COMPUTING MULTIVARIATE STATISTICS

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1 COMPUTING MULTIVARIATE STATISTICS PLAMEN KOEV Abstract. Many multivariate statistics are expressed as functions of the hypergeometric function of a matrix argument, or more generally, as series of Jac functions. This wor is a collection of formulas, identities, and algorithms useful for the computations of these statistics in practice. Numerical examples are presented.. Definitions.. Partitions and hoo lengths. For an integer we say that, 2,... is a partition of denoted if 2 are integers such that The quantity is also called the size. We introduce a partial ordering of partitions and say that µ λ if µ i λ i for all i, 2,.... Then λ/µ is called a sew shape and consists of those boxes in the Young diagram of λ that do not belong to µ. Clearly, λ/µ λ µ. The sew shape /µ is a horizontal strip when µ 2 µ 2 [3, p. 339]. The upper and lower hoo lengths at a point i, j in a partition i.e., i j, j i are h i, j j i + i j + ; h i, j j i + + i j. The products of the upper and lower hoo lengths are denoted, respectively, as H h i, j and H h i, j i,j i,j and the product of the two plays an important role in what follows, thus we define. j H H..2. Pochhammer symbol and multivariate Gamma function. For a partition, 2,..., n and >, the Generalized Pochhammer symbol is defined as.2 a a i n + j i a i n + j a i, i i,j i j where a aa + a + in the rising factorial. For a partition in only one part a a is independent of. The multivariate Gamma function of parameter is defined as m.3 Γ m c π mm 2 Γ c i for Rc > m. i i Date: July, 23.

2 2 PLAMEN KOEV.3. Jac function and hypergeometric function of a matrix argument. Definition. Jac function. The Jac function J X J x, x 2,..., x m is a symmetric, homogeneous polynomial of degree in the eigenvalues x, x 2,..., x m of X. It is defined recursively [29, Proposition 4.2]:.4 J x, x 2,..., x n, if n+ > ; J x x + + ; J x, x 2,..., x n J µ x, x 2,..., x n x /µ n β µ, n 2, µ where the summation in.4 is over all partitions µ such that /µ is a horizontal strip, and { i,j.5 β µ B µi, j h i,j µ Bµ µi, j, where Bν µi, j ν i, j, if j µ j ; h ν i, j, otherwise. Remar.2. There are several normalizations of the Jac function: C The normalization J Theorem.]. C 2 X is the zonal polynomial and P, J, P, and Q, related as in X is such that for n the coefficient of x x 2 x n in J X is n! [29, X Q X s λ X is the Schur function. Definition.3 Hypergeometric function of a matrix argument. Let p and q be integers, and let X be an m m complex symmetric matrix with eigenvalues x, x 2,..., x m. The hypergeometric function of a matrix argument X and parameter > is defined as.6 pf q a,..., a p ; b,..., b q ; X! a b a p b q C X. The hypergeometric function of two matrix arguments X and Y, and parameter > is defined as.7 pf q a,..., a p ; b,..., b q ; X, Y! a b a p b q C XC Y C I.4. Random matrix ensembles Wishart, Laguerre, and Jacobi. An m m matrix is distributed as β-wishart with n n m degrees of freedom and covariance matrix Σ denoted A W m n, Σ if the joint density function of its eigenvalues λ, λ 2,..., λ m is.8 det Σ n K m n m λ n m+ i i j< where 2/β, L diagλ, λ 2,..., λ m, and.9 K m a m!ma π mm λ λ j 2 F L, Σ, m aγ m m Γ m. For β >, a > m, and 2 β, a matrix A is said to be β Laguerre distributed denoted A L m a if A W m a, I. The joint density of the eigenvalues λ, λ 2,..., λ m of A is then []:. K m a m λ i Γ m a i e λ i j< λ λ j 2..

3 COMPUTING MULTIVARIATE STATISTICS 3 For β >, 2 m β, and a, b >, a matrix C is said to be β Jacobi distributed denoted C J m a, b if its joint eigenvalue density is m m a. λ m b S m i λ i λ j λ 2, a, b i j< where S m a, b is the value of the Selberg Integral [27]: m! m S m Γ m a, b π mm Γ m Γ m a Γ m b. Γ m a + b.5. Empirical models for the classical random matrix ensembles. It is convenient to use uniform notation for real, complex, and quaternion normal distributions: N, for 2,, 2, respectively. Definition.4 Real, complex, and quaternion Wishart ensembles. Let Σ be an m m symmetric semidefinite matrix and let Z N, I n Σ by an n m real, complex, or quaternion Gaussian random matrix for 2,, 2, respectively. Then the matrix A ZD Z is said to have a real, complex, or quaternion m m central Wishart distribution with n degrees of freedom and covariance matrix Σ. The notation is A W m n, Σ for 2,, and 2, respectively [23, Definition 3..3, p. 82], [26]. For 2,, 2, the density of A W m n, Σ is [26, Definition 3.], [23, 8, p. 62]:.2 detσ n Γ m n e tr Σ A det A n m+. and the eigenvalues of A are distributed as.8 [23, Theorem 9.4., p. 388], [26, 3.2]: Definition.5 Real, complex, and quaternion Jacobi ensembles. Let the matrices A W m n, Σ and B W m n 2, Σ be independent real, complex, or quaternion Wishart matrices where 2,, or 2, respectively. Then C AA + B is called a real, complex, or quaternion Jacobi matrix, respectively. Remar.6. The eigenvalues of C are unaffected by Σ, which can thus be assumed to equal I without loss of generality. Theorem.7. For the matrix C from Definition.5, we have C J m n, n2 for 2,, 2. Proof. Theorem in Muirhead [23] for 2 and repeating the same argument for and Empirical models for the classical random matrix ensembles. The complex normals a N, are generated as a 2 a + ia 2, where a, a 2 N 2,, and the quaternion normals are generated as a 2 a + ia 2 + ja 3 + a 4, where a, a 2, a 3, a 4 N 2,. A Wishart matrix is empirically generated as Σ /2 Z D Z Σ /2, where Z is n m and z ij are i.i.d. N,. Definition.8 β-laguerre matrix model. For β >, let 2 β and define the m m matrix L as L β BBT, where χ 2a χ βm χ 2a β B......, a > β m. 2 χ β χ 2a βm The notation Z D stands for the quaternion conjugate transpose of the matrix Z and reduces to the Hermitian transpose Z H when Z is complex and the transpose Z T when Z is real.

