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arXiv:1309.4328v1 [math.PR] 17 Sep 2013
Random Matrices: Theory and Applications
c World Scientific Publishing Company
The Beta-MANOVA Ensemble with General Covariance
Alexander Dubbs
Mathematics, MIT, Massachusetts Ave. and Vassar St.
Cambridge, MA 02139, United States of America,
alex.dubbs@gmail.com
Alan Edelman
Mathematics, MIT, Massachusetts Ave. and Vassar St.
Cambridge, MA 02139, United States of America,
edelman@math.mit.edu
Received September 18, 2013
Revised ?
We find the joint generalized singular value distribution and largest generalized singular
value distributions of the β-MANOVA ensemble with positive diagonal covariance, which
is general. This has been done for the continuous β > 0 case for identity covariance
(in eigenvalue form), and by setting the covariance to I in our model we get another
version. For the diagonal covariance case, it has only been done for β = 1, 2, 4 cases
(real, complex, and quaternion matrix entries). This is in a way the first second-order βensemble, since the sampler for the generalized singular values of the β-MANOVA with
diagonal covariance calls the sampler for the eigenvalues of the β-Wishart with diagonal
covariance of Forrester and Dubbs-Edelman-Koev-Venkataramana. We use a conjecture
of MacDonald proven by Baker and Forrester concerning an integral of a hypergeometric
function and a theorem of Kaneko concerning an integral of Jack polynomials to derive
our generalized singular value distributions. In addition we use many identities from
Forrester’s Log-Gases and Random Matrices. We supply numerical evidence that our
theorems are correct.
Keywords: Finite random matrix theory; Beta-ensembles; MANOVA.
Mathematics Subject Classification 2000: 22E46, 53C35, 57S20
1. Introduction
The first β-ensembles were introduced by Dumitriu and Edelman [1], the β-Hermite
ensemble is defined to be a real
ensemble and the β-Laguerre ensemble. A βrandom matrix with a nonrandom continuous tuning parameter β > 0 such that
when β = 1, 2, 4, the βensemble has the same joint eigenvalue distribution
as the real, complex, or quaternionic
-ensemble. For β not equal to 1, 2, 4 its
eigenvalue distribution interpolates naturally among the β = 1, 2, 4 cases. A βcircular ensemble and four β-Jacobi ensembles shortly followed [2], [3], [4], [5]. The
extreme eigenvalues of the β-Jacobi ensembles were characterized by Dumitriu and
1
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Koev in [6]. More recently, Forrester [7] and Dubbs-Edelman-Koev-Venkataramana
[8] separately introduced a β-Wishart ensemble with diagonal covariance, which
generalizes the β-Laguerre ensemble by adding the covariance term.
This paper introduces the β-MANOVA ensemble with diagonal covariance,
which generalizes the β-Jacobi ensembles by adding the covariance term. When
β = 1 this amounts to finding the distribution of the cosine generalized singular
values of the pair (Y, XΩ), where X is m × n Gaussian, Y is p × n Gaussian, and
Ω is n × n diagonal pds. Note that forcing Ω to be diagonal does not lose any
generality; using orthogonal transformations, were Ω not diagonal we could replace
it with its diagonal matrix of eigenvalues and preserve the model. Our β-MANOVA
ensemble also generalizes the real, complex, and quaternionic MANOVA ensembles
(the last of which has never been studied). [9], [10], and [11] independently solved
the β = 1 identity-covariance case, [12] solved our problem in the β = 1, generalcovariance case, and [13] solved our problem in the β = 2, general-covariance case.
We find the joint eigenvalue distribution of the β-MANOVA ensemble, and generalize Dumitriu and Koev’s results in [6] by finding the distribution of the largest
generalized singular value of the β-MANOVA ensemble. We also set the covariance
to the identity to add a fourth β-Jacobi ensemble to the literature in Theorem 3.1.
Our β-MANOVA ensemble is unique in that it is not built on a recursive procedure,
rather it is sampled by calling the sampler for the β-Wishart ensemble.
Generalizations of our results exist in the β = 1, 2 cases by adding a mean
matrix to one of the Wishart-distributed parameters. The β = 1 case is from [12,
p. 1279] and the β = 2 case is from [13, p. 490].
