Maximal Invariants Over Symmetric Cones

0.15 MB·English

by  Emanuel Ben-David

Maximal Invariants Over Symmetric Cones
EmanuelBen-David
StanfordUniversity
2 August2010
1
0
2
Abstract
n
a
In this paper we consider some hypothesistests within a family of Wishart distributions,
J
where both the sample space and the parameter space are symmetric cones. For such testing
2
problems,wefirstderivethejointdensityoftheorderedeigenvaluesofthegeneralizedWishart

] distributionandproposeateststatisticanalogtothatofclassicalmultivariatestatisticsfortest-
T inghomoscedasticityofcovariancematrix. InthisgeneralizationofBartlett’stestforequality
S ofvariancestohypothesesofreal,complex,quaternion,Lorentzandoctoniontypesofcovari-
.
h ancestructures.
t
a
m
1 Introduction
[

1 Consider astatistical modelconsisting ofasamplespace Xandunknown probability measures P ,
θ
v
with θ in the parameter space Θ. In this setting, the statistical inference about the model is often
4
1 concentrated on parameter estimation and hypothesis testing. In the latter, we typically test the
5 hypothesis H : θ ∈ Θ ⊂ Θvs. the hypothesis H : θ ∈ Θ\Θ . Systematically, this meansto finda
0 0 0
0
suitable test statistic t from Xtoameasurable space Y,and the distribution of tP , the transformed
. θ
1
measurewithrespectto P undert.
0 θ
ForaGaussian model, wherethe sample space isRn,the probability measures aremultivariate
2
1 normal distribution N (0,Σ)and the parameter space isPD (R),the cone ofn×npositive definite
n n
v: matricesoverR,someclassical examplesofsuchhypotheses arethesphericity hypothesis:
i
X
H :Σ = σ2I vs. H : Σ , σ2I , (1.1)
0 n n
r
a
forsomeσ > 0;thecomplexstructure hypothesis
A −B
H :Σ = ∈ PD (C) vs. Σ ∈ PD (R), (1.2)
0 B A ! k 2k
forsomerealk×kmatrices A,B,with2k = n;andthequaternion structure hypothesis
A −B −C −D
B A −D C
H : Σ =   ∈ PD (H) vs. H : Σ ∈ PD (R), (1.3)
0 CD −DC AB −AB m 4m
forsomerealm×mmatrices A,B,C,D,with4m = n. Incidentally, ineachofthesehypotheses the
parameter space in the full model is PD (R), the parameter space in the submodel is a subcone of
n
1
PD (R),andthehypothesisisinvariantunderanactionofasubgroupoftheorthogonallineargroup
n
ofRn,O(n,R).
Moregenerally, supposeΩisasymmetricconeand
M = {P ∈ P(X): σ ∈ Ω} (1.4)
σ
isastatistical modelwithparameterspaceΩ,and
M = {P ∈ P(X): σ ∈ Ω } (1.5)
0 σ 0
is asubmodel of M, where the parameter set Ω is a(symmetric) subcone of Ω. In this set up, we
0
wishtotestthehypothesis
H : σ ∈ Ω vs. H : σ ∈ Ω\Ω . (1.6)
0 0 0
In the presence of a group action, testing the hypothesis (1.6) leads one to the study of a maximal
invariant withrespect toagroup[13,19,3,23].
The maximal invariants that arise in testing problems like those stated in (1.1), (1.2) and (1.3)
are indeed functions of the eigenvalues of the sample covariance matrix [3, 1, 13, 19]. Therefore,
in most situations, the problem is reduced to finding the distribution of the ordered eigenvalues of
a distribution (often Wishart) over the cone of positive definite matrices over the real, complex, or
quaternion fields.
In this respect, studies on the distribution of the ordered eigenvalues of a random matrix with
valuesinaclassicalsymmetricconeoverlapwithsimilarstudiesinRandomMatrixTheory(RMT)
[7,8,18]. Therefore, applications ofsuchstudiesarenotmerelylimitedtothetestingproblemswe
mentioned. Other applications of the distribution of the ordered eigenvalues of a random matrix,
specially a Wishart matrix, can be found in physics, e.g. [18], principle component analysis, e.g.