4 4 PLAMEN KOEV Then L L m a [8, 28]. Remar.9. The above definition differs from the one in [8] in that we have the additional factor of β. This way W m n, I and L m n have the same eigenvalue distributions for 2,, 2. Proposition. Sutton s β-jacobi matrix model. Let C Z T Z, where c m s m c m Z c m s... m... s2 c, c s with c Beta a m c Beta, a + b 2m Then C J m a, b for any > [3, Section 5]., b m, s c 2 ;, s c Connection between β Laguerre and β Jacobi. If λ i are the eigenvalues of a β Jacobi matrix distributed as J m a, b, then λ bλ i i b λ i are jointly distributed as., the eigenvalues of a Laguerre matrix distributed as L m a as we now prove. Proposition.. Let L m a; Λ be the joint eigenvalue density of the β Laguerre ensemble. and J m a, b; Λ be the joint eigenvalue density of the β Jacobi ensemble.. Then lim b b m J m a, b; ΛbI + Λ L m a; Λ. Proof. Directly from.: J m a, b; ΛbI + Λ b m b ma m m a λ S m i + a, b i π mm Γ m Γ m a + b m ma m!γ m λ m Γ m aγ m bb ma i By taing the limit as a 2 we get.. i b λ i 2 a b λ j λ 2 m a j< + λ i 2 a b λ j λ 2. b j<

5 COMPUTING MULTIVARIATE STATISTICS 5 2. Identities involving p F q a :p ; b :q ; X, Y Let X and Y be m m Hermitian matrices with eigenvalues x,..., x m and y,..., y m, respectively, and let I be the m m identity matrix. Also define the m m matrices X diagx 2 x, x 3 x,..., x m x and Ȳ diagy 2 y, y 3 y,..., y m y. 2.. Pure identities. 2. F F X e tr X ; 2.3 F a; X deti X a, X < I; X, Y e tr X F X, Y I e ytr X+xtr Y mxy F F a; X, Y deti X a F a; XI X, Y I; F m ; X, Y i,j x iy j ; F a; c; X e tr X F c a; c; X; F ; m ; X F ; m ; X ti et ; 2F, n; m ; X tn 2F, n; m ; tx + ti, t ; 2F a, b; c; I Γ m cγ m c a b Γ m c aγ m c b ; 2. a, b; c; xi PfA see [3] for the definition of A; 2F a, b; c; X 2 F c a, b; c; XI X 2. deti X b F 2 m ; m ; X, Ȳ ; 2 F c a, c b; c; X deti X c a b ; 2F a, b; c; X 2 F a, b; c; I 2 F a, b; a + b + + m c; I X, a or b Z ; pf q, a 2:p; b :q ; xi p F q m, a 2:p; b :q ; x; pf q a :p ; b :q ; X i<j x i x j det x m j i pf q a :p j + ; b :q j + ; x i m ; i,j i i,j pf q a :p ; b :q ; X, Y c det pf q a :p m + ; b :q m + ; x i y j m i<j x i x j i<j y i y j m i! q j where c b j m + m i p j a ; j m + m i 2F m Γ m m +, r; X pf q a :p ; b :q ; X lim + + r Γ m m c p+ F q The references for the above identities are: + det X r a :p, c; b :q ; c mr C, r, X, r Z ;! X lim c p F q+ a :p; b :q, c; cx. 2. [, 3.3, p. 593] also [26, p. 444] and [23, p. 262] for and 2, respectively; 2.2 [2, section 6], also [2] for 2. Reproven in Theorem 5.; 2.3 [, p. 593, 3.4] also [26, p. 444] and [23, p. 262] for and 2, respectively; 2.4 [2, 6.29] both conjectures used appear to have been proven in literature; 2.4 [2, 6.]; 2.6 [, p. 596, 3.6] also [23, 6, p. 265] for the case 2; 2.7 New result, see Theorem 5.3. For m 3, 2 also proven by Bingham [3, Lemma 2.];

6 6 PLAMEN KOEV 2.8 This is a result of Raymond Kan, see Theorem 5.3; 2.9 [, 3.4, p. 594] also [4, p. 24, 5] for the case 2; 2. [3]; 2. [, Proposition 3..6, p. 595]; see also [23, p. 265, 7] for the case 2; 2.2 [, Proposition 3..7, p. 596]. The condition a or b Z implies that the series expansion for 2F terminates. 2.3 New result, see Theorem 5.2. See also [2, 5.3] for ; 2.4 [25, 33, p. 28]; 2.5 [2, Theorem 4.2] there is a typo in [2, 4.8]: βn should be β n, see also [25, 34, p. 28]; 2.6 New result, see Theorem 5.4; 2.7 [, 3.5]. Trivially implied by lim x x x Integral identities. Let X be an m m real matrix and dh be the invariant normalized measure on the orthogonal group Om. The identities 2.2 and 2.2 have complex equivalents, see, e.g., James [4]. The identities 2.8 and 2.9 are valid only for { 2,, 2}. In those cases the matrix Y is m m real symmetric, complex Hermitian, or symplectic self-dual for 2,, 2, respectively. Define Γ m c Then F m a, c F a; c; X F m a, c 2F a, b; c; X F m a, c I Γ m aγ m c a. e trxy det Y m a deti Y where Ra > m m m, Rc >, Rc a > I deti XY b det Y m a where X <, Ra > m m, Rc a > ; 2F 2 a 2, a+ 2 ; m 2 ; XXH deti XH a dh; F n ; XX H 2 F 2 X, Y p+f q a :p, a; b :q ; Y p+f Om U n Om c mγ m a q+ a :p, a; b :q, b; Y F m a, b c m C β Y e tr XH+XH dh, for,2; etrhxh T Y dh; R m + [,] p F q m Y a+p c β m Γ β m a + p a + p β Where p m β/2 +, c m π mm 2 m! m Γ i i Γ The references for the above identities are: 2.8 [23, p. 264, 4] for 2; [9] for { 2, }; 2.9 [23, p. 264, 5] for 2; [9] for { 2, }; 2.2 [24]; m c a ; deti Y dy, m c a dy, e tr X pf q a :p ; b :q ; X, Y X a p X 2 dx; a :p ; b :q ; X, Y X a p I X b a p X 2 dx F β R + m, cβ X, Y X a C β X X β dx; m c m2/β, and X i<j x i x j.