The sampler for the β-Wishart ensemble of Forrester [7] and Dubbs-EdelmanKoev-Venkataramana [8] is the following algorithm:
Beta-Wishart (Recursive) Model Pseudocode
Function Σ := BetaWishart(m, n, β, D)
if n = 1 then
1/2
Σ := χmβ D1,1
else
Z1:n−1,1:n−1 := BetaWishart(m, n − 1, β, D1:n−1,1:n−1 )
Zn,1:n−1 := [0, . . . , 0]
1/2
1/2
Z1:n−1,n := [χβ Dn,n ; . . . ; χβ Dn,n ]
1/2
Zn,n := χ(m−n+1)β Dn,n
Σ := diag(svd(Z))
end if
The elements of Σ are distributed according to the following theorem of [7] and
[8]:
Proposition 1.1. The distribution of the singular values diag(Σ) = (σ1 , . . . , σn ),
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σ1 > σ2 > · · · > σn , generated by the above algorithm is equal to:
n
2n det(D)−mβ/2 Y (m−n+1)β−1 2 β (β)
1 2 −1
− Σ ,D
σi
∆ (σ) 0 F0
dσ,
(β)
2
Km,n
i=1
(β)
where 0 F0(β) and Km,n are defined in the upcoming section, Preliminaries.
To get the generalized singular values of the β-MANOVA ensemble with
general covariance, in diagonal C, we use the following algorithm which calls
BetaWishart(m, n, β, D). Let Ω be an n × n diagonal matrix.
Beta-MANOVA Model Pseudocode
Function C := BetaMANOVA(m, n, p, β, Ω)
Λ := BetaWishart(m, n, β, Ω2 )
M := BetaWishart(p, n, β, Λ−1 )−1
1
C := (M + I)− 2
Our main theorem is the joint distribution of the elements of C,
Theorem 1.1. The distribution of the generalized singular values diag(C) =
(c1 , . . . , cn ), c1 > c2 > · · · > cn , generated by the above algorithm for m, p ≥ n
is equal to:
(β)
2n Km+p,n
(β)
(β)
Km,n Kp,n
det(Ω)pβ
n
Y
(p−n+1)β−1
ci
n
Y
(1 − c2i )−
i=1
i=1
× 1 F0(β)
p+n−1
β−1
2
Y
|c2i − c2j |β
i<j
m+p
· β; ; C 2 (C 2 − I)−1 , Ω2 dc.
2
(β)
where 1 F0(β) and Km,n are defined in the upcoming section, Preliminaries.
We also find the distributions of the largest generalized singular value in certain
cases:
Theorem 1.2. If t = (m − n + 1)β/2 − 1 ∈ Z≥0 ,
P (c1 < x) = det(x2 Ω2 ((1 − x2 )I + x2 Ω2 )−1 )
×
nt
X
X
k=0 κ`k,κ1 ≤t
pβ
2
1
(β)
(pβ/2)(β)
(1 − x2 )((1 − x2 )I + x2 Ω2 )−1 , (1.1)
κ Cκ
k!
(β)
(β)
where the Jack function Cκ and Pochhammer symbol (·)κ
upcoming section, Preliminaries.
are defined in the
These expressions can be computed by Edelman and Koev’s software, mhg, [14].
It is actually intuitive that BetaMANOVA(m, n, p, β, Ω) should generalize the
real, complex, and quaternionic MANOVA ensembles with diagonal covariance using the “method of ghosts.” The method of ghosts was first used implicitly to derive
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β-ensembles for the Laguerre and Hermite cases in [1], was stated precisely by Edelman in [15], and was expanded on in [8]. To use the method of ghosts, assume a
given ensemble is full of β-dimensional Gaussians, which generalize real, complex,
and quaternionic Gaussians and have some of the same properties: they can be
left invariant or made into a χβ ’s under rotation by a real orthogonal or “ghost
orthogonal” matrix. Then apply enough orthogonal transformations and/or ghost
orthogonal transformations to the ghost matrix to make it all real.
In the β-MANOVA case, let X be m × n real, complex, quaternion, or
ghost normal, Y be p × n real complex, quaternion, or ghost normal, and let Ω
be n × n diagonal real pds. Let ΩX ∗ XΩ have eigendecomposition U ΛU ∗ , and
ΩX ∗ XΩ(Y ∗ Y )−1 have eigendecomposition V M V ∗ . We want to draw M so we
1
can draw C = gsvd(cosine) (Y, XΩ) = (M + I)− 2 . Let ∼ mean “having the same
eigenvalues.”
