E L E. s is called the quadratic reresentation. It is known see Casalis and Letac that if belongs to, then F ; E 4 is the natural exonential family of Wishart distributions with,

8 580 G. LETAC AND H. MASSAM scale arameter and variance function 1 4. VF m. m., when m varies in the domain of the means MF E. To measure the imortance for quadratic regression of the air 3., let us rove the following statement which will be needed later in the roofs of Proosition 3.1 for the first art and Theorem.4 for the second.. PROPOSITION.1. Let Q be the linear sace of functions Q: E LE which are G-invariant, that is, such that for all g in G one has 5. Q g x.. gq x. g* and such that Q e. is in L E. s. Then Q has dimension, and the air x x x and defines a basis of Q. Furthermore, if x E is such that there exists a in with x. ax x, then either x 0 or a 1 and x is the multile of some rimitive idemotent. PROOF. It has been observed in Casalis and Letac after 6.1. that these two olynomials x x x and x. belong to Q. To see that they are linearly indeendent, we aly them to e and get x x x. e. x² x, e: and x. e. x x. The results are clearly indeendent. To see that Q has dimension, consider the linear ma : Q Ls E., Q Q. Q e.. Then Qe is K-invariant, since Q e. Q k e. kq e. k*. This comes from 5. and the definition of K. Thus, from Proosition 6.1 of Casalis and Letac or from Olkin and Rubin 196., Lemma 1 6., there exists, in such that Q e. ide e e. This imlies that the dimension of the image of is since it is included in the lane of L E. generated by id and e e s E. Finally, is one-to-one: Let us take Q in the kernel of. Since G acts transitively on E, let x be an arbitrary oint of E and g in G such that g e. x. Thus we have Q x. Q ge. gq e. g* g0 g* 0, that is, Q is0on E and is one-to-one. Thus Q and the image of have the same dimension. To rove the second art of the roosition, assume that x. ax x and that x is not 0. Then there exists a sequence of orthogonal rimitive idemotents c,..., c. such that 1 v x c c, 1 1 v v

9 WISHART DISTRIBUTIONS 581 with in 04 i for i 1,..., v. Since for all h in E one has x. h. ax² h, x :, let us take h c 1. Thus we get c1 x. c1. a c1 vcv. 1, which imlies that v 1 and that a 1. The roosition is roved. REMARK. Note that, in the revious roosition, imosing that Qe be symmetric imlies that Q x. is also symmetric for all x in E. Relaxing this condition leads to a surrise: The dimension of Q is still excet in the case r, d., 1., that is, in the case of,. real matrices, for which the dimension of Q is 3 see Letac and Massam 1997., where the above result is roved by Neher in an aendix. We now state the main results of the aer, Theorem. the quadratic regression roerty., Theorem.4 the characterization by quadratic regression., Theorem.5 the inverse regression roerty. and Theorem.7 the characterization by inverse regression.: THEOREM.. With the above notation, let be in E, let and q be in and let U and V be indeendent random variables with Wishart distributions, and q,, resectively. Then if 6. A. d 1 d. and b11 b1 7. b A. A q.., b b 1 one has U U U V. b11 U V. U V. b1 U V., 8. U U V b U V. U V. b U V.. by and Then 1 COROLLARY.3. Under the hyothesis of Theorem., define X and Y in Q X u. u u u. d Y u. u u u.. d. X U. U V. X U V., q. q d. 1. Y U. U V. Y U V.. q. q 1.