7 COMPUTING MULTIVARIATE STATISTICS U n stands for the orthogonal group of size n for 2 and the unitary group of size n for [4, 27, p. 479, and 9, p. 488], direct consequence of 2.2 via a limit, perhaps also due to Herz 955 [24]; 2.22 [23,, p. 39]; 2.23 [2, 6.2]; 2.24 [2, 6.2] [2, Conjecture C, section 8]. Proven in [2, section 6], but the scaling constant in Macdonald is the correct one Identities involving the Jac function C X tr X ; J xi m x m J X j! C X HQ X H P X; J X detx J X m, 2,..., m m i + + i, m > C I + X C I σ C σ X σ C σ I ; P ˆµ X detxn P µ X, where µ N and ˆµ i N µ m+ i, i, 2,..., m. For partitions in only one part, C I m 2.32 ; C C X + ti C I t s s X m 2.33 s C s I s The references for the above identities are: i s t s m s C s X. s 2.26 This is the definition of the normalization C X [29, Theorem 2.3]; 2.27 [29, Theorem 5.4]; 2.28 [7, 6], [22,.22, p. 38]. Then P Q s, the Schur function [29, Proposition.2]; 2.29 [29, Propositions 5. and 5.5]; 2.3 [7, 33]. This also serves as a definition for the generalized binomial coefficient σ ; 2.3 [2, 4.5]. ˆµ is the complement of µ in the partition N n ; 2.32 Directly from 2.27; 2.33 Directly from Multivariate Gamma function identities. Γ m a e tr A m a 2.34 det A da for, 2, 2.35 Γ m Γ m x x A> where A is Hermitian or symmetric positive definite, respectively; Γx Γ. x m The references for the above identities are:

8 8 PLAMEN KOEV 2.34 [4, 56 and 83]; 2.35 The 2 version is due to Pierre-Antoine Absil. In general, from the definition.3: Γ m x m Γ x i Γ m x Γ x i Γx Γ. x m i 3. Evaluating p F q Most densities and distributions in this paper are expressed in terms of the hypergeometric functions of the form 3. fx p F q a :p ; b :q ; xf x! a a p C b b q F x c, where F is a given matrix and c! a b a p b q C F. If fx is to be evaluated at many points x, then it maes perfect sense to evaluate the coefficients c only once. Then fx can be evaluated at as many points as necessary at a negligible cost using Horner s method. For example, in order to evaluate s p F q a :p ; b :q ; X we use the algorithms of [9]. We first choose a truncation limit M the summation in 3. will be only for M and call [s, c] mhgm,, a :p, b :q, x :m. The array c returns the marginal sums for partitions of size,,..., M, so that s sumc. When X xi is a multiple of the identity, we call 3.2 [s, c] mhgi[m, K],, a :p, b :q, m, x. The second output c c, c 2,..., c m+ is such that s c + c 2 x + + c m+ x m. The optional parameter K is such that it truncates the summation to be over only those partitions whose parts do not exceed K, i.e., i. This is particularly useful in evaluating the expressions 4.3 and 4.9. The parameter K defaults to M if unspecified. 4. Distributions of the eigenvalues of the random matrix ensembles 4.. Wishart. Let A W m n, Σ, 2,, 2, be a real, complex, or quaternion Wishart matrix Extreme eigenvalues. For ease of notation, define the common expressions: t n m+, Then G m a m Γ m Γ m + and F a + m m +, a Γ m m +. m a Γ P λ max A < x G m n det x Σ n F n ; n+m + ; x Σ G m n det x Σ n e tr x Σ F P λ min A < x e tr x Σ m t m! C x Σ, t Z, n+m + ; + ; x Σ ; F m a det x Σ t e tr x Σ 2 F m +, t; x Σ, t Z.

9 COMPUTING MULTIVARIATE STATISTICS 9 For the density we differentiate 4.2 and 4.3 with respect to x to obtain: dens λmaxax G m n x mn detσ n e tr x Σ x + mn tr x Σ c ; dens λminax e tr mt x Σ tr Σ + x tr x Σ where m C, t Σ.! C Σ!, t Z, c n+m + The expression 4.2 appears to be numerically more stable than 4. and is obtained using 2.6. The expressions 4. and 4.3 come from [23, Theorems 9.7. and 9.7.4] for 2 and from [26, Corollaries 3.3 and 3.5] for ; 4.4 is a new result see 2.6 and Theorem Trace. 4.5 dens tr A x det x Σ n e x z Γ nm + z x z n! C I zσ, where z is arbitrary. Muirhead [23, p. 34] suggests the value z 2σ σ m /σ +σ m, where σ σ m are the eigenvalues of Σ. The second sum in 4.5 is the marginal sum of F n ; I zσ thus the truncation of 4.5 for M can be computed using mhg as follows. If s is the vector of eigenvalues of Σ, these marginal sums for say through M are returned in the variable c by the call: [f,c]mhgm,,n/,[],-z./s, maing the remaining computation trivial. Empirically the trace can be generated as 2 m i t iσ i, where t i χ 2 2n/ are i.i.d. The density of the trace of the Wishart matrix was obtained by Muirhead [23, p. 34] in the real, 2, case. For other > this is a new result, see Theorem 6.2. [Moments of det of Wishart] [23, p. 5] β-laguerre. The distributions of the extreme eigenvalues of a β Laguerre matrix L L m a are given by , where n a and Σ I [8, Theorem.2., p. 46], [2]. For t a m Z the density of λ min L is nown from [8, Theorem.., p. 47] note the β factor in our definition of a β Laguerre matrix, but the scaling factor is a new result see Theorem 6.3: 4.6 dens λminlx m Γ m m + Γ m a x tm e mx 4.3. Jacobi. Let C J m a, b be a β Jacobi matrix. Define t b m and D m,a,b Γ m a + bγ m m + Γ m a + m + Γ m b. Then 4.7 P λ max C < x D m,a,b xma 2F a, t; a + m 4.8 dens λmaxcx D 2F m +, t; x I m. + ; xi; m,a,b ma xt x ma 2F a m, t; a + + ; xi m.

10 PLAMEN KOEV When t Z, 2.2 gives alternative and seemingly numerically much more stable expressions: mt P λ max C < x x ma, t a C xi ;! dens λmaxcx H x t x ma 2F a, t; t 2 ; xi m, where H Γ a + b m Γb + Γ Γ b m 2 Γt + Γ m. Γa The density and distribution of the smallest eigenvalue follow easily from the corresponding expressions for the largest ones. If C a,b J m a, b, then I C a,b J m b, a. Therefore P λ min C a,b < x P λ max C b,a < x dens λminc a,b x dens λmaxc b,a x. The results in this section are due to Constantine [7, 6] 4.7 for 2, Absil, Edelman, and Koev [], 4.8 for 2, and Dumitriu and Koev [9] for the generalization to any. The 2 version of 4.9 is also implied by [23, 37, p. 483]. [Density is unproven anywhere!] Trace. [In progress] The Laplace transform of the distribution of the trace of Jacobi is given in Muirhead [23, 29, p. 479]: Gt F 2 r 2 ; r+n 2 ; ti t! r 2 2 r+n 2 2 C I. [Moments of the determinant of noncentral Jacobi] [23, p. 455] Numerical experiments. Numerical experiments for the distributions in the previous sections are in Figure. 5. New results We now prove 2.2, 2.7, 2.8, 2.6, 2.3, and 4.5. Identity 2.8 identity 5.6 below is due to Raymond Kan. Theorem 5.. With the notation from Section 2. the identity 5. F X, Y e tr X F X, Y I e ytr X+xtr Y mxy F m ; m ; X, Ȳ holds for any >. Proof. For the first part, following Kaneo [7, Section 5.], denote by gµλ the normalized coefficient of J X in the expansion J µ XJ λ X j λ g µλj X, where j λ is defined in.. Since gµλ unless µ + λ [29, Corollary 6.4], the above along with 2.28 imply 5.2 C µ XC µ! λ! λ X j µ j λ! g µλc X.

