On Singular Wishart and Singular Multivariate Beta Distributions Author(s): Harald Uhlig Source: The Annals of , Vol. 22, No. 1 (Mar., 1994), pp. 395-405 Published by: Institute of Mathematical Statistics Stable URL: http://www.jstor.org/stable/2242460 . Accessed: 09/05/2013 18:20

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This content downloaded from 128.135.47.203 on Thu, 9 May 2013 18:20:26 PM All use subject to JSTOR Terms and Conditions The Annals of Statistics 1994,Vol. 22, No. 1, 395-405

ON SINGULAR WISHART AND SINGULAR MULTIVARIATE BETA DISTRIBUTIONS

BY HARALD UHLIG PrincetonUniversity

This paper extendsthe studyof Wishart and multivariatebeta distri- butionsto the singularcase, wherethe rankis belowthe dimensionality. The usual conjugacyis extendedto this case. A volumeelement on the space ofpositive semidefinite m x m matricesof rank n < m is introduced and some transformationproperties established. The densityfunction is foundfor all rank-nWishart distributions as wellas therank-1 multivari- ate .To do that,the Jacobianfor the transformationto the singularvalue decompositionof general m x n matricesis calculated. The resultsin thispaper are usefulin particularfor updating a Bayesian posteriorwhen tracking a time-varyingvariance-covariance .

1. Introduction. Considern drawsYj, j = 1, . . , n, froma normaldistri- butionAf(O, E), whereE is m x m and positivedefinite. The randomvariable X = Y271YjYJYhas a Wishartdistribution Wm(n, E). Usually,Wishart distribu- tionsare studiedonly for n > m- 1. This paperextends the study of Wishart as well as multivariatebeta distributionsto the singularcase, where0 < n < m, n an integer,that is, wherethe rank of the is belowits dimen- sionality.The usual conjugacybetween Wishart and beta distributions[see Muirhead(1982), Theorem 3.3.1] is extendedto thiscase (see Theorems1 and 7). A volumeelement on the space ofpositive definite m x m matricesof rank n < m is introduced(see Theorem2) and sometransformation properties es- tablished(see Theorems3 and 4-this is necessaryin orderto talk sensibly about densitieson that space. The densityis foundfor the rank-nWishart distributionfor all integersn, 0 < n < m (see Theorem6) and therank-1 mul- tivariatebeta distribution(see Theorem7). To do that,the Jacobianfor the transformationto the singularvalue decompositionof general m x n matrices is calculated(see Theorem5). This paperthus extends the resultsestablished by Fisher(1915), Wishart(1928), James (1954), Khatri(1959), Olkinand Roy (1954) and Olkin and Rubin (1964). Their resultsare presentedclearly and conciselyin bookform in Muirhead(1982): We willfollow his terminologyand formulationsclosely. The resultsin this paperare, in particular,fundamental and usefulfor up- datinga Bayesianposterior when tracking a time-varyingvariance-:that, in fact,motivated this investigation [see Uhlig(1992)]. Imagine the followingtime series model,which is a simplemultivariate state space alternativeto the popularARCH models.[For an overviewof the extensive

ReceivedJuly 1992; revisedJune 1993. AMS 1991 subjectclassifications. Primary 62H10; secondary62E15. Key wordsand phrases. Wishartdistribution, beta distribution,, singular matrixdistributions, conjugacy. 395

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ARCH literature,see Bollerslev,Chou and Kroner(1992). For a univariate state space specification,see Shephard(1994).] Thereis an unobservablepre- cisionmatrix Pt, evolvingover time according to