ΩX ∗ XΩ(Y ∗ Y )−1 ∼ ΛU ∗ (Y ∗ Y )−1 U ∼ Λ((U ∗ Y ∗ )(Y U ))−1 ∼ Λ(Y ∗ Y )−1 ,
which we can draw the eigenvalues M of using BetaWishart(p, n, β, Λ−1 )−1 . Since
Λ can be drawn using BetaWishart(m, n, β, Ω2 ), this completes the algorithm for
BetaMANOVA(m, n, p, β, Ω) and proves Theorem 1.1 in the β = 1, 2, 4 cases.
The following section contains preliminaries to the proofs of Theorems 1.1 and
1.2 in the general β case. Most important are several propositions concerning Jack
polynomials and Hypergeometric Functions. Proposition 2.1 was conjectured by
Macdonald [16] and proved by Baker and Forrester [17], Proposition 2.3 is due to
Kaneko, in a paper containing many results on Selberg-type integrals [18], and the
other propositions are found in [19, pp. 593-596].
2. Preliminaries
Definition 2.1. We define the generalized Gamma function to be
n(n−1)β/4
Γ(β)
n (c) = π
n
Y
Γ(c − (i − 1)β/2)
i=1
for <(c) > (n − 1)β/2.
Definition 2.2.
(β)
Km,n
=
(β)
2mnβ/2
π n(n−1)β/2
·
Γn (mβ/2)Γβn (nβ/2)
.
Γ(β/2)n
Definition 2.3.
∆(λ) =
Y
(λi − λj ).
i<j
If X is a diagonal matrix,
∆(X) =
Y
i<j
|Xi,i − Xj,j |.
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As in Dumitriu, Edelman, and Shuman, if κ ` k, κ = (κ1 , κ2 , . . . , κn ) is
nonnegative, ordered non-increasingly, and it sums to k. Let α = 2/β. Let
Pn
ρα
κ =
i=1 κi (κi − 1 − (2/α)(i − 1)). We define l(κ) to be the number of nonzero
elements of κ. We say that µ ≤ κ in “lexicographic ordering” if for the largest
integer j such that µi = κi for all i < j, we have µj ≤ κj .
Definition 2.4. As in Dumitriu, Edelman and Shuman, [20] we define the Jack
(β)
polynomial of a matrix argument, Cκ (X), as follows: Let x1 , . . . , xn be the eigen(β)
values of X. Cκ (X) is the only homogeneous polynomial eigenfunction of the
Laplace-Beltrami-type operator:
Dn∗ =
n
X
x2i
i=1
∂2
+β·
∂x2i
X
1≤i6=j≤n
x2i
∂
·
,
xi − xj ∂xi
ρα
k
with eigenvalue
+ k(n − 1), having highest order monomial basis function in
lexicographic ordering (see Dumitriu, Edelman, Shuman, Section 2.4) corresponding
to κ. In addition,
X
Cκ(β) (X) = trace(X)k .
κ`k,l(κ)≤n
Definition 2.5. We define the generalized Pochhammer symbol to be, for a partition κ = (κ1 , . . . , κl )
κi l Y
Y
i−1
(β)
β+j−1 .
(a)κ =
a−
2
i=1 j=1
Definition 2.6. As in Koev and Edelman [14], we define the hypergeometric function p Fq(β) to be
(β)
p Fq (a; b; X, Y ) =
∞
X
(a1 )βκ · · · (ap )βκ
X
(β)
k=0 κ`k,l(κ)≤n
(β)
(b1 )κ · · · (bq )κ
(β)
(β)
·
Cκ (X)Cκ (Y )
(β)
.
k!Cκ (I)
The best software available to compute this function numerically is described in
Koev and Edelman, mhg, [14]. p Fq(β) (a; b; X) = p Fq(β) (a; b; X, I).
We will also need two theorems from the literature about integrals of Jack
polynomials and hypergeometric functions.
Conjectured by MacDonald [16], proved by Baker and Forrester [17] with the
wrong constant, correct constant found using Special Functions [21, p. 406] (Corollary 8.2.2):
Proposition 2.1. Let X be a diagonal matrix.