[19,14],andsignalprocessing, e.g. [21,20,22].
Inthispaperwediscussthestatisticalanalysisofmodelsoftheform(1.4). Therearemanymod-
elsthatfallinthiscategory,suchastheWishart,invertedWishart,betaandhyperbolicdistributions.
In our analysis, we extensively exploit the algebraic and geometric structures of symmetric cones,
described by Jordan algebras and Riemannian symmetric spaces, respectively (see [12, 9, 11]). In
thestatisticalinferenceofthesemodelstwoproblemsofinterestaretheestimationoftheparameter
σ,andtestingahypothesis oftheform(1.6). Herewefocusmainlyonthelatter.
In approaching these problems, webegin inSection 2with giving ashort review ofsymmetric
cones and their analysis based on [9]. Essential parts of this section are the classification of irre-
ducible symmetric cones, or equivalently, simple Jordan algebras, and Pierce decomposition. In
Section 3 we study special functions over symmetric cones. Our main goal in this section is to
define zonal polynomials over a symmetric cone. Sections 4, 5, and 6 present our contribution to
the theory of maximal invariants over symmetric cones, introducing a new family of non-central
Wishart distributions oversymmetric cones, deriving the joint distribution ofthe ordered eigenval-
uesofsuchdistributions, andproposing ananalog ofBartlett’s testfortestinghomoscedasticity of,
roughly speaking, thecovariance matrix.
2
2 The structure of symmetric cones
Inthissectionweprovidethereaderwithacondensereviewofthestructuralanalysisofsymmetric
coneswhichisessentialinunderstandingthegeneralizednotions,suchashypergeometricfunctions,
zonal polynomials and the Wishart distribution, we discuss later. More thorough presentation of
thesetopicscanbefoundin[9]andelsewhereinthisvolume.
2.1 Symmetric cones
Let (J,h·,·i) be a Euclidean vector space and letGL(J) be the set of regular linear transformations
on J. A convex cone Ω ⊂ J is a set satisfying αx+βy ∈ Ω for any α,β > 0 and x,y ∈ Ω. Let Ω
denote the closure of Ω in J. The convex cone Ω is proper if Ω∩−Ω = {0}, full if Ω−Ω = J. A
homogeneous cone Ω is a full proper open convex cone with this property that the automorphism
groupofΩ,definedbyG(Ω)= {g ∈ GL(J) : gΩ = Ω},andconsequently theconnected component
of the identity in G(Ω), denoted by G, act transitively on Ω. The homogeneous cone Ω is called
symmetric if it is homogeneous and self-dual, i.e., Ω∗ = {y ∈ V : hx|yi > 0,∀x ∈ Ω\{0}} = Ω.
Everysymmetricconeisthefiniteproductofirreduciblesymmetriccones,whereasymmetriccone
isirreducible ifitisnot thedirect product oftwoormoresymmetric cones. Afamiliar example of
anirreducible symmetricconeisPD (R).
r
A well-known property of a homogeneous cone Ω is that the stabilizer of each x ∈ Ω,G , is a
x
maximal compact subgroup of G. For a symmetric cone Ω, if we set K = G ∩O(J), where O(J)
is the group of orthogonal transformation of J, then K is the stabilizer of an element of Ω and,
consequently, amaximalsubgroup ofG.
AnotherinterestingpropertyofahomogeneousconeΩisthatifwedefineϕ(x) = exp(−hx|yi)dy
Ω∗
for each x ∈ Ω, then ϕ(x)dx isaG-invariant measure on Ω. Thisfollows from thRe factthat for any
g ∈Gwehaveϕ(gx) = |detg|−1ϕ(x).