10 58 G. LETAC AND H. MASSAM PROOF. It is easily seen that the eigenvalues and the line eigenvectors of the matrix b are given by d. 1,. b 1,., q. q d. 1. d, 1. b d, 1.. q. q 1. To obtain the two identities in the corollary, we multily 8., written in matrix form as U U U V. U V. U V b U. U V. by the two eigenvectors. REMARKS. Xu 0 if and only if u is roortional to a rimitive idemotent, as we have seen in Proosition.1. Note that if d and if U is d, distributed, then U is almost surely roortional to some rimitive idemotent see Casalis 1990., and XU 0 almost surely. Note also that under the hyothesis of Theorem., one has U trace U U V. b U V. trace U V. b U V., 11 1 U U V. b1 U V. trace U V. b U V. use u e u and aly 8. to e. The next theorem is the converse of Theorem.; its hyothesis is the generalization to Wishart distributions of statement in Section 1. THEOREM.4. Let U and V be two indeendent non-dirac random variables taking their values in E. Assume that U V is not concentrated on a half line. Assume also that there exists a such that 9. U U V. au V. and that there exists a,. matrix b such that the two equalities 8. holds. Then there exist, q in and in E such that U and V have distributions and. Furthermore, a q., q, and b A.A q.. as in 7.. THEOREM.5. With the above notation, let be in E, let nr and q be in and let U and V be indeendent random variables with Wishart distributions, and q,, resectively. Then for q nr q c, c1, nr nr. q. 10. q nr. c, nr. q.

11 WISHART DISTRIBUTIONS 583 one has 11. U U V c U V, 1 E U U U V cid c U V U V. COROLLARY.6. Under the hyothesis of Theorem.5 one has q Cov U, U. U V. id E. nr. q. PROOF. By Huyghens formula, for any random variables X and Y taking their values in a Euclidean sace, we have Cov X, Y. X Y. X. Y.. Alying this formula for the conditional covariance of X, Y U, U. given U V, using 10., 11. and U U V. U V., q we obtain the desired result. The last theorem is the converse of Theorem.5; its hyothesis is the generalization to Wishart distributions of statement of Section 1. THEOREM.7. Let U and V be two indeendent non-dirac random variables taking their values in E. Assume that U V is not concentrated on a half line and that U. exists. Assume also that there exist a, c, c1 and c such that 9. and the two equalities 11. hold. Then there exist nr, q in and in E such that U and V have, and q, distributions. Furthermore, a q. and c, c and c are as in To comlete this section, we make a small digression about 9., generalizing Rao s result as follows: PROPOSITION.8. Let E be a Euclidean sace and let U and V be two indeendent non-dirac random variables taking their values in E. Assume that the two sets U., defined as the interior of ; L ex², U : U 4, and V., defined similarly, have a nonemty intersection. Assume also that there exists a such that U U V. au V.. Then 0 a 1, U V and a 1a. 1. L. L.. U V

12 584 G. LETAC AND H. MASSAM PROOF. For in U. V., we have L. L. U ex², U V : U V. a U V. ex², U V :. alu V.. Thus L L U v a. a. L. L. U The rincile of maximal analyticity Kawata 197. shows that U. V. from 13.. Now, if a were equal to 0 or 1, 13. would imly that U or V are Dirac. Finally, since the differential of L. L. U U must be ositive, then 1 a and a have the same sign and therefore 0 a 1. And 1. follows immediately from Proof of Theorem.. The roof of Theorem. relies on the following two roositions, which are of interest in their own right. PROPOSITION 3.1. Let : L E. L E. s s be the linear ma defined by 3.1. x x. x. for all x in E. Then 3.. x. d d x x 1 x.. ž / PROPOSITION 3.. Let U be a random variable taking its values in E with Wishart distribution, as defined in Section or as given in Then 3.3., U U.., d d U.. ž / PROOF OF PROPOSITION 3.1. The existence and uniqueness of were roved in Lemma 6.3 of Casalis and Letac 1996.; the notation that we adot here relaces by. We observe that the ma x Q x. x.., E Ls E. is G-invariant in the sense of.5. To see this, we take g in G and write 3.4. g x. g* g x.. g x. g x.. g x x. g*.. The first equality in 3.4. is a standard roerty of, the second is by definition of and the third is obvious. Now, since the set x x; x E4 generates the linear sace L E., we obtain, from 3.4. by linearity, that 3.5. g f. g* gfg*.. s From 3.5. now alied to f x., we get g x. g* g x. g* g x... V