(β)
(β) (β)
−1
c(β)
)
n Γn (a + (n − 1)β/2 + 1)(a + (n − 1)β/2 + 1)κ Cκ (Y
Z
(β)
a (β)
β
= |Y |a+(n−1)β/2+1
0 F0 (−X, Y )|X| Cκ (X)|∆(X)| dX,
X>0
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(β)
where cn = π −n(n−1)β/4 n!Γ(β/2)n
Qn
i=1
Γ(iβ/2).
From [19, p.593],
Proposition 2.2. If X < I is diagonal,
(β)
1 F0 (a; ; X)
= |I − X|−a .
Kaneko, Corollary 2 [18]:
Proposition 2.3. Let κ = (κ1 , . . . , κn ) be nonincreasing and X be diagonal. Let
a, b > −1 and β > 0.
Z
n
Y
a
(β)
β
Cκ (X)∆(X)
xi (1 − xi )b dX
0<X<I
i=1
= Cκ(β) (I) ·
n
Y
Γ(iβ/2 + 1)Γ(κi + a + (β/2)(n − i) + 1)Γ(b + (β/2)(n − i) + 1)
Γ((β/2) + 1)Γ(κi + a + b + (β/2)(2n − i − 1) + 2)
i=1
.
From [19, p. 595],
Proposition 2.4. Let X be diagonal,
(β)
2 F1 (a, b; c; X)
= 2 F1(β) (c − a, b; c; −X(I − X)−1 )|I − X|−b
= 2 F1(β) (c − a, c − b; c; X)|I − X|c−a−b .
From [19, p. 596],
Proposition 2.5. If X is n × n diagonal and a or b is a nonpositive integer,
(β)
2 F1 (a, b; c; X)
= 2 F1(β) (a, b; c; I)2 F1(β) (a, b; a + b + 1 + (n − 1)β/2 − c; I − X).
From [19, p. 594],
Proposition 2.6.
(β)
(β)
2 F1 (a, b; c; I)
=
(β)
Γn (c)Γn (c − a − b)
(β)
(β)
Γn (c − a)Γn (c − b)
.
3. Main Theorems
Proof of Theorem 1.1. Let m, p ≥ n. We will draw M by drawing Λ ∼
P (Λ) = BetaWishart(m, n, β, Ω2 ), and compute M Rby drawing M P (M |Λ) =
BetaWishart(p, n, β, Λ−1 )−1 . The distribution of M is P (M |Λ)P (Λ)dλ. Then we
1
will compute C by C = (M + I)− 2 . We use the convention that eigenvalues and
generalized singular values are unordered. By the [8] BetaWishart described in the
introduction, we sample the diagonal Λ from
P (Λ) =
n
det(Ω)−mβ Y
(β)
n!Km,n
i=1
m−n+1
β−1
2
λi
Y
i<j
β
|λi − λj | 0 F0
(β)
1
−2
− Λ, Ω
dλ,
2
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(β)
Km,n
=
7
(β)
2mnβ/2
·
π n(n−1)β/2
Γn (mβ/2)Γβn (nβ/2)
.
Γ(β/2)n
Likewise, by inverting the answer to the [8] BetaWishart described in the introduction, we can sample diagonal M from
P (M |Λ) =
n
det(Λ)pβ/2 Y
(β)
n!Kp,n
− p−n+1 β−1
µi 2
i=1
Y
|µ−1
i
(β)
β
µ−1
j | 0 F0
−
i<j
1 −1
− M , Λ dµ.
2
To get P (M ) we need to compute
n
det(Ω)−mβ Y
(β)
(β)
n!2 Km,n Kp,n i=1
Z
det(Λ)pβ/2
×
λ1 ,...,λn ≥0
n
Y
− p−n+1
β−1
2
µi
Y
−1 β
|µ−1
i − µj | dµ
i<j
m−n+1
β−1
2
λi
i=1
Y
|λi −λj |β 0 F0 (β)
i<j
1 −2
Ω , −Λ
2
1 −1
(β)
F
−
M
,
Λ
dλ.