2.2 Euclidean Jordanalgebras
A Jordan algebra J over Ris a real vector space equipped with amultiplication satisfying xy = yx
and x(x2y) = x2(xy) for any x, y in J. An associative bilinear form B on J is a bilinear form
satisfying
B(xy,z)= B(x,yz) forany x,y,z ∈ J.
AJordanalgebrawithnonon-trivialidealiscalledsimple. TypicalexamplesofJordanalgebrasare
so-called special Jordan algebras. If Aisanassociative algebra, then thespecial Jordan algebra A+
isthevectorspace Aequipped withtheJordanproduct x◦y= (xy+yx)/2. Inparticular M (R),the
r
setofr×rmatrices,andS (R),thesetofr×rsymmetricmatrices,arespecialJordanalgebraswith
r
theJordanproduct.
A finite dimensional Jordan algebra J over R is called Euclidean if there is an associative inner
producton J. EveryEuclideanJordanalgebrahasamultiplicativeidentityelemente. Thequadratic
representation of J is the map P : J → End(J), where P : x 7→ 2L(x)2 −L(x2)and L(x) is the left
multiplication mapby x,i.e., L(x)(y) = xy.
3
2.3 Spectral decomposition
Anon-zeroelementcin Jisidempotentifc2 = c. Anidempotentisprimitiveifitcannotbewritten
as the sum of two idempotents. Two idempotents c and c are orthogonal if c c = 0. A Jordan
1 2 1 2
frame is a maximal set of orthogonal primitive idempotents c ,...,c . If J is simple, then for any
1 r
Jordanframec ,...,c wehavec +···+c = eandthenumberofelementsinthisJordanframe,r,
1 r 1 r
isinvariantforeveryJordanframe. ThiscommonnumberiscalledtherankofJ. Moreover,foreach
x in J there is a Jordan frame c ,...,c such that ξ c +···+ξ c = x, where ξ ,...,ξ are called
1 r 1 1 r r 1 r
the eigenvalues of x. The trace and determinant of x, denoted by tr(x) and det(x), respectively, are
thesumandproduct ofalleigenvalues of x.
2.4 Euclidian Jordanalgebrasand symmetriccones
ForaEuclidean Jordan algebra J thesetofsquared elements x2 where x ∈ J isaclosed cone. The
interiorofthiscone,denotedby J ,isasymmetricconeandiscalledtheconeofpositiveelements
+
of J. Equivalently, J isthesetofall x2 suchthat xisregular, i.e.,det(x) , 0. Conversely, ifΩisa
+
symmetric cone and ǫ ∈ Ω, then one can construct aEuclidean Jordan algebra J such that J = Ω
+
andǫ istheidentityelementof J.
2.5 Classificationofirreducible symmetriccones
Theirreduciblesymmetricconesareinone-to-onecorrespondence withsimpleEuclideanalgebras,
whichareclassifiedintofourfamiliesofclassicalJordanalgebrastogetherwithasingleexceptional
Jordanalgebra.
The first three families of classical Jordan algebras are matrix spaces. More specifically, let
D = R,CorthequaternionH. Denoteby xtheconjugateof xinD,ℜxtherealpartof x,andH (D)
r
thesetofr×r Hermitian matricesoverD. Recallthat X∗,theadjoint ofamatrix X,isobtained by
taking the conjugate of each entries and then transposing the matrix. A matrix X is Hermitian if it
is equal to X∗. This space equipped with the Jordan product X ◦Y = (XY +YX)/2 and the scaler
product hX|Yi = ℜtr(XY)isasimple Euclidean Jordan algebra ofrankr andcorresponds totheir-
reducible symmetricconeofpositivedefinitematricesoverR,CorH,respectively. Thedimension
of H (D) over R is r +r(r −1)d/2, where d, the Peirce constant, is equal to the dimension of the
r
spaceDoverR. WhenDisthesetofrealnumbersH (R)isindeedS (R)andS (R) isPD (R).
r r r + r
The fourth class of Jordan algebras is a Minkowski space. This is the vector space R×Rn−1,
n > 2, equipped with the product (ζ,x)(ξ,y) = (ζξ + x·y,ζy+ξx). It corresponds to the Lorentz
coneL = {(ζ,x)∈ R×Rn−1 : ζ > kxk}.