13 Thus we have roved that WISHART DISTRIBUTIONS 585 x x.., E Ls E. is G-invariant in the sense of 5.. Proosition.1 imlies that there exist and in such that, for all x in E, 3.6. x. x. x x. Since both members of 3.6. are quadratic olynomials, this equality 3.6. is also true for all x in E. To comute and, we use 6.1. in Casalis and Letac by taking x e in 3.6. and we get 3... PROOF OF PROPOSITION 3.. Writing A U U. and B U. and using A. as defined in 6., identities 3.3., to be roved, can be written as A A.. B. From., we have that, for in E., L det and k log L. For any quadratic olynomial Q defined on E, it is a general roerty of Lalace transforms that d 3.7. H Q u. ex², u: du. Q k. Q k.. L.. ž ž / / E d We aly 3.7. to Qu u u and note that, in this case, Qdd j j for any C -function j defined on an oen set of E, and that k and k...; we obtain the first identity in 3.3., for A, by taking.. The second identity in 3.3. for B is more delicate and will use Proosition 3.1. Let us aly 3.7. to Q. To simlify the notation, we write k. log det. for E. We get 3.8. H u. ex², u: du. E d ž ž / / k... det... d To comute dd. k. in 3.8., we now use in Casalis and Letac 1996., which says that as defined in Proosition 3.1 satisfies dd. j. j.. Since k..., Proosition 3.1 can be alied and, with the hel of 3.8. with., yields the second formula of COMMENTS. Recalling the definition of the variance function as given in 4., the first formula in 3.3. is nothing but a multivariate version of Huyghens formula linking the second moment and the variance. The second formula of 3.3. could also be deduced from Theorem 1 of Casalis 199., whose roof is not really shorter.

14 586 G. LETAC AND H. MASSAM We now rove Theorem.. First, we note that, in order to rove Q U. U V Q U V. 1 for given functions Q and Q 1, it is enough to show that, for in some nonemty oen set, 3.9. Q U. ex², U V : Q U V. ex², U V :. We aly this rincile to the air Take in E u u Q u., u. u u Q1 u. A. A q... u. and, for simlicity, denote Using 3.10, 3.3 can be rewritten as 3.1. Q U.. A.,. changing into in 3.1. gives 1 Q U. ex², U :. A. LU.,. Q U V ex², U V : 1. A. LUV... Formula 3.9 follows immediately and Theorem. is roved. 4. Proof of Theorem.4. We first show the imlication 3,. as given in Section 1. The roof of Theorem.4 will be an extension of this simle case. U As usual, we write L e. and k log L ; k exists on,0. U U U U, since U 0. We write k U V. The generalization of Rao s theorem, given in Proosition.8, and 1.. imly that 4.1. ku a. Thus, from 1.3., we derive that, for 0, a. a. L. L. L. U V U V U ex U V... b U V. ex U V... b. b.. LU V... This gives a b b a. Since 0 a 1 see Proosition.5, a b b a 0 is imossible. Since U V is non-dirac, a b 0 is

15 imossible. Thus, if b a. a b., one has WISHART DISTRIBUTIONS 587 The equality 4.. is identical to 11. in Casalis and Letac It says that the variance function of the natural exonential family NEF. generated by the distribution of U V is V m m on its domain of the means. Thus U V, U and V follow the gamma distributions with shae arameters, a and 1 a. q, resectively; the scale arameter is the same for all variables. Let us now give the roof of Theorem.4. We write k U V. The roof imitates the atterns of the roof of 3. and aims to show that there exists some in defined by 1. such that, on some oen subset of E, , thus showing that the variance function of the NEF generated by the law of U V is V m. m. on its domain of the means; that is, U V is Wishart distributed with shae arameter since the domain of existence E of the Lalace transform of the law of U V is oen, any robability generating the NEF must belong to the NEF.. Rao s theorem Proosition.8. and 9. imly that, for in E, one has 4.4 k a U note that U concentrated on E imlies that U. E. Now, from 8. and 4.4., we obtain 4.5. Similarly, 4.6. a a. LU V U U ex², U V :. ž ž / / d b11. b1. L UV. d ž ž / / d a a. LU V d U ex², U V : ž ž / / d b1. b. L UV. d If we eliminate dd between 4.5 and 4.6, we get a a trace b det b b a 1 a a a b ab det b. 1 11

