On Singular Wishart and Singular Multivariate Beta Distributions
On Singular Wishart and Singular Multivariate Beta Distributions Author(s): Harald Uhlig Source: The Annals of Statistics, 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 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp
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The Annals of Statistics
1994,Vol.22,No. 1, 395-405
ON SINGULAR WISHART AND SINGULAR MULTIVARIATE BETA DISTRIBUTIONS BY HARALD UHLIG
PrincetonUniversity This paperextendsthe studyofWishartand multivariate beta distributionsto the singularcase, wherethe rankis belowthe dimensionality. The usual conjugacyis extendedto this case. A volumeelementon the space ofpositivesemidefinite m x m matricesofrankn < m is introduced and some transformation established.The densityfunction properties is foundforall rank-nWishartdistributions as wellas therank-1multivariTo do that,the Jacobianforthe transformation ate beta distribution. to the singularvalue decomposition ofgeneralm x n matricesis calculated. The resultsin thispaperare usefulin particularforupdatinga Bayesian posterior whentrackinga time-varying variance-covariance matrix.
1. Introduction. Considern drawsYj,j = 1, . . ,n, froma normaldistributionAf(O, E), whereE is m x m and positivedefinite. The randomvariable X = Y271YjYJY has a Wishartdistribution Wm(n, E). Usually,Wishartdistributionsare studiedonlyforn > m- 1. ThispaperextendsthestudyofWishartas well as multivariate beta distributions to the singularcase, where0 < n < m, n an integer, thatis, wheretherankoftherandommatrixis belowits dimensionality.The usual conjugacybetweenWishartand beta distributions [see Muirhead(1982),Theorem3.3.1] is extendedto thiscase (see Theorems1 and 7). A volumeelementon the space ofpositivedefinitem x m matricesofrank n < m is introduced(see Theorem2) and sometransformation propertiesestablished(see Theorems3 and 4-this is necessaryin orderto talk sensibly about densitieson that space. The densityis foundforthe rank-nWishart distribution forall integersn, 0 < n < m (see Theorem6) and therank-1multivariatebeta distribution (see Theorem7). To do that,the Jacobianforthe transformation to the singularvalue decomposition ofgeneralm x n matrices is calculated(see Theorem5). This paperthusextendstheresultsestablished byFisher(1915),Wishart(1928),James(1954),Khatri(1959), Olkinand Roy (1954) and Olkin and Rubin(1964). Theirresultsare presentedclearlyand conciselyin bookformin Muirhead(1982): We willfollowhis terminology and formulations closely. The resultsin thispaperare,in particular, fundamental and usefulforupdatinga Bayesianposterior whentrackinga time-varying variance-covariance matrix:that,in fact,motivatedthisinvestigation [see Uhlig(1992)]. Imagine the followingtime series model,whichis a simplemultivariatestate space alternativeto the popularARCH models.[For an overviewofthe extensive ReceivedJuly1992;revisedJune1993. AMS 1991subjectclassifications. Primary62H10; secondary62E15. Key wordsand phrases.Wishartdistribution, beta distribution, Stiefelmanifold,singular matrixdistributions, conjugacy. 395
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H. UHLIG
396
ARCH literature,see Bollerslev,Chou and Kroner(1992). For a univariate see Shephard(1994).] Thereis an unobservableprestate space specification, cisionmatrixPt, evolvingovertimeaccordingto (1)
Pt =+u(Pt11)'Qtu(Pt p
l),
whereU(Pt-1) is the upper-triangular Choleskyfactor,that is, that uppertriangularmatrixT withpositivediagonalelementswhichsatisfiesPt-, = T'T. Suppose that the Qt are drawni.i.d. froma multivariatebeta distribua researcherstartswith the priorthat tion 3m(p/2,1/2).Suppose further, Pt-, - Wm(A-', AS7-'), whereA = 1/(p+ 1), so Pt-, is Wishartdistributed, thatE[Pt-1]-1= St_,.If the usual conjugacybetweenWishartand multivariate beta distributions holds,thenthepriorforPt is a WishartWm(p, ST-X1/p). Supposenowthattheresearcherobservesa singledrawYtfroma multivariate normaldistribution withthatprecisionmatrix,Yt - K(O,P7'). The posterior forPtis thengivenbya Wishart Wm(A-1, AS-'), where St = AYtYt+(l- A)St_j so that E[PtI-L = St and the game can begin anew. To findthe parameter p governingthe degreeof timevariation,the explicitlikelihoodfunctionis needed.The problemwiththeseargumentsis thatthe singularmultivariate beta distributions 1/2) have yet to be definedand the "usual conju!3m(p/2, gacy"betweenWishartand this multivariatebeta distribution has yetto be established.To do that,singularWishartdistributions have to be analyzedas well since theyare fundamentalforthe studyof singularmultivariatebeta in orderto statethelikelihoodfunction distributions. Furthermore, explicitly, the densityfunction fora L3m(p/2, randomvariableQ has yet 1/2)-distributed to be found,sinceIm- Q is ofrank1 and thusis singularalmostsurely.Solvingtheseproblemsis thepurposeofthispaper.In thecourseofdoingso,some generallyusefultheoremsforthe analysisofmultivariaterandomvariables are established. 2. Results. Unless statedotherwise, all ournotation,definitions and terfollow minology Muirhead(1982). First,we generalizethe definition ofmultivariatebeta distribution B1m(n/2,p/2) to integers0 < n < m. Let m and n be positiveintegers,p > m - 1 and E be ofsize m x m and positivedefinite.Recall Definition3.1.3 in Muirhead(1982), that a randomvariableA is ifA can be writtenas Wm(n, E)-distributed
A = E,Y)Y,
withYj, A(O,E) i.i.d.