n
The exceptional Jordan algebra can be described as follows. First notice that H× H with the
product (x ,y )(x ,y ) = (x x − y y ,y x + y x ) is a non-associative algebra, called octonion
1 1 2 2 1 2 2 1 1 2 2 1
O. Any element in O can be written as x+ jy, where j = (0,1) and x,y ∈ H. The conjugation in
O is defined by x+ jy = x− jy. The exceptional Jordan algebra is H (O) and its cone of positive
3
elementsisdenoted byPD (O).
3
Thefollowingtablesummarizestheinformation aboutsimpleEuclidean Jordanalgebras[9,6].
4
Table1: Classification ofsimpleJordan algebras
J J G K dimJ rankJ d
+
S (R) PD (R) GL+(R) SO (R) 1r(r+1) r 1
r r r r 2
H (C) PD (C) GL (C) U (C) r2 r 2
r r r r
H (H) PD (H) GL (H) Sp (C) r(2r−1) r 4
r r r 2r
R×Rn−1 L R×SO+ (R) SO ,(R) n 2 n−2
n 1,n−1 n−1
H (O) P (O) R×E F 27 3 8
3 3 6 4
2.6 Additional properties
In the remainder of this section, and throughout the rest of the paper, we will assume that Ω is an
irreducible symmetric cone ofpositive elements ofasimple Jordan algebra J ofdimension n,rank
randthePeirceconstant ddefinedbyn = r+r(r−1)d/2.
a)Foreach x ∈ Ωandg ∈Gwehave
1. detL(x) = det(x)n/r,
2. detP(x)= (detx)2n/r,
3. detgx= det(g)r/ndet(x)foreachg∈G,
4. detP(y)(x) = (dety)2detx,
5. (gx)−1 = g∗−1x−1,whereg∗ istheadjointofg,
6. P(x)−1 = P(x−1),
7. P(x)∗ = P(x),i.e.,P(x)isHermitian.
b)K actstransitively onthesetofprimitiveidempotents andthesetofallJordanframes.
3 Special functions on symmetric cones
The rich geometric structure of symmetric cones has naturally motivated many research studies in
the field of harmonic analysis. In this section we give a short description of special functions on
symmetriccones.
3.1 Gammafunction
Let fix a Jordan frame c ,...,c in J. For each 1 ≤ j ≤ r we define the idempotent e = c +
1 r j 1
···+c . Ingeneral, it can be shown that if c isan idempotent element of J, then the only possible
j
eigenvalues ofthelineartransformation L(c)are0,1/2,1.Onecanseethattheeigenspace, J(e ,1),
j
corresponding totheeigenvalue1ofL(e )isaJordanalgebrawiththemultiplicationinheritedfrom
j
J. Let Ω be the cone of positive elements and det(j) the determinant with respect to this Jordan
j
algebra. Let P : J → J(e ,1)betheorthogonal projection on J(e ,1). Theprincipal minor, ∆ (x),
j j j j
5
is a homogeneous polynomial of degree j on J defined by ∆ (x) = det(j)(P (x)). We extend this
j j
definitionasfollows. Foreach s= (s ,...,s )∈ Cr weset
1 r
∆ (x) = ∆ (x)s1−s2∆ (x)s2−s3···∆ (x)sr.
s 1 2 r
For each s ∈ Cr the gamma function is ΓΩ(s) = Ωexp{−tr(x)}∆s(x)det(x)−nrdx. This integral is
absolutely convergent ifℜs > (j−1)d/2, for j=R1,...,r. Moreover,
j
r
d
n−r
ΓΩ(s)= (2π) 2 Γ(sj−(j−1) ).