16 588 G. LETAC AND H. MASSAM We note that the coefficient of in 4.7. is the characteristic olynomial of the matrix b evaluated at a. Let us show that b1 is not 0 and that a is not an eigenvalue of b. If b 0, then 4.5. imlies a b b a. But since 0 a 1 see Proosition.8., a b would imly 0 and U V would be Dirac: this is imossible. Hence 4.8. would imly, with b a. a b This, in turn, would imly that has rank 1 everywhere and therefore that U V is concentrated on a line; this contradicts the hyothesis of Theorem.4. Therefore b1 is not 0. Assume now that a is an eigenvalue of b. From 4.7., since b1 0, we get that there exists a constant such that From the second art of Proosition.1 and from 4.9., we see that. is a multile of some rimitive idemotent c.. Now, rimitive idemotents are on the extremal lines of the convex cone E. Since. is the exectation of some element of the NEF generated by the distribution of U V, this imlies that U V is concentrated on an extremal line of E, and this contradicts the hyothesis of Theorem.4. Therefore a cannot be an eigenvalue of b. We have just roved that the coefficients of and. in 4.7. cannot be 0. Therefore from 4.7. it follows that there exist in 04 and in such that, for in E, It has been roved in Casalis and Letac see discussion following that imlies that the distribution of U V generates a genuine NEF i.e., not concentrated on some affine hyerlane. and that 0. Thus 4.3. and Theorem.4 are roved. 5. A roof of Olkin and Rubin s theorem. In this section, we will rove the following Proosition 5.1. From this roosition and Theorem.4, we obtain a roof of Olkin and Rubin s theorem, as exressed in the form given in Casalis and Letac 1996., Theorem 3.. The roof of our Theorem.4 borrows a number of features from Casalis and Letac essentially Proosition 6.1, for our Proosition.1; Lemma 6.3, for our Proosition 3.1; and the tedious discussion following Furthermore, the roof of Proosition 5.1 will use Lemma 5.1 and Proosition 6. of Casalis and Letac However, the resent aroach is basically simler and seems to be the normal route, as mountain climbers say, toward Olkin and Rubin s result. This elegant eak has not been very much visited after its discovery: The Science Citation Index acknowledges only three quotations before 1994, and one of them is confusing it with another aer by the same authors and in the same journal, but ublished in Srivastava says that the extension of the result to comlex Wishart laws is under investigation. This aim seems to have been reached only later, in Casalis and Letac