(1) Pt =+u(Pt11)'Qtu(Pt l), p whereU(Pt-1) is the upper-triangularCholesky factor, that is, that upper- triangularmatrix T with positivediagonal elementswhich satisfies Pt-, = T'T. Suppose that the Qt are drawni.i.d. froma multivariatebeta distribu- tion 3m(p/2,1/2).Suppose further,a researcherstarts with the priorthat Pt-, is Wishartdistributed, Pt-, - Wm(A-', AS7-'), whereA = 1/(p + 1), so thatE[Pt-1]-1 = St_,. If the usual conjugacybetween Wishart and multivari- ate beta distributionsholds, then the priorfor Pt is a WishartWm (p, ST-X1/p). Supposenow that the researcher observes a singledraw Yt from a multivariate normaldistribution with that matrix, Yt - K(O,P7'). The posterior forPt is thengiven by a WishartWm (A-1, AS-'), whereSt = AYtYt+(l- A)St_j so that E[PtI-L = St and the game can begin anew. To findthe parameter p governingthe degreeof time variation,the explicitlikelihood function is needed.The problemwith these argumentsis that the singularmultivariate beta distributions!3m(p/2, 1/2) have yet to be definedand the "usual conju- gacy"between Wishart and this multivariatebeta distributionhas yet to be established.To do that,singular Wishart distributions have to be analyzedas well since theyare fundamentalfor the studyof singularmultivariate beta distributions.Furthermore, in orderto statethe likelihoodfunction explicitly, the densityfunction for a L3m(p/2,1/2)-distributed Q has yet to be found,since Im - Q is ofrank 1 and thusis singularalmost surely. Solv- ingthese problems is thepurpose of this paper. In thecourse of doing so, some generallyuseful theorems for the analysis ofmultivariate random variables are established.

2. Results. Unless statedotherwise, all ournotation, definitions and ter- minologyfollow Muirhead (1982). First,we generalizethe definitionof mul- tivariatebeta distributionB1m(n/2,p/2) to integers0 < n < m. Let m and n be positiveintegers, p > m - 1 and E be of size m x m and positivedefi- nite.Recall Definition3.1.3 in Muirhead(1982), that a randomvariable A is Wm(n,E)-distributed if A can be writtenas A = E,Y)Y, withYj, A(O,E) i.i.d. For a positivedefinite matrix S, letU(S) denotethe upper-triangular Cholesky factor,that is, thatupper-triangular matrix T withpositive diagonal elements whichsatisfies S = T'T.

DEFINITION 1. A randomvariable X is 3m(n/2,p/2)-distributed,if it can be writtenas

X = U(A +B)'-'AU(A +B)-

This content downloaded from 128.135.47.203 on Thu, 9 May 2013 18:20:26 PM All use subject to JSTOR Terms and Conditions ON SINGULARWISHART AND BETA DISTRIBUTIONS 397 whereA Wm(nf,Im)and B Wm(p,Im)with A representedas abovefor E = I and the Yj,j = 1, .. ., n, independentfrom B. X is Bm(n/2,p/2)-distributedif A Wm((p, Im,)and B - Wm(n,Im,) instead.

This definitionis moldedafter Theorem 3.3.1 in Muirhead(1982) and is thereforecontained as a special case in Definition3.3.2 in Muirhead(1982), forn > m - 1. However,for n < m - 1, this definitionis new and Theorem 3.3.1 in Muirhead(1982) needsto be establishedfor these parameters as well. This is done in the followingtheorem, which is stated"backwards" from the versionin Muirhead(1982) to make it particularlysuitable for the purposeof posteriorupdating alluded to in the Introduction.

THEOREM1. Let m and n be positiveintegers and letp > m - 1. Let H Wm(p+ n,E) and Q ' Bmf(p/2,n/2) be independent.Then G=-U(H)'QU (H) -Wm(P,2 E)

PROOF. The theoremfollows from the followingsomewhat broader claim:

CLAIM. Let A Wm(pjIm),B = Ejn=YjYjl,with Yj Af(O,Im)i.i.d., and H Wm(p + n, E), whereA, Yj,j = 1, ... , n, and H are independent.Define C A + B, Q - U(C)'-1AU(C)-', G _ U(H)'QU(H)and D =-H - G. Then C Wm(p + n,Im),G Wm(p,E) and D = EjnZjZj withZ; KO, E), where C, G and Zj, j = 1, . . ., n, are independent.