2
Yj=1
Inparticular, identifying z ∈ Cwith(z,...,z) ∈ Cr,wehave
ΓΩ(z) = exp{−tr(x)}det(x)z−nrdx
Z
J+
isabsolutely convergent ifℜz> n/r−1. Ifthisisthecase,then
r
n−r
ΓΩ(z) = (2π) 2 Γ(z−(j−1)d/2).
Yj=1
3.2 Beta functions
Thebetafunction onasymmetricconeΩisdefinedbytheintegral
BΩ(a,b) = ∆a−n(x)∆b−n(e− x)dx,
Z r r
Ω∩(e−Ω)
wherea,b ∈ Cr,ande−Ω = {e−x : x ∈ Ω}.Thisintegralconvergesabsolutelyifℜa > (j−1)d/2
j
Γ (a)Γ (b)
Ω Ω
andℜb > (j−1)d/2. Inthiscase B (a,b) = ,and
j Ω Γ (a+b)
Ω
∆a−n(y)∆b−n(x−y)dy = BΩ(a,b)∆a+b−n(x).
Z r r r
Ω∩(x−Ω)
3.3 The spaceofpolynomials
A partition is a finite sequence of non-negative integers λ = (λ ,··· ,λ ) in decreasing order
1 r
λ ≥ ··· ≥ λ . The weight of λ is |λ| = λ + ··· + λ . If |λ| = k, then we say λ is a partition
1 r 1 r
ofk.
Afunction f : J → Risapolynomial on J ifthereisabasis{v ,...,v }of J andapolynomial
1 n
p ∈ R[t ,...,t ]suchthatforanylinearcombination x= n ξv,
1 n i=1 i i
P
f(x) = p(ξ ,...,ξ ).
1 n
Onecan check that this definition isindependent ofthe choice ofbasis. Theset ofallpolynomials
over J isdenoted byP(J). Theaction ofG on J can benaturally extended toanaction onP(J)by
defininggp(x) = p(g−1x). LetP (J)bethesubspace ofP(J)generated bypolynomials g∆ ,g ∈G.
λ λ
Thenevery pinP (J)isahomogeneous polynomialofdegree|λ|.
λ
6
3.4 Spherical polynomials
Recall that K, the stabilizer of the identity e, is a compact Lie subgroup of G, thus there exists a
Haarmeasureon K. Foreachpartition λthespherical polynomial Φ is
λ
Φ (x) = ∆ (kx)dµ (k),
λ λ K
Z
K
where µ is the normalized Haar measure on K. The function Φ is indeed a homogeneous poly-
K λ
nomial of degree |λ| and is invariant under the action of K, i.e., Φ (kx) = Φ (x) for any k ∈ K
λ λ
and x ∈ J. Moreover, the spherical polynomial Φ is, up to a constant factor, the only K-invariant
λ
polynomial inP (J). Moreprecisely, if pisaK-invariant homogeneous polynomial inP (J),then
λ λ
p(kx)dµ (k) = p(e)Φ (x). (3.1)
K λ
Z
K
Consequently, foranyg ∈G
Φ (gkx)dµ (k) = Φ (ge)Φ (x). (3.2)
λ K λ λ
Z
K
If x ∈ Ωandℜγ > (r−1)d/2, thenforanyyinΩandg ∈Gwehave
e−tr(xy)Φλ(gx)det(x)γ−nrdx = ΓΩ(λ+γ)∆−γ(y)Φλ(gy−1). (3.3)
ZΩ
3.5 Zonal polynomials
ThePochhammersymbolforΩisdefinedby
Γ (s+λ)
(s) = Ω , s∈ C.
λ Γ (s)
Ω
Thisdefinition generalizes theclassical Pochhammersymbol. Thezonalpolynomial Z isahomo-
λ
geneous K-invariant polynomial on J definedby
|λ|!