0 0
2
Expanding the hypergeometric function, this is
n
det(Ω)−mβ Y
(β)
(β)
n!2 Km,n Kp,n

Z
×
− p−n+1 β−1
µi 2
i=1
n
Y
λ1 ,...,λn ≥0 i=1
Y
|µ−1
i
−
β
µ−1
j |
i<j
m−n+p+1
β−1
2
λi
Y
(β)
∞ X
X
Cκ − 21 M −1
dµ
(β)
k!Cκ (I)
k=0 κ`k
|λi − λj |β 0 F0 (β)
i<j

1 −2
Ω , −Λ Cκ(β) (Λ) dλ .
2
Using Proposition 2.1,
n
det(Ω)−mβ Y
(β)
(β)
n!2 Km,n Kp,n
− p−n+1 β−1
µi 2
i=1
(β)
"
Y
|µ−1
i
−
β
µ−1
j |
i<j
n!Km+p,n
× (m+p)nβ/2
2
m+p
β
2
(β)
∞ X
X
Cκ − 21 M −1
(β)
det
κ
1 −2
Ω
2
− m+p
2 β
dµ
k!Cκ (I)
k=0 κ`k
(β)
#
Cκ(β)
2Ω
2
.
Cleaning things up,
(β)
n
det(Ω)pβ Km+p,n Y
(β)
(β)
n!Km,n Kp,n
− p−n+1 β−1
µi 2
i=1
Y
|µ−1
i
−
β
µ−1
j |
i<j
"
×
m+p
·β
2
(β)
∞ X
X
Cκ − 12 M −1
(β)
k=0 κ`k
(β)
Cκ(β)
dµ
k!Cκ (I)
#
2Ω2 .
κ
By the definition of the hypergeometric function, this is
(β)
n
Y
det(Ω)pβ Km+p,n Y
m+p
β−1
− p−n+1
(β)
−1
−1 β
−1
2
2
µi
|µi −µj | ·1 F0
β; ; −M , Ω dµ.
(β)
(β)
2
n!Km,n Kp,n i=1
i<j
(3.1)
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Converting to cosine form, C = diag(c1 , . . . , cn ) = (M + I)−1/2 , this is
(β)
2n Km+p,n
det(Ω)pβ
(β)
(β)
n!Km,n Kp,n
n
Y
(p−n+1)β−1
ci
i=1
n
Y
(1 − c2i )−
p+n−1
β−1
2
× 1 F0 (β)
|c2i − c2j |β
i<j
i=1
Y
m+p
· β; ; C 2 (C 2 − I)−1 , Ω2 dc. (3.2)
2
Theorem 3.1. If we set Ω = I and ui = c2i , (u1 , . . . , un ) obey the standard βJacobi density of [2], [3], [4], and [5].
(β)
n
Y
Km+p,n
(β)
(β)
n!Km,n Kp,n i=1
n
Y
p−n+1
β−1
2
ui
(1 − ui )
m−n+1
β−1
2
i=1
Y
|ui − uj |β du.
(3.3)
i<j
Proof. Proposition 2.2 works from the statement of Theorem 1.1 because C 2 (C 2 −
I)−1 < I (we know that M > 0 from how it is sampled, so 0 < C 2 = (M +I)−1 < I,
likewise C 2 − I).
(β)
n
2n Km+p,n Y
(β)
(β)
n!Km,n Kp,n
(p−n+1)β−1
ci
n
Y
(1 − c2i )−
p+n−1
β−1
2
i=1
i=1
Y
|c2i − c2j |β
i<j
× det(I − C 2 (C 2 − I)−1 )−
m+p
2 β
dc,
or equivalently
(β)
n
2n Km+p,n Y
(β)
(β)
n!Km,n Kp,n
(p−n+1)β−1
ci
n
Y
(1 − c2i )
m−n+1
β−1
2
i=1
i=1
Y
|c2i − c2j |β dc.
i<j
If we substitute ui = c2i , by the change-of-variables theorem we get the desired
result.