The proofmimics the proofof Muirhead[(1982), Theorem3.3.1]. Define Zi = U(H)'U(C)'-'Yj and note that D = Ej lZjZj. It followsfrom Muirhead [(1982),Theorem 2.1.4] that (dA) A (dH) A (dY1)A ...A (dYn) = (dC) A (dH) A (dY1)A .*. A (dYn) = (detH)-n/2(detC)n/2(dC)A (dH) A (dZ1)A ... A (dZn) = (detH)-n/2(detC)n/2(dC) A (dG) A (dZ1) A .. A (dZn), exploitingG = H - D forthe last equality.Writing out the densities,it now followsthat

(2mP/2rm(p/2))-l etr(-A/2)(detA)(P-m- 1)/2

? (2m(n+p)/2rm((n +p) /2) (det E)(n+p)/2) -1 etr(-E-'H/2) (detH)(n+p-m-1)/2 ? (27r)-mn/2etr(-B/2) (dA) A (dH) A (dY1) A ... A (dYn) -1 = (2m(n+P)/2Irm((n +p)/2)) etr(-C/2) (det C)(n+p-m-l)/2

x (2mP/2rm(p/) (det)p/2)/ etr(-D (d-1G/2) (detG)(P-m-1)/2 ? (2ir) -mn/2(det E) -n/2 etr(-D/2) (dC) A (dG) A (dZj) A ... A (dZn),

This content downloaded from 128.135.47.203 on Thu, 9 May 2013 18:20:26 PM All use subject to JSTOR Terms and Conditions 398 H. UHLIG exploitingdetA = detC detQ, H = G + D and detH = detG/ det Q. Inspecting thelatter density finishes the proof. 0

Let m > n > 0 be integers.For the computation of a likelihoodfunction, say,a densityon a spaceof appropriate dimensionality is needed. Densities do notexist for V Wm(ln,E) orX B13m(p, n/2) on the space of symmetric mx m matrices,since V and Im - X are singularand ofrank n almostsurely [see Muirhead(1982), Theorem 3.1.4]. As shownbelow, however, densities do exist onthe (mn - n(n- 1)/2)-dimensionalmanifold of rank-n positive semidefinite m x m matricesS withn distinctpositive eigenvalues; denote that manifold by Sm+,,n A naturalglobal coordinate system for this manifold is to use the decompositionS = H1LH',where L is n x n,diagonal, L = diag(1,... , ln)with 11 > 12 > ... > 0 and whereH1 E Vn,m,the (mn- n(n+ 1)/2)-dimensional Stiefelmanifold of m x n matricesH1 with orthonormal columns, H'H1 = I, This parameterizationis unique up to the assignmentof n arbitrarysigns to the columnsof H1. The task is to definethe volume element (dS): with the chosenparameterization, (dS) needsto be definedas somefunction of H1 and L multipliedwith (Hl dH1) A Ai=1dli. [We followMuirhead (1982), page56, in ignoringsigns of the overall differential and defining only positive integrals.For the definition of(H' dH1),see Muirhead(1982), page 63 andthe discussionfollowing page 67.] Note that dS = dHj LH' + H1dLH' + HL dH'. Findan m x (m- n) matrixH2, so thatH _ [H1tH2]E O(m), thatis, so that H'H = Im,and let hi be thei-th column of H. LetR = H' dSH and calculate that

R=H'dSH= [ o]b where Ra H= dH1L + dL + (H dH,L)' di, (11 -12)h/ dhl .. (I, -ln)hndhl- ... (1 - 12)k2 dhl d12 (12- ln) h dh2

(11- n)hn dhi (12-1n)hn dh2 dln [exploitingthe skew symmetryof H, dHj, see Muirhead(1982), bottomof page 64] and Rb =H dH,L

llhn'+1dhl ... lnh+1 Clhn

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Appealingto theanalogy that (dS) wouldbe theexterior product of all distinct, nonzeroentries in R by Theorem2.1.6 in Muirhead(1982), if S were a full- ranksymmetric matrix, we define(dS) to be the exteriorproduct of all entries on and belowthe maindiagonal of Ra and ofall entriesin Rb timesa factor2-n to correctfor the factthat each matrixS is the image of 2n decompositions H1LH' due to the arbitraryassignment of signs to the columnsof H1. We thereforeget the followingtheorem.