For a positivedefinite matrixS, letU(S) denotetheupper-triangular Cholesky factor,thatis, thatupper-triangular matrixT withpositivediagonalelements whichsatisfiesS = T'T. DEFINITION 1. A randomvariableX is 3m(n/2,p/2)-distributed, if it can
be writtenas
X = U(A +B)'-'AU(A +B)-
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ON SINGULAR WISHART ANDBETADISTRIBUTIONS
397
whereA Wm(nf,Im) and B Wm(p,Im)withA represented as aboveforE = I and the Yj,j = 1, .. ., n, independentfromB. X is Bm(n/2,p/2)-distributed if A Wm( (p,Im,)and B - Wm(n, Im,)instead. This definition is moldedafterTheorem3.3.1 in Muirhead(1982) and is therefore containedas a special case in Definition3.3.2 in Muirhead(1982), forn > m - 1. However,forn < m - 1, this definition is new and Theorem 3.3.1 in Muirhead(1982) needsto be establishedfortheseparametersas well. This is done in the following theorem,whichis stated"backwards"fromthe versionin Muirhead(1982) to makeit particularly suitableforthepurposeof posteriorupdatingalludedto in the Introduction. THEOREM1. Let m and n be positiveintegersand letp > m - 1. Let H Then Wm(p+ n,E) and Q Bmf(p/2, n/2)be independent. '
G=-U(H)'QU(H) -Wm(P,2E) PROOF. The theoremfollowsfromthe following somewhatbroaderclaim: CLAIM. Let A
withYj Wm(pjIm),B = Ejn=YjYjl,
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 proofmimicsthe proofof Muirhead[(1982), Theorem3.3.1]. Define Zi = U(H)'U(C)'-'Yj and note that D = Ej lZjZj. It followsfromMuirhead [(1982),Theorem2.1.4] 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.Writingout the densities,it now followsthat (2mP/2rm(p/2))-l
etr(-A/2)(detA)(P-m-
? (2m(n+p)/2 rm ((n+p)/2)(detE)(n+p)/2)
1)/2
-1
etr(-E-'H/2)
(detH)(n+p-m-1)/2
? (27r)-mn/2 etr(-B/2) (dA) A (dH) A (dY1) A ... A (dYn) = (2m(n+P)/2Irm((n +p)/2))
-1
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 (detE) -n/2 etr(-D/2) (dC) A (dG) A (dZj) A ... A (dZn),
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398
exploiting detA= detC detQ, H = G + D and detH= detG/detQ. Inspecting thelatterdensity finishes theproof.0 ofa likelihood Forthecomputation Let m > n > 0 be integers. function, ona spaceofappropriate is needed.Densitiesdo say,a density dimensionality mx m notexistforV Wm(ln, E) orX B13m(p, n/2)onthespaceofsymmetric matrices, sinceV andIm- X are singularand ofrankn almostsurely[see 3.1.4].As shownbelow,however, densities doexist Muirhead (1982),Theorem manifold ofrank-n onthe(mn- n(n- 1)/2)-dimensional semidefinite positive S withn distinct denotethatmanifold m x m matrices positive eigenvalues; A coordinate natural for this manifold is to use the bySm+, global system ,n = x = L diag(1,... , ln)with S H1LH',whereL is n n,diagonal, decomposition 11> 12 > ... > 0 and whereH1 E Vn,m, the(mn- n(n+ 1)/2)-dimensional H1withorthonormal Stiefelmanifold ofm x n matrices columns, H'H1 = I, This parameterization is uniqueup to theassignment ofn arbitrary signs to the columnsofH1. The taskis to definethevolumeelement(dS): with the chosenparameterization, (dS) needsto be defined as somefunction of H1 and L multiplied with(Hl dH1) A Ai=1 dli. [We followMuirhead(1982), page56,inignoring signsoftheoveralldifferential anddefining onlypositive integrals. Forthedefinition of(H' dH1),seeMuirhead (1982),page63 andthe discussion following page67.]NotethatdS = dHj LH' + H1dLH' + HL dH'. Findan m x (m- n) matrix H2,so thatH _ [H1tH2] E O(m), thatis, so that H'H = Im,andlethi be thei-thcolumnofH. LetR = H' dSH andcalculate that [
R=H'dSH=
o]b
where
Ra H= dH1L + dL + (H dH,L)' di,
(11 -12)h/
(1 - 12)k2 dhl
(11-
n)hn dhi
dhl
d12
..