17 WISHART DISTRIBUTIONS 589 PROPOSITION 5.1. Let U and V be two indeendent random variables concentrated on E such that U V belongs to E almost surely and is not concentrated on some half line. Let g: E G be a division algorithm, that is, a measurable ma such that g x. x. e for all x in E. Finally, assume that Z g U V. U. is indeendent of U V and that the distribution of Z is invariant by K. Then there exist a in and a,. real matrix b such that U U V. au V. and 8. holds. PROOF. Since the distribution of Z is K-invariant, Z. is K-invariant. Then, from Lemma 5.1 of Casalis and Letac 1996., we know that there exists a in such that ae Z.. Hence ae g U V. U. U V. g U V. U U V... Since g x. x. e for all x, we have g x.. e. x. Hence au V. U U V.. Similarly, for, in, Q x. x x x. defines a G-invariant quadratic olynomial taking its values in L E.. Hence QZ. s is a K- invariant element of L E. s and, from Proosition 6.1 of Casalis and Letac 1996., there exist and in such that Thus id e e Q Z.. E. id e e g U V. Q U. g U V. * U V E g U V. Q U. U V. g U V..*.. Using gu V e U V again, we get 5.1. Q U. U V. U V. U V. U V., since gu V.. g U V...* U V., by alying gx. g x. g* see 3.8. in Casalis and Letac to x e and g gu V... Since and are linear functions of,., 5.1. roves that there exists b such that 8. holds, and the roof is comlete. 6. Proof of Theorem.5. The roof of Theorem.5 relies on the following roosition: PROPOSITION 6.1. Let U be a random variable on E with Wishart distribu- tion. Then U. is finite if and only if nr 1 dr 1.,. Furthermore, for nr one has 6.1 U, nr U U. ide.. nr

18 590 G. LETAC AND H. MASSAM REMARKS. Equation 6.1. is well known from symmetric real matrices see Muirhead 198, ages 97 and 113; note that on age 113, 3.6 b,v should be relaced by V. However, the roof is not that easy, and ours is rather different from the one given by Muirhead. Note also that 1 is the condition for a gamma random variable U in order that 1U. is finite. Finally, 6.1. and 6.. imly that, as in Corollary.6, Cov U, U. id E. nr PROOF. where Suose that nr. For x in E, denote nr g x.. det x., E nr d. d r 1... E Then the density of U with resect to the Lebesgue measure on the Euclidean sace E and restricted to E. is ² : 6.4. ex, xg x. det... A way to see that U exists is to observe that, for x in E, x det x is actually a olynomial taking its values in E. This comes from the very definition of the determinant of x by the equality r r r x x trace x e det x 0. Since nr, this olynomial divided by det x is integrable on E with resect to ex ², x: g x. dx. Let us assume that belongs to E. Since E is Euclidean, we shall write the differential of a real differentiable function on E as a gradient, that is, as an element of E. Clearly, the function x ex², x: g x. has differential ² : x ex, x g x. nr. x, since the differential of x log det x is x x. Furthermore, this function is 0 on the boundary E E. An alication of Stokes formula then gives H ² : 6.5. ex, x g x. nr. x dx 0. E For and given the density 6.4., 6.5. gives now Taking the differential of the first member of 6.5. with resect to yields H E E. ² : ex, x g x. nr. x x x id dx 0.

19 WISHART DISTRIBUTIONS 591 Again, for and given the density 6.4., the receding equality and H ex ², xg : x. det. xdx, E which is the exectation of U, yield 6... Finally, from 6.1., we get 6.6. H E nr ² : ex, x det x trace x dx trace. det. E., nr which, as a function of, is a Lalace transform of a ositive function. The second member of 6.6. has a ole at nr, and, from 6.3., we see that the integral does not exist for nr. The roof of Proosition 6.1 is comlete. PROOF OF THEOREM.5. We aly the rincile of equality 3.9. to the air Qu u and Q cq, with c defined by 10.. Relacing in by. will give the first of the two equalities 11.. One roceeds in the same way with 6.., to get the second equality in 11., by dealing with the air Qu u u and Q1 cid 1 E cq, where c1 and c are defined by 10.. The roof of Theorem.5 is then comlete. To end this section, we observe that an analogue of Proosition.1 for the inverse is available here. We ski its roof, which imitates the first art of Proosition.1. Here again, the symmetry of Qe is imortant. PROPOSITION 6.. Let Q be the linear sace of functions Q: E LE 1 such that for all g in G one has 6.7. Q g x.. g* Q x. g* and such that Q e. is in L E. s. Then Q1 has dimension, and the air x x x and x ide defines a basis of Q Proof of Theorem.7. Since 9. is true, 1. and 4.1. hold. Re- call that 0 a 1. For in E, we denote h U ex², U :. U. Then the first equality of 11. imlies that 7.1. hu. LV. chuv.. Taking the differential of both sides of 7.1., we get 7.. h L h L ch U V U V UV.. The second equality of.11 imlies that 7.3 h L c L id c h. U V 1 UV E UV