THEOREM 2. Let m > n > 0 be integers.On the space S., n of positive semidefinitem x m matricesS ofrank n withn distinctpositive eigenvalues, thevolume element (dS) is n n n (2) (dS) =_2-n H Im-n rj (li-I j) (H' dH,) /\A l i=l i<.j i=1

withS representedas S = H,LH', H1 E Vn,mand L = diag(l1, ln,I),11 > 12>

We aim at expressingthe densitieswith respectto this volumeelement subjectto the restrictionthat the densityis the same whenevertwo pairs (H1,L) and (H1,L) differonly in the assignmentof signs to the columnsof H1 and H1. The followinguseful theorem also justifiesdefining (dS) in termsof the nonzeroand distinctentries of R.

THEOREM 3. Let X, Y E S.,n be relatedby X = QYQ', whereQ E O(m). Then(dX) = (dY), where(dX) and (dY) are thevolume elements on S., ndefined above.

PROOF. DecomposeY = H1LH', H1 E Vn,mand L = diag(ll, ... 11 > 12> ... > 0. Note that G1 -=QH1 e Vn,m,so thatX = G1LG' is the decompo- sitionfor X. Since (H' dHi) is invariantto multiplicationon the leftwith an orthonormalm x m matrix,that is, since (G'1dG1) = (H' dH,) [see Muirhead (1982), bottomof page 69], the resultfollows. 0

The followingtheorem is a versionof Muirhead's Theorem 2.1.6 [Muirhead (1982)] forthe case n = 1. The generalrank-n case is an open problem.

THEOREM4. Let X, Y E Sm,1 be relatedby X = BYB', whereB is m x m and of full rank. Form the representationsX = G1KG', Y = H1LH', where G1,Hl E Vi,mand K,L E R. Then

(dX) = IGBHi Imdet(B) (dY).

Since Xv = (KG'v)Gl = (LH'B'v)BH,, forany v E R", and since JIG,11= 1 (wherewe use 11 to denotethe normof the m x 1 vectorconstituting G1),

This content downloaded from 128.135.47.203 on Thu, 9 May 2013 18:20:26 PM All use subject to JSTOR Terms and Conditions 400 H. UHLIG we have G1 = BH/ IIBHiII and K = IIBH1I2L,and thus (3) G'BH, =11BH1 II=(K/L)1/2 = 1/j G enablingexplicit calculation of the expressionin the theorem.We conjecture thatthe formulafor the generalrank-n case is givenby

(dX) = det(G/BHi)m+l-n det(B)n(dY).

PROOF OF THEOREM 4. We firstshow this for the case thatB = D is diagonal. Find m x (m - 1) matricesG2 and H2, so that G _ [G1.G2] and H _ [H1.H2] satisfyG'G = Im and H'H = Im,that is, G,H E 0(m). Let gi and hi denotethe ith columnsof G and H. Let E = G'DH and notethat G'dXG =EH'dYHE',

where,for example,

H'dYH= (H2'dhlL)' HH [dLdh L 0

Thus, the firstcolumn of E H'dYH E' has

m e1leildL + (elleij + eile11)hjdhl L j=2

as its ith entry.Taking the exteriorproduct over all these entries,ignoring the overall sign fornow and using the abbreviationfil = ellei,, fij = elleij + eilelj yields m m\ (dx)=A 1dL + E fijhjdhiL) i=l j=2 m \m = E sgn(a) Eia(i) Lm-ldLA AhJdhl aEI(m) i=l j=2 =(det F) (dY),

usingthe skew symmetryof the operatorA, where11(m) is the set ofall per- mutations of (1, ... , m) [cf. Horn and Johnson (1985), page 8] and where F is m the matrix[fi=l, The firstcolumn of F is thefirst column of E, multipliedwith e1l = GIDH1 , 0. Thejth columnof F is thesum of thejth column of E multipliedwith ell and the firstcolumn of F multipliedwith ejl/el1. By the rules about calculating withdeterminants, it followsthat

det F = det(eulE) = eTmdet E = eTmdet D.