Proof of Theorem 1.2. Let H = diag(η1 , . . . , ηn ) = M −1 . Changing variables from
(3.1) we get
(β)
n
det(Ω)pβ Km+p,n Y
(β)
(β)
n!Km,n Kp,n
p−n+1
β−1
2
ηi
i=1
Y
|ηi − ηj |β · 1 F0 (β) ((m + p)β/2; ; H, −Ω2 )dη.
i<j
Taking the maximum eigenvalue, following mvs.pdf,
(β)
P (H < xI) =
Z
×
det(Ω)pβ Km+p,n
(β)
(β)
n!Km,n Kp,n
n
Y
Y
p−n+1
β−1
ηi 2
|ηi − ηj |β · 1 F0 (β) ((m + p)β/2; ; H, −Ω2 )dη,
H<xI i=1
i<j
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Letting N = diag(ν1 , . . . , νn ) = H/x, changing variables again we get
(β)
P (H < xI) =
×
det(Ω)pβ Km+p,n
·x
npβ
2
(β)
(β)
n!Km,n Kp,n
Z
n
Y
Y
p−n+1
β−1
|νi
νi 2
N <I i=1
i<j
− νj |β · 1 F0 (β) ((m + p)β/2; ; N, −xΩ2 )dν,
Expanding the hypergeometric function we get
(β)
P (H < xI) =
det(Ω)pβ Km+p,n
(β)
(β)
n!Km,n Kp,n

Z
×
·x
npβ
2
·
∞ X
(β) (β)
X
((m + p)β/2)κ Cκ (−xΩ2 )
(β)
k!Cκ (I)
k=0 κ`k
n
Y

p−n+1
β−1
2
νi
N <I i=1
Y
|νi − νj |β · Cκ(β) (N )dν  . (3.4)
i<j
Using Proposition 2.3,
n
Y
Z
p−n+1
β−1
2
νi
N <I i=1
=
=
(β)
Cκ (I)
Γ(β/2 + 1)n
Y
|νi − νj |β · Cκ(β) (N )dν
i<j
·
n
Y
Γ(iβ/2 + 1)Γ(κi + (β/2)(p + 1 − i))Γ((β/2)(n − i) + 1)
Γ(κi + (β/2)(p + n − i) + 1)
i=1
n
(β)
(β)
(β)
Cκ (I)Γn ((nβ/2) + 1)Γn (((n − 1)β/2) + 1) Y Γ(κi + (β/2)(p + 1 − i))
·
n(n−1)β
Γ(κi + (β/2)(p + n − i) + 1)
π 2 Γ(β/2 + 1)n
i=1
Now
n
Y
i=1
Γ(κi + (β/2)(p + 1 − i)) =
n
Y
Γ((β/2)(p + 1 − i))
i=1
= π−
κi
Y
((β/2)(p + 1 − i) + j − 1)
j=1
n(n−1)
β
4
Γn(β) (pβ/2)
κi
Y
((β/2)(p + 1 − i) + j − 1)
j=1
n(n−1)
= π − 4 β Γn(β) (pβ/2)(pβ/2)(β)
κ
κi
n
n
Y
Y
Y
Γ(κi + (β/2)(p + n − i) + 1) =
Γ((β/2)(p + n − i) + 1)
((β/2)(p + n − i) + j)
i=1
i=1
= π−
j=1
n(n−1)
β
4
Γn(β) ((p + n − 1)β/2 + 1)
κi
Y
((β/2)(p + n − i) + j)
j=1
= π−
n(n−1)
β
4
Γn(β) ((p + n − 1)β/2 + 1)((p + n − 1)β/2 + 1)(β)
κ .
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Therefore,
Z
n
Y
p−n+1
β−1
2
νi
N <I i=1
=
|νi − νj |β · Cκ(β) (N )dν
i<j
(β)
(β)
Cκ (I)Γn ((nβ/2)
π
Y
n(n−1)β
2
(β)
(β)
(β)
+ 1)Γn (((n − 1)β/2) + 1)Γn (pβ/2)
(β)
Γ(β/2 + 1)n Γn ((p + n − 1)β/2 + 1)
·
(pβ/2)κ
(β)
((p + n − 1)β/2 + 1)κ
Using (3.4) and the definition of the hypergeometric function we get
(β)
(β)
(β)
det(Ω)pβ Km+p,n Γ(β)
n ((nβ/2) + 1)Γn (((n − 1)β/2) + 1)Γn (pβ/2)
·
n(n−1)β
(β)
(β)
(β)
n!Km,n Kp,n
π 2 Γ(β/2 + 1)n Γn ((p + n − 1)β/2 + 1)
npβ
m+p p p+n−1
(β)
2
2
×x
· 2 F1
β, β;
β + 1; −xΩ .