Z (x) = d Φ (x), (3.4)
λ λ(n) λ
r λ
where d is the dimension of the vector space P (J), and Φ is the spherical polynomial. Zonal
λ λ λ
polynomials are K-invariant polynomials normalized bytheproperty
tr(x)k = Z (x) x∈ Ω. (3.5)
λ
|Xλ|=k
Notice that the function p(x) = tr(x)k on Ω is a K-invariant homogeneous polynomial in P (J). It
λ
followsfromEquation(3.2)thatforeach x∈ J andy ∈ Ω
Z (y)Z (x)
Zλ(P(y12)kx)dµK(k) = λ λ . (3.6)
ZK Zλ(e)
7
3.6 Hypergeometric functions
Let a ,...,a and b ,...,b be real numbers with a −(i−1)d/2 ≥ 0,b −(j−1)d/2 ≥ 0 and
1 p 1 q i j
x,y ∈ J. Thehypergeometric function F isdefinedby
p q
∞ (a ) ···(a ) Z (x)Z (y)
F (a ,...,a ,b ,...,b ,x,y) = 1 λ p λ λ λ .
p q 1 p 1 q (b ) ···(b ) k! Z (e)
Xk=1|Xλ|=k 1 λ q λ λ
Fory= eweset F (a ,...,a ,b ,...,b ,x)= F (a ,...,a ,b ,...,b ,x,e).Thismeansthat
p q 1 p 1 q p q 1 p 1 q
∞ (a ) ···(a ) Z (x)
F (a ,...,a ,b ,...,b ,x) = 1 λ p λ λ .
p q 1 p 1 q
(b ) ···(b ) k!
Xk=1|Xλ|=k 1 λ q λ
This series converges absolutely if p ≤ q and diverges if p > q. Furthermore, F (x) = etr(x) and
0 0
F (b,x) = det(e− x)−b.
0 1
Suppose x∈ Ωandg∈G. ByusingtheintegralinEquation(3.6)for p ≤ qweget
F (a ,...,a ,b ,...,b ,g(kx))dµ (k)= F (a ,...,a ,b ,...,b ,x,ge). (3.7)
p q 1 p 1 q K p q 1 p 1 q
Z
K
Similarly,ifℜγ > (r−1)d/2, y ∈ Ωand p < q,ory ∈ e−Ωand p = q,thenbyapplying Equation
(3.3)weget
e−tr(xy)pFq(a1,...,ap,b1,...,bq,gx,z)det(x)γ−nrdx =
ZΩ
Γ (γ)det(y)−γ F (a ,...,a ,γ,b ,...,b ,gy−1,z).
Ω p+1 q 1 p 1 q
WerecordthefollowingProposition, laterusedinderiving thebetatypedistribution, from[9].
Proposition 3.1. Suppose that p ≤ q+1, ℜη > (r−1)d/2 andℜη > (r−q)d/2. Ifg ∈ G such
1 2
thatge ∈ Ω∩(e−Ω),then
pFq(a1,...,ap,b1,...,bq,gx)det(x)η1−nr det(e− x)η2−nrdx (3.8)
ZΩ∩(e−Ω)
= B (η ,η ) F (a ,...,a ,η ,b ,...,b ,η +η ,ge).
Ω 1 2 p+1 q+1 1 p 1 1 q 1 2
Suppose y ∈ J and x ∈ Ω. We define x⋆y = P(x21)y and Eig(y|x) equal to the r-tuple of the
orderedeigenvaluesofywithrespectto x,indecreasingorder. Noticethattheseeigenvalues arethe
roots of the polynomial p(ℓ) = det(ℓx−y). In particular, for x = e, we have Eig(y) = Eig(y|e) is
theorderedeigenvalues ofy. NotethatforaspecialJordanalgebra, suchasthespaceofsymmetric
1 1
matrices or Hermitian ones, the operation x⋆y is x2yx2. Here welist some interesting properties
of⋆operation.
Lemma3.1. Let xandybeinΩ. Then
a) x⋆e= e⋆x = x, x⋆x−1 = x−1⋆ x= eand(x⋆y)−1 = x−1⋆y−1.
b) x⋆y and y⋆ x have the same eigenvalues, therefore, any k-invariant function on Ω returns the
samevalueonboth x⋆yandy⋆x. Inparticular, det(x⋆y) = det(y⋆x)= det(x)det(y), tr(x⋆y) =
tr(y⋆x) = tr(xy)andZ (x⋆y) = Z (y⋆x).