...
(I, -ln)hndhl-
(12- ln) h dh2
(12-1n)hn dh2
dln
[exploitingthe skew symmetry of H, dHj, see Muirhead(1982), bottomof page 64] and Rb =H dH,L llhn'+1dhl
...
lnh+1 Clhn
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ON SINGULARWISHARTAND BETA DISTRIBUTIONS
399
Appealingtotheanalogythat(dS) wouldbe theexterior productofall distinct, nonzeroentriesin R by Theorem2.1.6 in Muirhead(1982), ifS were a fullranksymmetric matrix,we define(dS) to be theexteriorproductofall entries on and belowthemaindiagonalofRa and ofall entriesinRbtimesa factor2-n to correctforthe factthat each matrixS is the image of2n decompositions H1LH' due to the arbitraryassignmentof signs to the columnsofH1. We therefore get the following theorem. THEOREM2. Let m > n > 0 be integers.On the space S., n ofpositive m x m matricesS ofrankn withn distinctpositiveeigenvalues, semidefinite thevolumeelement(dS) is
(2)
n
n
n
i=l
i 12>
We aim at expressingthe densitieswithrespectto this volumeelement subjectto the restriction that the densityis the same whenevertwo pairs (H1,L) and (H1,L) differ onlyin the assignmentofsignsto the columnsofH1 and H1. The following usefultheoremalso justifiesdefining (dS) in termsof the nonzeroand distinctentriesofR. THEOREM 3. Let X, Y E S.,n be relatedbyX = QYQ', whereQ E O(m). Then(dX) = (dY), where(dX) and (dY) are thevolumeelements 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 decompositionforX. Since (H' dHi) is invariantto multiplication on the leftwithan orthonormal m x m matrix,thatis, since (G'1dG1) = (H' dH,) [see Muirhead (1982),bottomofpage 69], the resultfollows.0
The following theoremis a versionofMuirhead'sTheorem2.1.6 [Muirhead (1982)] forthe case n = 1. The generalrank-ncase is an openproblem. THEOREM4. Let X, Y E Sm, 1 be relatedbyX = BYB', whereB is m x m and offull rank.Form the representations X = 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 normofthe m x 1 vectorconstituting G1),
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H. UHLIG
400
we have G1 = BH/ IIBHiII and K = IIBH1I2L,and thus
(3)
BH1 II=(K/L)1/2 = 1/j G'BH, =11
G
enablingexplicitcalculationofthe expressionin the theorem.We conjecture thatthe formulaforthe generalrank-ncase is givenby (dX) = det(G/BHi)m+l-n det(B)n(dY). PROOF OF THEOREM 4.
We firstshowthisforthe 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,thatis, G,H E 0(m). Let gi and hi denotethe ith columnsofG and H. Let E = G'DH and notethat G'dXG =EH'dYHE', where,forexample,
(H2'dhlL)' [dL 0 dh L
H'dYH= HH
Thus,the firstcolumnofE H'dYH E' has m
e1leildL + (elleij + eile11)hj dhl L j=2
as its ith entry.Takingthe exteriorproductoverall these entries,ignoring the overallsign fornow and using the abbreviation fil = ellei,, fij = elleij+ eilelj yields m
m\
(dx)=A 1dL +E i=l
=
j=2
E
aEI(m) =(det
fijhjdhiL) m
sgn(a)
Eia(i)
i=l
\m
Lm-ldLA
AhJdhl
j=2
F) (dY),
ofthe operatorA, where11(m)is the set ofall perusingthe skew symmetry mutations of (1, ... , m) [cf.Horn and Johnson(1985), page 8] and where F is
the matrix[fi=l,m The firstcolumnofF is thefirstcolumnofE, multipliedwithe1l = GIDH1, 0. Thejth columnofF is thesumofthejthcolumnofE multipliedwithell and the firstcolumnofF multipliedwithejl/el1.By the rules about calculating withdeterminants, it followsthat det F = det(eulE) = eTmdet E = eTmdet D.