11 COMPUTING MULTIVARIATE STATISTICS pdf λ min W 5,I cdf λ max W 3 3,diag.5,.6, pdf λ min CW 2 5,I pdf λ min β Laguerre n,β,a5,.5, pdf λ min Jacobi, n,n 2,m6,4, pdf of tra, AW 3 4,diag.25,.4, cdf λ max CW 2 2,diag.5, cdf λ max β Laguerre m,β,a3,2, cdf λ max Jacobi, n,n 2,m4,5, cdf λ max complex Jacobi, n,n 2,m6,6, Figure. Numerical experiments Also from Kaneo [7, Proposition 2], 5.3 µ λ j λ j µ g µλ. Since µ unless µ, the summation in 2.3 can be thought of as being over all partitions σ. We use this fact along with 5.3 to transform the left hand side of 5.: 5.4 F X, Y + I! C XC Y + I C I! C X C µ Y µ µ C µ I gµλ C µ Y j λ j µ C µ I.! C X µ λ

12 2 PLAMEN KOEV Next, we transform the right hand side of 5. using 2. and 5.2: F X, Y e tr X µ! C µ Y C µ X C µ I µ µ µ λ λ C µ Y C µ I C µ Y C µ I µ! λ! C j µ j λ λ λ! C λ µ XC X λ g µλc X!, which is the same as 5.4, thus proving 5.. For the second part, let I be the identity matrix of size. Since C X x I m C X and analogously for Y, from 2.2 F X, Y e ymtr X F X, Y y I m X e ytr X+xtr Y mxy F X x I, Y y I m e ytr X+xtr Y mxy! C X x I m C Y y I m C I m e ytr X+xtr Y mxy! C I m C XC Ȳ C I m C I m e ytr X+xtr Y mxy F m ; m ; X, Ȳ. The next two theorems use the fact that Theorem 5.2. for partitions in more than one part. pf q, a 2:p; b :q ; xi p F q m, a 2:p; b :q ; x. Proof. Using 2.32, pf q, a 2:p; b :q ; xi! a 2 a p C b b q xi! a 2 a p x m b b q p F q m, a 2:p; b :q ; x. Theorem 5.3. Let the matrices X and I be m m. Let t and n be real numbers and let >. The following identities hold: 5.5 F ; m ; X + ti F ; m ; X et ; 5.6 2F, n; m ; X tn 2F, n; m ; tx + ti, t.

13 COMPUTING MULTIVARIATE STATISTICS 3 Proof. We transform the left hand side of 5.5 using 2.32 and 2.33: F ; m ; X + ti We use 2.33 again to obtain 5.6:! m C X + ti! C X + ti C I t s C s X! s s C s I i i F C i X C i I i! m ji ; m ; X et. t j i j! j i C X t j i j i! t n 2F, n; m ; tx + ti t n! n m C t n n C t s s! s s ji tx + ti tx C s t n C i X t j i i C ti i I j! ji i! C i X C I n i t n+i i 2 F, n; m ; X. i l I j i n j t l l! n + i l } {{ } Next, we prove that 2.6 is true for any X. Theorem 5.4. For r Z : 5.7 2F m Γ m m +, r; X + + r Γ m m + det X r mr, r C X.! Proof. From 2., 2F a, r; c; I X 2 F c a, r; c; I X det X r.

14 4 PLAMEN KOEV For r Z we have from 2.2: Γ m cγ m c a + r Γ m c aγ m c + r 2F a, r; a r + + m c; X Γ m cγ m a + r Γ m aγ m c + r 2F c a, r; a r + + m ; X det X r. Cancelling terms on both sides we get: Γ m c a + r Γ m c a Plugging in a m Γ m c m + r Γ m c m 2F cmr 2F a, r; a r + + m c; X Γ m a + r 2F Γ c a, r; a r + + m m a ; X det X r. +, replacing X by cx, and then moving a cmr factor to the left we get: m +, r; r c; cx Γ m m + + r Γ m m + 2F c m, r; r; c X det X r. By letting c and taing limits on both sides using 2.7 and Γz + zγz to conclude that we get 5.7. lim c Γ m c m + r Γ m c m cmr, 6. Eigenvalue distributions Theorem 6.. For the extreme eigenvalues of a Wishart matrix A W m n, Σ we have: P λ max A < x m Γ m Γ m n+m + + det x P λ min A < x e tr mt x Σ, t Σ n F C x Σ! n ; n+m + ; x Σ, t n m+ Z. Proof. For the first part, the condition λ max < x is equivalent to λ i [, x], i, 2,..., n. To obtain the desired distribution, we integrate the joint density.8 over [, x] n, mae a change of variables L xz, then use dl x m dz and 2.24 to obtain P λ max < x [,x]m det Σ n x mn K m n det Σ n K m n m λ n m+ i i j< m z n m+ i [,] m i j< detxσ n K m n m! m i Γ i π mm 2 Γ Γ m λ λ j 2 F L, Σ dl z z j 2 F Z, x Σ dz m n Γ m m + Γ m n+m + Simplifying the last expression using.3 and.9 we obtain 6.. F n ; n+m + ; x Σ.