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Since (dX) has a positivesign, the absolutevalue of ell needs to be taken, demonstratingthe claimfor B = D diagonal. For generalB, writeB as B = P'DQ, whereP, Q E 0(m) and D is diagonal [see TheoremA9.10 in Muirhead(1982), page 5931.Let G1 = PG1,H1 = QH1, X = PXP' and Y = QYQ'. Withthe aid ofthe previoustheorem and the proof above fordiagonal matrices D, it followsthat

(dX) = (dk) = (b'Dfi))m(det D)(dY) = (GIBHi)m(detB)(dY),

as claimed. O

The followingtheorem is an extremelyuseful cousin of Muirhead's Theorem 2.1.13 [Muirhead(1982), page 63]. The proofis not a straightforwardgener- alizationof the proofof that theorem, but it proceedsalong similarlines. Let Z be an m x n (m > n) matrixof rank n and with distincteigenvalues of Z'Z. Using the nonsingularpart of the singularvalue decomposition,write Z = H1DP', whereH1 E Vn,m,D is diagonalwith D1l > D22 > ... > Dnn > 0 and P e 0(n): this decompositionis unique up to the arbitraryassignment of signs to columnsof P as can be seen upon examinationof, for example, The- orem7.3.5 and its proofin Horn and Johnson[(1985), page 414, withA = Z' there].

THEOREM5. Let Z be an m x n matrixand Z = H1DP' the nonsingular part ofthe singular value decomposition,where H1 e Vn,m,D is diagonal with Dui >D22 > * > Dnn> 0 and P E0(n). Then

(4) (dZ)=2 n(det D)m-n J(Ds -D )(HIdHj) A (dD) A (P dP), i<,/ where n (5) (dD) = AdDii. i=l

PROOF. Find an m x (m - n) matrixH2 so thatH _ [H1:H2]E 0(m). Since

dZ = dHiDP' +H1dDP' +HlDdP, it followsthat

(6) H' dZP = [Hi dHj D + dD + (PI dPD)']

Since (dZ) = (H'dZP) by Theorem2.1.4 in Muirhead(1982), calculatingthe exteriorproduct of the differentialforms on the right-handside of equation

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(6) deliversthe solution. The exteriorproduct of all elementsin thebottom partH2 dHj D is n m (7) (H2dHlD)=(detD)m-nA A '?hi'i i=lj=n+l usingTheorem 2.1.1 in Muirhead(1982). The top part T = H, dHj D + dD + (P dPD)' is an n x n matrixof differential forms with entries (dDiip ii =j, - 720= < h'dhjDj Djjp' dpj ifi >j, hjdhi Dj, - Diipj dpi, ifi < j, exploitingthe skew symmetry ofH' dH1 andP' dP. To calculatethe exterior productof these elements, write that product conveniently as n (8) (T)=ATii AATij ATji i=1 i

hj dhi Ahjdhi = ZHkjHlj(dHkiA dH1i + dH1jA dHki) k

Tij A Tji= (DM- D)hj dhi A pj dpi Combiningequations (7) and (8),the overall exterior product of the elements ofthe right-hand side of equation (6) afterappropriate reordering (and again ignoringthe overall sign) is thus (H2dH1D) A (T) n m n n n n n = (detD)m.nA A hbdhif(DM-D2) AdDiiA AA hjdhiAA A pjdpi i=l j=n+l i

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Allowingfinally for arbitrary assignment of signs to the columnsof P makes Z the image of 2n decompositionsZ = H1DP', so that the densityhas to be dividedby that numberin orderfor the integrationover the entirespace P E 0(n) to yieldthe correctresult. 0

THEOREM6. Let m > n > 0 be integers.The density for a Wm(n,E)-distribu- tion on the space Sm+,nof rank-npositive semidefinite m x m matriceswith distinctpositive eigenvalues with respect to the volumeelement (dS) defined above is givenby

( ) ~2mn/2rn(n/2) (det E)n/2 ( /)( ) whereL = diag(l1, . . 1,In), S = H1LHl.