20 59 G. LETAC AND H. MASSAM Eliminating h. between 7.. and 7.3. and using 4.1. U V, we get 7.4. c c. h. c 1 a. h.. cc L. id 0. U U 1 U E Observe that.11 imlies that c 0, since U and U V. are in E. Having c c 0in 7.4. imlies c 1 U U E h L id, 1 a and this is imossible since the left-hand side of the equation has rank 1 and the right-hand side has rank 0 if c1 0 and rank n if c1 0. Thus c c 0. We adat to 7.4. the method of resolution of one-dimensional linear and nonhomogeneous ordinary differential equations, by considering the function w on E, taking its values in E and defined by hu LU w, where c 1 a. c. c c. a Then 7.4. becomes 7.5. w cc 1 1 LU id E. c We now rove that the second member of 7.5., denoted by id E, must be a constant with resect to on E. This comes from the fact that w is a Hessian and must be symmetric; that is, the function on E defined by h, k. w. h, k. ²., h: k is symmetric in h and k. Clearly, this imlies that. 0onE. Since E is connected, is a real constant and is equal to 0 if and only if c1 0. In this case, from 7.5., it follows that w is a constant belonging to E. Thus, for any h in E orthogonal to w, we have ² U, h: ex², U :. 0 for all : This imlies that U is concentrated on w, which imlies, in turn, that U is concentrated on w. Proosition.8 imlies that the same is true for U V, and this contradicts the hyothesis of Theorem.7. Thus 0, and must be equal to 1, which is equivalent to c ac. Thus 7.5. becomes w. c 1 aid., and there exists a constant c in E such that w. 1 E 0 c 1 a. c Since hu LUw is in E for in E, this imlies that c1 0 and that c0 is in E. We write c0 and introduce the ositive numbers and q defined by nr a. c1 and a q.. So, for in E we now have 7.6. hu. LU., nr and we are going to deduce from this that the distribution of U is,. We introduce the ositive measure on E such that the distribution of U is 7.7. ex ², xg : x. dx., where g has been defined in the roof of Proosition 6.1. We want to show that is roortional to the Lebesgue measure.

21 WISHART DISTRIBUTIONS 593 Using density 7.7 for comuting hu in 7.6 and relacing by in E, we obtain that H 7.8. ex², x: g x. dx. 0. E Since ex², x: g x., as a function of x, is 0 on the boundary of E, using the Schwartz derivative, we see that 7.8. imlies H ex², x: g x. dx. 0 E for all in E. Thus g x. dx. 0, and is roortional to the Lebesgue measure. The roof is now comlete. 8. Further comments. Recall the following result by Huang, Li and Lo Huang 1994 : THEOREM. Let U and V be ositive, nondegenerate random variables such that U. is finite. Then there exists a and b 0 such that U U V au V, U U V b U V if and only if there exist, q 0 and 0 such that U and V have distributions and., q, Huang, Li and Lo Huang add the hyothesis V.. This is not necessary if one roceeds as follows for the roof: Define for 0 the function f U ex U... One gets easily U 8.1 f L af, U V UV 8. f L bf. U V UV Derive 8.. and eliminate f with 8.1.; then derive 8.1. U V and use fuv f UL V. One gets the two equations ž / b U V U V U V U V 1 f L f L 0, 1 a f L f L 0, a and the elimination of L, L. V V between them leads to a differential equation for f U. Now, a referee has asked for the generalization to the Wishart distributions of the above result. It seems to be an interesting and difficult roblem which needs new methods. As a artial answer, we shall content ourselves here to mention the following direct result: THEOREM 8.1. Let U and V be indeendent random variables with Wishart distributions, and q, on the simle Euclidean Jordan algebra E, with Pierce constant d and rank r. Assume that A 1 dr Denote also A dr 1., B 1 dr and consider