2
2
2
Rewriting the constant we get
P (H < xI) =
(β)
(β)
(β)
Γ(β/2)n Γn ((m + p)β/2) Γn ((nβ/2) + 1)Γn (((n − 1)β/2) + 1)
·
.
(β)
(β)
(β)
n!Γn (mβ/2)Γn (nβ/2)
Γ(β/2 + 1)n Γn ((p + n − 1)β/2 + 1)
Commuting some terms gives
!
(β)
(β)
(β)
Γ(β/2)n Γn ((nβ/2) + 1)
Γn ((m + p)β/2)Γn (((n − 1)β/2) + 1)
.
·
(β)
(β)
(β)
n!Γ(β/2 + 1)n Γn (nβ/2)
Γn (mβ/2)Γn ((p + n − 1)β/2 + 1)
The left fraction in parentheses is
Qn
Qn
Γ((nβ/2) + 1 − (i − 1)β/2)
Γ((iβ/2) + 1)
1
1
i=1
Qn
Qn
· Qn
= Qn
· i=1
= 1.
i=1 (iβ/2)
i=1 Γ((nβ/2) − (i − 1)β/2)
i=1 (iβ/2)
i=1 Γ(iβ/2)
Hence
(β)
(β)
P (H < xI) = det(Ω)pβ ·
×x
Γn ((m + p)β/2)Γn (((n − 1)β/2) + 1)
(β)
(β)
npβ
2
Γn (mβ/2)Γn ((n + p − 1)β/2 + 1)
m+p p p+n−1
· 2 F1 (β)
β, β;
β + 1; −xΩ2 . (3.5)
2
2
2
Now H = M −1 and C = (M + I)−1/2 , so equivalently,
(β)
P (C < xI) = det(Ω)pβ ·
×
x2
1 − x2
npβ
2
(β)
Γn ((m + p)β/2)Γn (((n − 1)β/2) + 1)
(β)
(β)
Γn (mβ/2)Γn ((n + p − 1)β/2 + 1)
m+p p p+n−1
x2
(β)
2
· 2 F1
β, β;
β + 1; −
· Ω . (3.6)
2
2
2
1 − x2
Remark. Using U = diag(u1 , . . . , un ) = C 2 and setting Ω = I this is
(β)
P (U < xI) =
(β)
Γn ((m + p)β/2)Γn (((n − 1)β/2) + 1)
(β)
(β)
Γn (mβ/2)Γn ((n + p − 1)β/2 + 1)
npβ
2
x
m+p p p+n−1
x
(β)
×
· 2 F1
β, β;
β + 1; −
·I ,
1−x
2
2
2
1−x
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11
so by using Proposition 2.4, this is
(β)
P (U < xI) =
(β)
Γn ((m + p)β/2)Γn (((n − 1)β/2) + 1)
(β)
(β)
Γn (mβ/2)Γn ((n + p − 1)β/2 + 1)
npβ
n−m−1
p p+n−1
(β)
2
· 2 F1
β + 1, β;
β + 1; xI ,
×x
2
2
2
which is familiar from Dumitriu and Koev [6].
Now back to the proof of Theorem 1.2. If we use Proposition 2.4 on (3.6) we get
P (C < xI) = det(x2 Ω2 ((1−x2 )I+x2 Ω2 )−1 )
× 2 F1 (β)
pβ
2
(β)
(β)
·
Γn ((m + p)β/2)Γn (((n − 1)β/2) + 1)
(β)
(β)
Γn (mβ/2)Γn ((n + p − 1)β/2 + 1)
n−m−1
p p+n−1
2 2
2
2 2 −1
β + 1, β;
β + 1; x Ω ((1 − x )I + x Ω )
.
2
2
2
(3.7)
Using the approach of Dumitriu and Koev [6], let t = (m−n+1)β/2−1 in Z≥0 . We
can prove that the series truncates: Looking at (3.7), the hypergeometric function
involves the term
κi n Y
Y
i−1
(β)
(−t)κ =
−t −
β+j−1 ,
2
i=1 j=1
which is zero when i = 1 and j − 1 = t, so the series truncates when any κi has
κi − 1 ≥ t, or just κ1 = t + 1. This must happen if k > nt. Thus (3.7) is just a finite
polynomial,
P (C < xI) = det(x2 Ω2 ((1−x2 )I+x2 Ω2 )−1 )
×
nt
X
(β)
X
pβ
2
Γn ((m + p)β/2)Γn (((n − 1)β/2) + 1)
(β)
(β)
Γn (mβ/2)Γn ((n + p − 1)β/2 + 1)
(β)
((n − m − 1)β/2 + 1)κ (pβ/2)κ
(β)
k=1 κ`k,κ1 ≤t
(β)
(β)
·
((p + n − 1)β/2 + 1)κ
·Cκ(β) (x2 Ω2 ((1−x2 )I +x2 Ω2 )−1 ).