PROOF. Let Y = [Y1 ... Yn], where Yi -.'K(0, E) i.i.d., Y is m x n. Let S = YY'. It is easy to checkthat S has n distinctpositive eigenvalues almost surely,and we will assume so fromhere onward.The densityfor Y is given by

(10) (27r mn/2(det E)-n/2 etr(-E-S-1S2) (dY) DecomposingY = H1DP' as in the previoustheorem results in S = H1LH', the desiredparameterization, where L =_D2 and li _ Lsi. Note that

n n n Adli =2n]7Dii AdDii i=l i=1 i=1 and that det D = (det L)1/2. Replacing(dY) by the right-handside of(4) and integratingover (P'dP) withCorollary 2.1.16 in Muirhead(1982), one there- foreobtains the densitystated in the theorem.O

The followingtheorem is a versionof Muirhead's Theorem 3.3.1 [Muirhead (1982)] forthe case n = 1. The generalrank-n case is an open problem.[The refereesuggested the followinggeneral approach. Show that U in Theorem7 has thesame distributionas (A + BY'1/2A(A+ B)-1/2 and thatits momentgen- eratingfunction can be writtenin termsof a confluenthypergeometric func- tionof matrix argument; see Muirhead(1982). The matrixargument will be a matrixof the same (and thusreduced) rank as A. Reductionformulas in Herz (1955) can thenbe used to rewritethe momentgenerating function of U as a momentgenerating function of lower dimensionality.]

THEOREM 7. Let m > 1 be an integerand letp > m - 1. Let A and B be independent,where A is Wm(1,E) and B is Wm(p,E). Put A + B = T'T, where T is an upper-triangularm x m matrixwith positive diagonal elements. Let U be them x m symmetricmatrix defined by A = T'UT. ThenA + B and U are

This content downloaded from 128.135.47.203 on Thu, 9 May 2013 18:20:26 PM All use subject to JSTOR Terms and Conditions 404 H. UHLIG independent;A + B is Wm(p + 1,E) and thedensity function of U on thespace S+ 1 withrespect to thevolume element (dU) on thisspace definedabove is

(11) 7r(-m+l)/2 Fm((p + 1)/2) L-m/2 det(Im -U)(p-m-)/2 r7(1/2)rm(p/2) whereU = H,LH', H1 E Vl,mL E R.

We conjecturethat the densityin the generalrank-n case is givenby

+ 7r (-mn+n2)/2 rm((p n)/2)deL)nmU2dtIm- (pm12 rn(n/2)rm(p/2) (L) ( -

PROOFOF THEOREM7. The proofis almosta verbatimcopy of the proofof Muirhead'sTheorem 3.3.1 [Muirhead(1982)] and is statedhere for reasons of completeness.Find the representationA = G,KG', whereG1 E Vl,m,KE R. The joint densityof A and B is

7r(-m+1)/22-m(p+1)/2(detE)-(p+l)/2 etrt -1(A +B) r(1/2)Fm(p/2) K 2 XK-m/2(detB)(P-m-l)/2(dA) A (dB).

Let C = A + B and notethat (dA) A (dB) = (dA) A (dC). Set C = TIT, whereT is uppertriangular with positive diagonal elements, and A = T'UT. Find the representationU = H,LH', H1 E Vl,m,L E DR.Theorem 4 impliesthat

(dA) A (dC) = (K/L)m/2(detT) (dU) A (dC), rememberingthat T is a functionof C alone. Substitutinginto the density above and collectingterms yields the desiredconclusion. O

Analogouslyto Definition3.3.2 in Muirhead(1982), we have the following.

DEFINITION2. A matrixU withdensity function (11) is said to have the multivariatebeta distributionBm(1/2,p/2) with parameters 1/2 and p/2. A matrixV withdensity function (11) forU = Im- V is said to have the multivariatebeta distributionBm(p/2, 1/2) with parametersp/2 and 1/2(note that one thenneeds to decomposeIm - V = H1LHI XH1 E Vi,m, L E R).

It is clear fromTheorem 7 that,for n = 1, Definition1 is a special case of Definition2. Furthermore,by readingTheorem 7 "backwards"and switching the roles ofA and B, one obtainsanother proof for Theorem 1 forthe case n = 1.

Acknowledgment. I am gratefulfor useful comments from the referee.

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DEPARTMENTOF ECONOMICS PRINCETON UNIVERSITY PRINCETON, NEW JERSEY 08544

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