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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 irreducible 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) 9 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). 3 10