15 COMPUTING MULTIVARIATE STATISTICS 5 For the second part, analogously, the change of variables is L xi + Z with dl x m dz: P λ max > x det Σ n K m n x mn det Σ n K m n m λ t i [x, ] m i j< R m + e tr x Σ detxσ n K m n e tr x Σ detxσ n K m n Γ m m Γ + m n deti + Z t j< λ λ j 2 F L, Σ dl R m + m t m t z z j 2 F I + Z, x Σ dz t t! e tr x Σ det x Σ t m t C Z! R m + j< z z j 2 F Z, x Σ dz C Z z z j 2 F Z, x Σ dz j< t m! + C a x Σ, which along with 2.6 implies 6.2. We used 2.2 and 2.3 to go from 6.3 to 6.4 and 2.25 to go from 6.5 to 6.6. The following theorem generalizes Theorem in Muirhead [23, p. 339] to all >. Theorem 6.2. If A W m n, Σ, then dens tr A x det x Σ n e x λ where λ is arbitrary. Γ nm + x λ λ n! C I λσ, Proof. We follow the argument in Muirhead [23, Theorem 8.3.4, p. 339]. From.8, the moment generating function of tr A is: φt E[e tr tl ] det Σ n K m n R m + e tr tl m λ n m+ i i j< λ λ j 2 F tl, t Σ dl maing a change of variables L tl and using 2.23 we obtain det Σ n t mn c mγ m n K m n F n ; det tσ n deti t Σ n deti tσ n. t Σ

16 6 PLAMEN KOEV 6.7 For < λ < write φt deti tσ n tλ mn detλ Σ n det [I tλ mn detλ Σ n F tλ mn detλ Σ n ] n tλ I λσ n ; tλ I λσ n tλ! C I λσ, where t and λ are such that I λσ < tλ, so that the F function above converges. Since tλ r is the moment-generating function of the gamma distribution with parameters r and λ and density function g r,λ u e u λ u r λ r u >, Γr the moment-generating function 6.7 can be inverted term by term to obtain the density function of tr A: dens tr A x det x Σ n e x λ Γ nm + x λ λ n! C I λσ, where λ is arbitrary and, as in Muirhead [23, p. 34], can be chosen as λ 2λ λ λ +λ, where λ and λ are the largest and the smallest eigenvalues of Σ, respectively. We now derive the scaling constant in 4.6; the proportionality result is from [8, Theorem.., p. 46]. Theorem 6.3. For the smallest eigenvalue of a β Laguerre matrix A L m a, where r a m Z, we have: P λ min A < y Γ m m + e my Γ y m a mr 2F m +, r; y I; dens λminay m Γ m m + Γ m a Proof. Let C J m a, b. From 4.7, 4., and the identity 2., we have: e my y mr 2F m +, r; y I m. P λ min C < x D m,b,a xmb 2F b, a + m + ; b + m + ; xi Substituting y b x x D m,b,a xmb x mr 2F m +, r; b + m we get P λ min bci C < y D m,b,a b + y mr b mb y mr + y b 2 F m +, r; b + m + ; b y I. + ; xi. Using Proposition. and taing the limit as b we obtain the distribution of λ min. With the density we tae the same approach. From 4.8, 4.2, and the identity 2. we have: mr dens λmincx mb D m,b,a xmb x r 2F b m, r; b + + ; xi m mb D m,b,a xmb x mr 2F m m +, r; b + + ; x x x I m

17 COMPUTING MULTIVARIATE STATISTICS 7 Substituting y b x x we get: m dens λminbci C y m Γ m + Γ m a m a + b b mr b + m Γ Γ m + + y b 2 F m m +, r; b + + ; b y I m. Using Proposition. and taing the limit as b, we obtain the density result. mb+ y mr + y mr b 7. Largest eigenvalue asymptotics In this section we present numerical evidence for the convergence of the largest eigenvalues of the Wishart and Jacobi ensembles to the Tracy Widom limits. 7.. The largest eigenvalue of the Wishart ensemble. Let A W m n, I. If m, n in such a way that n/m γ, then [5, Theorem.] 7. where λ max A µ σ D W, 7.2 µ n + m 2, σ n + m n + m /3. Johnstone suggests a slight modification of the rescaling and recentering constants, which yields faster convergence see El Karoui [8]: 7.3 µ 2 n 2 + m 2, σ n 2 + m 2 n 2 + m 2 In Figure 2 we plot 7. for m 2, 3,..., 6, and n 4m for the constants 7.2 and 7.3. / The smallest eigenvalue of the complex Wishart ensemble. Let A W m n, Σ and let λ m be the smallest eigenvalue of A. Define the constants µ and σ as in [5, 5.2, p. 36]: µ n m 2, σ m n /3. m n Then, as n, m in such a way that n/m γ >, we have λ m µ σ D W 2. In Figure 3 we plot the limiting distribution Painlevé, for β 2, the formula 4.3, and a Monte-Carlo experiment for m 4. 2 I than Brian Sutton for pointing out that σn in [5] has an incorrect sign.

18 8 PLAMEN KOEV Tracy Widom limit m2 m3 m4 m5 m Tracy Widom limit m2 m3 m4 m5 m6 m Figure 2. Comparison of largest eigenvalue of the finite Wishart vs the Tracy Widom limit for various values of m the right tail for m 6 has not converged. In the left plot we use the rescaling and recentering constants 7.2 [5]. In the right plot we use the recentering and rescaling constants 7.3 [8] The largest eigenvalue of the Jacobi ensemble. Let C J p 2 q 2, n q 2, where p minq, n q. 3 Following [6], let the angles γ p φ p in, π be defined via the equations sin 2 γ p p 2 2 n, φp sin2 q 2 2 n. Let µ p+ sin 2 φp + γ p, σ 3 p+ 2 2n 2 2 sin4 φ p + γ p. sin φ p sin γ p Now if p, q, n in such a way that p/n, q/n converge to positive, finite constants, then λ max C µ p+ σ p+ D W. For our experiment we fix p 4t + 2, q 6t + 3, n 2t + 6, and let t, 2,... Thus p/n, q/n /3, /2 and t n p q /2 is an integer. On Figure 4 we plot the density of λ max C µ p+ σ p+ vs the Tracy Widom limit for t, 2,..., In other words, if A W p q, I and B W p 2 n q, I have independent real Wishart distributions, then the eigenvalues of C are distributed as the roots u i of the determinantal equation A u i A + B.

19 COMPUTING MULTIVARIATE STATISTICS Smallest eigenvalue of complex wishart matrix CW m 2m,I painleve Figure painleve Figure 4. Comparison of the finite Jacobi vs the Tracy Widom limit for various values of t. We have p 4t + 2, q 6t + 3, n 2t Incomplete expressions. It appears that an incomplete expression for the distribution of λ max C yields results that appear to converge faster see Figure 5. It appears that the largest eigenvalue in the finite case have a Tracy Widom component smaller partitions and a correction. It would be interesting to understand why such a truncation, essentially, is closer to the limit.