(3.8)
Let Z be a positive-definite diagonal matrix, and a real with || > 0. Define
f (Z, ) =
nt
X
(β)
X
k=1 κ`k,κ1 ≤t
(β)
((n − m − 1)β/2 + 1)κ (pβ/2)κ
((p + n − 1)β/2 + 1 +
(β)
)κ
· Cκ(β) (Z).
Using Proposition 2.5,
n−m−1
p p+n−1
(β)
f (Z, ) = 2 F1
β + 1, β;
β + 1 + ; I
2
2
2
n−m−1
p n−m−1
(β)
× 2 F1
β + 1, β;
β + 1 − ; I − Z
2
2
2
(3.9)
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Using the definition of the hypergeometric function and the fact that the series
must truncate,
n−m−1
p p+n−1
β + 1, β;
β + 1 + ; I
2
2
2
nt
X X ((n − m − 1)β/2 + 1)κ(β) (pβ/2)(β)
κ
×
Cκ(β) (I − Z)
(β)
((n
−
m
−
1)β/2
+
1
−
)
κ
k=1 κ`k,κ1 ≤t
f (Z, ) = 2 F1 (β)
(3.10)
Now the limit is obvious
p p+n−1
n−m−1
f (Z, 0) = 2 F1 (β)
β + 1, β;
β + 1; I
2
2
2
nt
X X
(β)
(pβ/2)(β)
×
κ Cκ (I − Z). (3.11)
k=1 κ`k,κ1 ≤t
Plugging this expression into (3.8)
P (C < xI) = det(x2 Ω2 ((1−x2 )I+x2 Ω2 )−1 )
pβ
2
(β)
·
(β)
Γn ((m + p)β/2)Γn (((n − 1)β/2) + 1)
(β)
(β)
Γn (mβ/2)Γn ((n + p − 1)β/2 + 1)
n−m−1
p p+n−1
× 2 F1 (β)
β + 1, β;
β + 1; I
2
2
2
nt
X X
(β)
2
2
2 2 −1
×
(pβ/2)(β)
).
κ Cκ ((1 − x )((1 − x )I + x Ω )
k=1 κ`k,κ1 ≤t
Cancelling via Proposition 2.6 gives
P (C < xI) = det(x2 Ω2 ((1 − x2 )I + x2 Ω2 )−1 )
×
nt
X
X
k=0 κ`k,κ1 ≤t
pβ
2
1
(β)
(pβ/2)(β)
(1 − x2 )((1 − x2 )I + x2 Ω2 )−1 . (3.12)
κ Cκ
k!
4. Numerical Evidence
The plots below are empirical cdf’s of the greatest generalized singular value as
sampled by the BetaMANOVA pseudocode in the introduction (the blue lines)
against the Theorem 1.2 formula for them as calculated by mhg (red x’s).
Acknowledgments
We acknowledge the support of the National Science Foundation through grants
SOLAR Grant No. 1035400, DMS-1035400, and DMS-1016086. Alexander Dubbs
was funded by the NSF GRFP.
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13
CDF of Greatest Generalized Singular Value
1
0.9
0.8
Cumulative Probability
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.4
0.5
0.6
0.7
0.8
Greatest Generalized Singular Value
0.9
1
Fig. 1. Empirical vs. analytic when m = 7, n = 4, p = 5, β = 2.5, and Ω = diag(1, 2, 2.5, 2.7).
CDF of Greatest Generalized Singular Value
1
0.9
0.8
Cumulative Probability
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.4
0.5
0.6
0.7
0.8
Greatest Generalized Singular Value
0.9
1
Fig. 2. Empirical vs. analytic when m = 9, n = 4, p = 6, β = 3, and Ω = diag(1, 2, 2.5, 2.7).
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