22 594 G. LETAC AND H. MASSAM the matrices and Then B. b b b b d A 1 1 A A B d 1 A B. B q b1 U V.., U U U V b U V U V 1 U b U V. U V.. b U V. A statement similar to Corollary.3 also holds by considering the line eigenvectors of the matrix B.. The roof of Theorem 8.1 is more difficult than the roofs of Theorems. and.5. It relies on the extension to the Jordan algebras of the formulas of von Rosen 1988., Theorem 3.1 ii., and Das Guta 1968., Lemma.4 ii.. The tool for this is a result on the Peirce decomosition of a Wishart variable obtained by Massam and Neher 1997., Theorem The roof of our Theorem 8.1 can be found in Letac and Massam 1997., which contains a general study of equalities of the tye. c Q. U. U V c Q. U V., where U and V are indeendent and have distribution in the same exonential disersion model e.g., the Wishart one., where Q belongs to some finite-dimensional linear sace of functions and where c ;. is a suitable family of automorhisms of. q Acknowledgment. We thank Muriel Casalis for several comments. REFERENCES CASALIS, M Familles exonentielles naturelles invariantes ar un groue. These, ` Univ. Paul Sabatier. CASALIS, M Un estimateur de la variance our une famille exonentielle `a fonction variance quadratique. C.R. Acad. Sci. Paris Ser. I, Math CASALIS, M. and LETAC, G The LukacsOlkinRubin characterization of the Wishart distributions on symmetric cones. Ann. Statist DAS GUPTA, S Some asects of discrimination function coefficients. Sankhya Ser. A FARAUT, J. and KORANYI, A Analysis on Symmetric Cones. Oxford Univ. Press.

23 WISHART DISTRIBUTIONS 595 HUANG, W.-J., LI, S.-H. and HUANG, M.-N. L Characterizations of the Poisson rocess as a renewal rocess via two conditional moments. Ann. Inst. Statist. Math KAWATA, Y Fourier Analysis and Probability. Wiley, New York. LETAC, G. and MASSAM, H Reresentations of an exonential disersion model with an aendix by E. Neher.. Technical reort, Univ. Paul Sabatier. LUKACS, E A characterization of the gamma distribution. Ann. Math. Statist MASSAM, H An exact decomosition theorem and a unified view of some related distributions for a class of exonential transformation models on symmetric cones. Ann. Statist MASSAM, H. and NEHER, E On transformations and determinants of Wishart variables on symmetric cones. J. Theoret. Probab MUIRHEAD, R Asects of Multivariate Analysis. Wiley, New York. OLKIN, I. and RUBIN, H A characterization of the Wishart distribution. Ann. Math. Statist RAO, C. R On a roblem of Ragnar Frish. Econometrica SRIVASTAVA, M. S On comlex Wishart distribution. Ann. Math. Statist VON ROSEN, D Moments for the inverted Wishart distribution. Scand. J. Statist WANG, F Extensions of Lukacs characterization of the gamma distribution. Analytic Methods in Probability Theory. Lecture Notes in Math Sringer, New York. WESOŁOWSKI, J A constant regression characterization of the gamma law. Adv. in Al. Probab LABORATOIRE DE STATISTIQUE DEPARTMENT OF MATHEMATICS ET PROBABILITES AND STATISTICS UNIVERSITE PAUL SABATIER YORK UNIVERSITY 3106 TOULOUSE CEDEX NORTH YORK, ONTARIO M3J 1P3 FRANCE CANADA letac@cict.fr AND DEPARTMENT OF STATISTICS UNIVERSITY OF VIRGINIA CHARLOTTESVILLE, VIRGINIA 903