20 2 PLAMEN KOEV One needs to sum up to pt 4t + 2t in 7.5. Instead we sum up to M as indicated. It is interesting that we need to increase M by about t as we move from t to t + ; thus M Ot 2 /2. If we loo at the expression 7.5 and simplify it, it does not seem to shed any light on why this phenomenon occurs P λ max C < x x pq 2 pt, t pt x 32t+2 t q 2 2 C 2 xi! 6t C 2 4t+2t x 32t+2 x t 4t+2t x 32t+2 x t xi! 6t+3 2 2!2 J 2 I j! 6t , j where j is defined in.. t,p,q,n,pt,m,6,9,8,6, t,p,q,n,pt,m2,,5,3,2,3 t,p,q,n,pt,m3,4,2,42,42, t,p,q,n,pt,m4,8,27,54,72, t,p,q,n,pt,m5,22,33,66,, t,p,q,n,pt,m6,26,39,78,56, t,p,q,n,pt,m7,3,45,9,2, t,p,q,n,pt,m8,34,5,2,272, t,p,q,n,pt,m9,38,57,4,342, t,p,q,n,pt,m,42,63,26,42, t,p,q,n,pt,m,46,69,38,56, t,p,q,n,pt,m2,5,75,5,6, Figure 5. Truncated polynomial expressions. Seems lie summing up to Ot 2 /2 converges to the limit faster.

21 COMPUTING MULTIVARIATE STATISTICS 2 8. The William Chen tables The MATLAB routine qmaxeigjacobi produces the entries of the William Chen tables [6, 5]. For example for m 7, n 4, s 6, and 2 2 means we are woring with real Jacobi matrices qmaxeigjacobi,2*m+s+,2*n+s+,s,.9,6 returns.9276 for the.9 percentage point to 6 digits, as expected. The matrices A and B in William Chen s papers are distributed as W s 2 2m + s +, I, and W s 2 2n + s +, I. Then the percentage points are computed for the largest eigenvalue of AA+B, which is distributed as J s 2 2m + s +, 2n + s What Jacobi implies about Laguerre when b This approach lead to discovering new results, e.g., 2.6 and Theorem 5.4. Let C J m a, b, C AA+B, where A L m a, I and B L m b, I are independently distributed Laguerre matrices. If λ is an eigenvalue of C then b λ λ is an eigenvalue of bab. Also, if b, the eigenvalues of bab approach those of A, a Laguerre matrix, see Proposition.. We transform 4.7 using P λ max C < x D m,a,b xma 2F a, b + m + ; a + m + ; xi D m,a,b xma x ma 2F a, a + b; a + m + ; x x I. We mae a change of variables y b x x to obtain the distribution of the largest eigenvalue of bab P λ max bab < y D m,a,b y b am 2F a, a + b; a + m y + ; b I Since lim b a + b D m,a,b b ma y ma 2F a, a + b; a + m + ; b y I. b, we have lim b 2 F a, a + b; a + m + ; b y I F a; a + m + ; y I. By taing the limit in 9. as b, we obtain 4. for Σ I and n a.. Any beta research We consider the joint eigenvalue density of the 2 2 matrix A X T X where [ ] [ ] χnβ G X β. χ n β σ The matrix X has the same distribution as an n 2 matrix of G β s, i.e., A has the same distribution as a 2 2 β-wishart matrix with n DOF and covariance matrix Σ diag, σ 2. For complex-values G β, we have [ ] [ ] [ ] [ ] χnβ χ X β, signgβ χ n β σ signgβ i.e., without loss of generality we may assume that [ ] [ ] χnβ χ X β χ n β σ [ ] [ ] χnβ σχ β x y, σχ n β x 2 where x χ nβ, x 2 χ n β, and y χ β are i.i.d. The joint eigenvalue density of A is given by.8 with m 2 and we will obtain it from direct arguments following Dumitriu s thesis [8].

22 22 PLAMEN KOEV If a is a random variable with density fx, then ca has density cf x c. The χ distribution has density Thus the joint density of X is dx 2 nβ 3 Γn β/2γnβ/2γβ/2 xnβ x e x2 /2 2 /2 Γ/2. x2 n β y β e x2 2 x2 2 2σ σ σ 2 y2 2σ 2 dx dx 2 dy. Going from here via the Jacobian J X A 2 4 x 2 x 2 /σ see [8, 4.2, p. 43], we obtain for the density of A da J X A dx 2 nβ 5 xnβ 3 Γn β/2γnβ/2γβ/2σnβ x n β 2 2 y β e x 2 2 x2 2 2σ 2 y2 2σ 2. From here we do eigendecomposition and get the joint density of the eigenvalues and eigenvectors. Integrating the eigenvectors out, this should jibe with the joint eigenvalue density.. Other See James [4, p. 496] for additional distributions. Khatri for moments of trace or determinant? Connection formula from Asymptotics of Special Functions? Reflection formula for Γ function? Change the values of t and r section 4.3 in software as well. Hypergeometric Functions of 2 2 matrix argument are expressible in terms of Appell s functions F 4. Tom Koornwinder and Sprinhuizen-Kuyper. Proceedings of the AMS vol. 7, no., 978. p See also [7,, p. 89]. Approximating the matrix Fisher and Bingham distributions: applications to spherical regression and Procrustes analysis Source Journal of Multivariate Analysis archive Volume 4, Issue 2 May 992 table of contents Pages: Year of Publication: 992 ISSN:47-259X Authors Ted Chang Donald Richards Christopher Bingham Publisher Academic Press, Inc. Orlando, FL, USA K.I. Gross and D.St.P. Richards, Hypergeometric functions on complex matrix space, Bull. Amer. Math. Soc, 24 99, Ua, b, z where Ma, b, z F a, b, z. π [ sin πb Ma, b, z M + a b, 2 b, z z b Γ + a bγb ΓaΓ2 b ],

23 COMPUTING MULTIVARIATE STATISTICS Faster versions of the algorithms in Koev and Edelman 26 [9] This section outlines a faster way to compute the updates in the computation of the hypergeometric function of a matrix argument in [9]. The results in this section are due to Raymond Kan and Vesselin Drensy. In what follows, it will be convenient to use certain additional definitions. We consider the partition, 2,..., h, where h is the number of nonzero parts of. When i > i+ we define the partition i, 2,..., i, i, i+,..., h. Also, let i i i and i i i. Next we express H in a way that does not involve the conjugate partition: Analogously, H h r c r + r c + r c h r j r j c j++ h H j j r h j r j r + r c + j r + r j + j j+ j j r + + r j. j j+ From these expressions, we obtain, for h, the number of parts of, H h H H h H h j h + h j j h +, h j h h + j j h The case X xi. Using 2.27 and 2.28 we have: where pf q a :p ; b :q ; X Q a a p b b q x Q, 2 j m Denoting δ ij i j and using 2. and 2.2 we get the following update for Q : p j a j + h q j b j + h Q Q h h m + h h h + j. δ ij δ ij + δ ij + δ ij +.