λ λ
8
Proof. a) We only prove the last equality. The other equalities are immediate from the definition
of x⋆y. Fromproperty (5)inSubsection 2.6weget
(x⋆y)−1 = (P(x12)y)−1 = P(x12)∗−1y−1 = P(x−12)y−1 = x−1⋆y−1.
b) Foranr-tuple (ξ ,...,ξ )withξ > ··· ξ > 0,wedefine
1 r 1 r
(ξ ,...,ξ )−1 = (ξ−1,...,ξ−1).
1 r r 1
Bythisnotation,onecancheckthatforanyx ∈ ΩwehaveEig(x−1) = Eig(x)−1. Clearly,theordered
eigenvalues of x⋆yareequal toEig(y|x−1). Onthe other hand, Eig(y|x)isequal toEig(x|y)−1 (ifℓ
isaneigenvalue of xwithrespect toy, then1/ℓ isaneigenvalue ofywithrespect to x). Therefore,
Eig(x⋆y) = Eig(y|x−1)= Eig(x−1|y)−1 = Eig(y−1 ⋆x−1)−1 = Eig((y−1 ⋆x−1)−1) = Eig(y⋆x). (cid:3)
Remark3.1. Thereisworthmentioninganinterestingfactaboutthe x⋆yoperationonΩ. Suppose
µ and ν are two K invariant measures on Ω. These can be lift up, respectively, to K bi-invariant
measuresµ♯ andν♯ onG. Byprojecting theconvolution measureµ♯⋆ν♯ onΩweobtainameasure
on Ω , denoted by µ ⋆ ν and called the convolution of µ and ν. Now if x and y are two random
vectorsinΩwithK invariant distributions µandν,thenthedistribution oftherandom vectorx⋆y
istheconvolution µ⋆ν[10].
4 The non-central Wishart distribution
In multivariate statistics, the so-called non-central Wishart distribution is the multivariate analog
of the non-central chi-square distribution and, similarly, it arises out of the sampling distribution
of a sample statistic, namely the sample covariance matrix of a multivariate normally distributed
population. Moregenerally, ifX isanN×rrandommatrixsuchthatX ∼ N (M,I ⊗Σ)withΣ ∈
N×r N
PD (R),thenthedensityofS = XTXwithrespecttothestandardLebesguemeasureisproportional
r
to
1 1 1 1
det(Σ)−N2 exp{−2tr(Σ−1S)}exp{−2tr(∆)}0F1(2N, 4∆Σ−1S)1PDr(R)(S).
Thusthenon-centralWishartdistribution isthetransformeddistribution withrespecttoamultivari-
atenormaldistribution underthequadratic mapping X 7→ XTX. Wecangeneralize thissituation as
follows.
Let J beasimpleEuclideanJordanalgebra, and(E,h·|·i )aEuclideanspace. A(Jordan)repre-
E
sentation ψof J over E isalinearmappingψ : J → End(E)suchthatψ(x2) = ψ(x)2 foreach x∈ J.
Therepresentation ψiscalled self-adjoint ifthe linear transformation ψ(x) isself-adjoint for every
x∈ J.
Assumeψisaself-adjoint representation of J ontheEuclideanvectorspace E. Foreachv∈ E,the
function x 7→ hψ(x)v|vi isalinear form on E. Therefore, there exists aquadratic map Q : E → J
E
such that hψ(x)v|vi = hx|Q(v)i . The fact that hQ(v)|x2i = kψ(x)vk2 > 0 implies that Q(v) ∈ Ω,
E J J E
foreachv ∈ E.