24 24 PLAMEN KOEV 2.2. Single matrix argument. In this case we use the Schur normalization of the Jac function S X Q X, defined in The choice to use Q as opposed to P, J, or C yields the simplest update for the coefficients σ µ below. Then pf q a :p ; b :q ; X Q S X, where From 2.2, the update for Q is Q Q h Q a a p p j a j + i q j b j + i b b q H. h h + j j h j h +. This update costs only 2p + q + 4h +, representing savings of h + 5h 2 arithmetic operations over the one in [9, Lemma 3.]. The coefficient update in the computation of the Jac function can also be done more efficiently than in [9]. From.4 we have: 2.3 S x,..., x n µ S µ x,..., x n x /µ n σ µ, where the summation is over all partitions µ such that /µ is a horizontal strip and see σ µ β µ H H µ i,j h i, j h i, j i,j µ h µi, j h µ i, j, where both products are over all i, j such that j µ j +. Following the assumptions of Lemma 3.2 in Koev and Edelman [9], let ν µ, where j µ j for j, 2,..., µ. Using 2.4 we update a more efficient way to obtain σ µ : σ ν σ µ r h r, µ h µ r, µ h r, µ h r µr, µ r + r µ µ r µ +. + r µ µ r r µ This update costs arithmetic operations assuming the µ i and i are stored and only updated, and the quantities r µ and µ r µ are computed only once and reused, representing savings of 2 + 6µ 7 arithmetic operations over the update in [9, Lemma 3.2]. We use 2.3 to compute S x,..., x i for i h +,..., n. For i h, we use a result of Stanley [29, Propositions 5. and 5.5] to obtain: S x,..., x h x x h h S where h I h, 2 h,..., h h. h I x,..., x h h h j i h i + + i j h i + i j +, 2.3. Two matrix arguments. We write the hypergeometric function of two matrix arguments.7 as: 2.5 pf q a :p ; b :q ; X, Y Q S XS Y,

25 COMPUTING MULTIVARIATE STATISTICS 25 where we define Q slightly differently from before: where we have used Q Q h Q a... a p b... b q C Y C I H m From 2. and 2.2, the updating scheme for Q is: p j a j + h q j b j + h H m H S Y. 2 h h m + h h + h + j, j h j h + j h + j h Complexity analysis. In this section we attempt to give some useful estimates of the cost of computing the hypergeometric function of a matrix argument. In practice, we truncate the hypergeometric series.6 by summing over partitions that do not exceed a certain size, M, for some M. Our experience shows that in order to reach convergence, one needs a rather large M, often several times the size of the matrix argument m. For such m M, we count the number of partitions M. If M, then i M i +, meaning M + m Λ M,m, m where Λ M,m { M, m+ } On the other side, if we choose an m-tuple, 2,..., m such that i M i +, i, 2,..., m, then the number of all such m-tuples such that [To be completed] 3. Cooley Tuey Ideas for Jac Functions The results of this section belong to Vesselin Drensy. 3.. Normalization of Jac polynomials. The Jac polynomials J x,..., x n satisfy the relation 3. J x,..., x n J µ x,..., x n x /µ n β µ, where the sum runs on all partitions µ such that /µ is a horizontal strip, i,j β µ B µi, j i,j µ Bµ µi, j, Bµi, ν j h νi, j ν j i + ν i j +, the upper hoo length, if j µ j and Bν µi, j h ν i, j ν j i + + ν i j, the lower hoo length, if j µ j +. We normalize J in the following way: 3.2 S In this way we obtain for that x,..., x n J x,..., x n i,j h i, j. S x,..., x n S x,..., x n,

26 26 PLAMEN KOEV the usual Schur function. The equation 3. goes to 3.3 S x,..., x n S µ x,..., x n x n /µ σ µ, under the same restrictions on the summation and where σ µ β µ i,j µ h µi, j i,j h i, j. Since the nominator and the denominator of β µ contain, respectively, as factors h and h µ for all j µ j, we obtain that 3.4 σ µ i,j h i, j h i, j i,j µ h µi, j h µ i, j, where the products are on all i, j such that j µ j +. This seems to be a simplification from computational point of view. We can rewrite 3.4 in the form 3.5 σ µ i,j µ h i, jh µi, j h i, jh µ i, j i,j /µ h i, j h i, j, where the first product is again on all i, j such that j µ j +. Using the formula a b a! b a b a b + we obtain i,j /µ a + ba + b + a + b +! h i, j h i, j µ n µ n /µ, n µ n i,j µ i,j µ h i, j h i, j h µi, j h µ i, j n n n n µ n n+ n µ n 3 3 n n 2 2 n n µ n 2 n n n µ n µ µ µ 3 n 2 µ n+ n µ n n µ n n µ n 2 n+ n µ n 3 µ 3 n 3 n 2 µ 3+ µ µ 3 2 µ 3 µ2 3 3 µ 3 3 µ µ 2, 2+ 2 µ 2 n n+ n µ n µ2 n n µ n µn n+ n µ n µ2 n n µ n µn n n µ n 3 µ 3 µ 2+ 2 µ 2, µ 2 2 µ 2

27 COMPUTING MULTIVARIATE STATISTICS Counting the Partitions. We order the partitions,..., n in the following way: µ if µ i+ i+,..., µ n n, µ i < i. We want to find the number r of the place of in the sequence of all partitions λ λ,..., λ n, λ N. If p n N is the number of all partitions of N in n parts, then the generating function satisfies p n t N p n Nt N n i t i. It is a standard tric to show for the generating function q n t N p n M t N p nt t, N M which gives the expression q n N for the number of all partitions in n parts with N. In order to find the exact number r of, let us assume that m, m+. Then r,..., m q m N + q m N m + + q m N m m + r m,..., m m because q m N m is the number of partitions µ,..., µ m with µ m Relations between the Coefficients. We shall write σ µ σ, µ σ,..., n, µ,..., µ n. The following relation is obvious follows from 3.4 because + µ + : 3.6 σ +,µ+ σ +,..., n +, µ +,..., µ n + σ,..., n, µ,..., µ n σ µ. The following also follows from 3.4: 3.7 n i σ +,..., n +, µ +,..., µ n +, n i++ i n i+ i+ n i n i +µ i+ n i+µ i σ,..., n, µ,..., µ n,. 4. Raymond Kan s Note 2 The results in this section are due to Raymond Kan. Let > and A be a real symmetric n n matrix. We define d A as the power series expansion of 4. I n ta Note that d J d Atr. A is a normalized version of the top-order Jac polynomial, it is related to C A and A by the following relation 4.2 d J A!! C A.

28 28 PLAMEN KOEV Lemma: Let x be a real scalar. We have d A + xi n F ; n ; A + xi n 2F, m; n ; A + xi n n + d j j Axj, ; n ; A, j e x F x m 2F, m; n ; x A. Proof: The first identity can be proven as follows: I n ta + xi n tx n I n t tx A d A + xi nt tx n d t i A tx d A + xi nt d A + xi nt d A + xi nt i i i d i At i tx n +i d i j At i i n + i + j t j x j j j n + d j j Axj t, i + j. Comparing the coefficient of t on both sides, we prove the first identity. When one of the numerator coefficients in p F q Using the fact that C has only part, we have is /, only the top-order jac polynomials are involved. c for partition that A when has more than one part, and 2 c 4.7 pf q, 2,, p ; β,..., β q ; A 2 p d β β q A. 4 For the case of 2, the first identity is given in Robbins 948, Eq.28.

29 COMPUTING MULTIVARIATE STATISTICS 29 Using this result and the first identity, we can prove the second identity as follows: F ; n ; A + xi d n A + xi n n 4.8 n + d j Axj j n j Γ n Γ jd n j j + j Axj j! Γ n Γ pd n j p + p A xj p j j! x j d p A j! n j p p e x F ; n ; A. There are some potential applications of this identity. For example, suppose A has two distinct eigenvalues λ and λ 2, with multiplicities n and n 2, respectively. Then putting x λ 2, we have 4.9 F ; n ; A e λ2 F ; n ; λ n λ 2 I n e λ2 F ; n ; λ λ 2 by using the fact that d xi n n x /!. This identity allows us to write the result as a hypergeometric function with scalar argument. The third identity can be similarly obtained. 2F, m; n ; A + xi m d n A + xi n n 4. This completes the proof. n + m d j Axj j n j Γ n Γ n j j + jm d j Axj j! Γ n Γ n j p + pm p+jd p A xj p j j! m p d p A m + p j x n j j! p p j m p d p A n x m+p p p x m 2F, m; n ; x A.

30 3 PLAMEN KOEV Let A xb for x, we can write the last identity as 4. 2F, m; n ; B x m 2F, m; n ; xb + xi n. This identity can be used for the purpose of dimension reduction. Suppose B has two distinct eigenvalues λ and λ 2, with multiplicities n and n 2, respectively. Without loss of generality, we assume λ 2. Letting x λ 2 / λ 2, we obtain 4.2 2F, m; n ; B + λ m 2 λ 2 λ 2 m 2 F 2F Acnowledgments, m; n ; λ λ 2 I n λ 2. n, m; n ; λ λ 2 λ 2 I than Iain Johnstone for his indispensable help, guidance, and advice in developing this wor. I also than Brian Sutton for his help in producing the results in Section 7.2. References. Pierre-Antoine Absil, Alan Edelman, and Plamen Koev, On the largest principal angle between random subspaces, Linear Algebra Appl , T. H. Baer and P. J. Forrester, The Calogero-Sutherland model and generalized classical polynomials, Comm. Math. Phys , no., Christopher Bingham, An antipodally symmetric distribution on the sphere, Ann. Statist , R. W. Butler and A. T. A. Wood, Laplace approximations for hypergeometric functions with matrix argument, Ann. Statist. 3 22, no. 4, William R. Chen, Table for upper percentage points of the largest root of a determinantal equation with five roots. 6., Some new tables of the largest root of a matrix in multivariate analysis: A computer approach from 2 to 6, 22, Presented at the 22 American Statistical Association. 7. A. G. Constantine, Some non-central distribution problems in multivariate analysis, Ann. Math. Statist , Ioana Dumitriu, Eigenvalue statistics for the Beta-ensembles, Ph.D. thesis, Massachusetts Institute of Technology, Ioana Dumitriu and Plamen Koev, The distributions of the extreme eigenvalues of beta Jacobi random matrices, SIAM J. Matrix Anal. Appl. 28, 6.. P. J. Forrester, Exact results and universal asymptotics in the Laguerre random matrix ensemble, J. Math. Phys , no. 5, , Log-gases and random matrices, London Mathematical Society Monographs Series, vol. 34, Princeton University Press, Princeton, NJ, Kenneth I. Gross and Donald St. P. Richards, Total positivity, spherical series, and hypergeometric functions of matrix argument, J. Approx. Theory , no. 2, Rameshwar D. Gupta and Donald St. P. Richards, Hypergeometric functions of scalar matrix argument are expressible in terms of classical hypergeometric functions, SIAM J. Math. Anal , no. 4, Alan T. James, Distributions of matrix variates and latent roots derived from normal samples, Ann. Math. Statist , Iain M. Johnstone, On the distribution of the largest eigenvalue in principal components analysis, Ann. Statist. 29 2, no. 2, , Notes on the largest canonical correlation, 24, Wor in progress. 7. Jyoichi Kaneo, Selberg integrals and hypergeometric functions associated with Jac polynomials, SIAM J. Math. Anal , no. 4, Noureddine El Karoui, On the largest eigenvalue of Wishart matrices with identity covariance when n, p and p/n tend to infinity, Plamen Koev and Alan Edelman, The efficient evaluation of the hypergeometric function of a matrix argument, Math. Comp , no. 254, P. R. Krishnaiah and T. C. Chang, On the exact distribution of the smallest root of the Wishart matrix using zonal polynomials, Ann. Inst. Statist. Math ,

31 COMPUTING MULTIVARIATE STATISTICS 3 2. Ian G. Macdonald, Hypergeometric functions I, handwritten notes. 22. Ian G. Macdonald, Symmetric functions and Hall polynomials, Second ed., Oxford University Press, New Yor, R. J. Muirhead, Aspects of multivariate statistical theory, John Wiley & Sons Inc., New Yor, , 25, private communication. 25. A. Yu. Orlov, New solvable matrix integrals, Proceedings of 6th International Worshop on Conformal Field Theory and Integrable Models, vol. 9, 24, pp T. Ratnarajah, R. Vaillancourt, and M. Alvo, Eigenvalues and condition numbers of complex random matrices, SIAM J. Matrix Anal. Appl /5, no. 2, Atle Selberg, Remars on a multiple integral, Nors Mat. Tidssr , Jac W. Silverstein, The smallest eigenvalue of a large-dimensional Wishart matrix, Ann. Probab , no. 4, Richard P. Stanley, Some combinatorial properties of Jac symmetric functions, Adv. Math , no., , Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, Brian Sutton, The stochastic operator approach to random matrix theory, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, 25.