Proposition 4.1. Let v ∼ N(µ,ψ(σ)) be a normally distributed random vector in E with mean
vector µ ∈ E and covariance ψ(σ) ∈ PD(E), where ψ is a self-adjoint representation of the simple
Jordanalgebra J over E,σ ∈ Ω,and N > 2(n−r). Thenthedensityofx = Q(v)withrespecttothe
canonical LebesguemeasureonΩis
N −1 1 N 1
px(x) = 2N2ΓΩ( )det(σ)2Nr exp{− tr(ǫ)}·det(x)2Nr−nr0F1( , (σ−1⋆ǫ)⋆x) (4.1)
2r 2 2r 4
(cid:18) (cid:19)
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whereǫ = σ−1⋆Q(µ).
Proof. Firstwecomputeg(z) = E[exp{−hz,Q(v)i }],theLaplacetransform of Q(v). Foreachz ∈ J
J
wehaveg(z)isequalto
−1 1
(2π)N2 det(ψ(σ))12 exp{−hψ(z)v,viE}exp{− hψ(σ)−1(v−µ),(v−µ)iE}dv
(cid:18) (cid:19) ZV 2
= (2π)N2 det(σ)2Nr −1· exp{−1 2hψ(z)v,viE +hψ(σ)−1(v−µ),(v−µ)iE }dv.
Z 2
(cid:16) (cid:17) V (cid:16) (cid:17)
Werewritetheexpression 2hψ(z)v,vi +hψ(σ)−1(v−µ),(v−µ)i as
E E
hψ(2z+σ−1)(v−ψ(2z+σ−1)−1ψ(σ−1)µ,(x−ψ(2z+σ−1)−1ψ(σ−1)µi
E
+hψ(2z+σ−1)−1ψ(σ−1)µ,ψ(σ−1)µi +hψ(σ−1)µ,µi ,
E E
andsubstitute itintheoriginalequations, obtaining
1 1 1 1
g(z) = 2−N2 det(σ)−2Nr det(z+ σ−1)−2Nr exp{− tr(ǫ)}0F0( (σ−1⋆ǫ)⋆(z+ σ−1)−1),
2 2 4 2
where ℜ(z+ 1σ−1) > 0. The density of x = Q(v) in Equation (4.1) then follows from the inverse
2
Laplacetransform ofg(z)(see[4]). (cid:3)
InlightofProposition4.1,Equation(4.1)canbetakenasthedensityofthenon-centralWishart
distribution overtheirreducible symmetricconeΩ.
Definition 4.1. A random vector x with values in an irreducible symmetric cone Ω is said to have
theWishartdistribution withparameters η > n−r,σ ∈ Ωandǫ ∈ Ω,ifitsprobability density with
respecttothestandard canonical Lebesguemeasureon J isgivenby
1 1 1
exp{− tr(ǫ)} F ( η, (σ−1⋆ǫ)⋆x)w (x|η,σ)1 (x),
0 1 Ω Ω
2 2 4
where
1 1
wΩ(x|η,σ) = exp{− tr(σ−1x)}det(x)21η−nr1Ω(x). (4.2)
212ηrΓΩ(2η)det(σ)12η 2
This is denoted by x ∼ W (η,σ,ǫ), where η, σ, and ǫ are, respectively, the shape, (multivariate)
Ω
scale, and non-centrality parameters. In particular, if ǫ = 0, the distribution is called the (central)
Wishartdistribution overasymmetriccone[5],denoted by
x∼ W (η,σ),
Ω
otherwise, itiscalledthenon-central Wishartdistribution overasymmetriccone.
Remark4.1. Ifx ∼ W (η,σ),thenitsprobability densityisw (x|η,σ).
Ω Ω
Remark 4.2. In accordance with classification of irreducible symmetric cones, the non-central
Wishart distribution over an irreducible symmetric cone can be categorized as : i) the real type
W (η,σ,ǫ) over PD (R), ii) the complex type CW (η,σ,ǫ) over PD (C), iii) the quaternion type
r r r r
HW (η,σ,ǫ) over PD (H), iv) the Lorentz type LW (η,σ,ǫ) over the Lorentz cone L , and v) the
r r n n
exceptional typeOW(η,σ,ǫ)overtheoctonion conePD (O).
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