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ON SINGULAR WISHART ANDBETADISTRIBUTIONS
401
Since (dX) has a positivesign,the absolutevalue ofell needs to be taken, demonstrating the claimforB = 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 previoustheoremand the proof above fordiagonalmatricesD, it followsthat (dX) = (dk) = (b'Dfi))m(det D)(dY) = (GIBHi)m(detB)(dY), as claimed. O The following theoremis an extremely usefulcousinofMuirhead'sTheorem 2.1.13 [Muirhead(1982), page 63]. The proofis not a straightforward generalizationofthe proofofthattheorem,but it proceedsalongsimilarlines. Let Z be an m x n (m > n) matrixof rank n and withdistincteigenvaluesof Z'Z. Using the nonsingularpart of the singularvalue decomposition, write Z = H1DP', whereH1 E Vn,m, D is diagonalwithD1l > D22 > ... > Dnn> 0 and P e 0(n): this decomposition is unique up to the arbitrary assignmentof signsto columnsofP as can be seen upon examinationof,forexample,Theorem7.3.5 and its proofin Hornand Johnson[(1985),page 414, withA = Z' there]. THEOREM5. Let Z be an m x n matrixand Z = H1DP' the nonsingular part ofthesingularvalue decomposition, whereH1 e Vn,m, D is diagonal with 0 Dui >D22 > * > Dnn> and P E0(n). Then
(4)
(dZ)=2
n(det D)m-n
J(Ds -D
)(HIdHj) A (dD) A (P dP),
i n > 0 be integers. m x m matriceswith of rank-npositivesemidefinite tionon the space Sm+,n distinctpositiveeigenvalueswithrespectto the volumeelement(dS) defined above is givenby
( )
~2mn/2rn (n/2)(detE)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 checkthatS has n distinctpositiveeigenvaluesalmost surely,and we will assume so fromhere onward.The densityforY is given by (10)
(27r mn/2 (det E)-n/2
etr(-E-S-1S2) (dY)
DecomposingY = H1DP' as in theprevioustheoremresultsin S = H1LH', the desiredparameterization, whereL =_D2 and li _ Lsi.Notethat n
n
n
Adli =2n]7Dii AdDii i=l
i=1
i=1
side of(4) and and that detD = (detL)1/2. Replacing(dY) by the right-hand over(P'dP) withCorollary2.1.16 in Muirhead(1982), one thereintegrating foreobtainsthe densitystatedin the theorem.O theoremis a versionofMuirhead'sTheorem3.3.1 [Muirhead The following (1982)] forthe case n = 1. The generalrank-ncase is an open problem.[The generalapproach.Show that U in Theorem7 refereesuggestedthe following + B)-1/2 and thatits momentgenhas thesame distribution as (A + BY'1/2A(A funchypergeometric eratingfunctioncan be writtenin termsofa confluent tionofmatrixargument;see Muirhead(1982). The matrixargumentwillbe a matrixofthe same (and thusreduced)rankas A. Reductionformulasin Herz ofU as a (1955) can thenbe used to rewritethe momentgeneratingfunction momentgeneratingfunction oflowerdimensionality.] THEOREM 7. Let m > 1 be an integerand letp > m - 1. Let A and B be whereA is Wm(1,E) and B is Wm(p,E). Put A + B = T'T, where independent, T is an upper-triangular m x m matrixwithpositivediagonalelements.Let U be them x m symmetric matrixdefinedbyA = T'UT. ThenA + B and U are
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H. UHLIG
404
A + B is Wm(p + 1,E) and thedensityfunction independent; of U on thespace S+ 1 withrespectto thevolumeelement(dU) on thisspace definedaboveis (11)
7r(-m+l)/2 Fm((p +
1)/2) L-m/2 det(Im
r7(1/2)rm(p/2) whereU = H,LH', H1 E Vl,mL E R.
-U)(p-m-)/2
We conjecturethatthe densityin the generalrank-ncase is givenby 7r(-mn+n2)/2
rm((p + n)/2)deL)nmU2dtIm-
rn(n/2)rm(p/2)
(pm12
(
(L)
-
PROOFOF THEOREM 7. The proofis almosta verbatimcopyofthe proofof Muirhead'sTheorem3.3.1 [Muirhead(1982)] and is statedhereforreasonsof A = G,KG', whereG1 E Vl,m,KE R. completeness.Find the representation The joint densityofA and B is 7r(-m+1)/22-m(p+1)/2 (detE)-(p+l)/2 etrt r(1/2)Fm(p/2)
K
-1(A +B) 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 uppertriangularwithpositivediagonalelements,and A = T'UT. Find the representation U = H,LH', H1 E Vl,m,L E DR.Theorem4 impliesthat (dA) A (dC) = (K/L)m/2(det T) (dU) A (dC), that T is a functionof C alone. Substitutinginto the density remembering above and collectingtermsyieldsthe desiredconclusion.O 3.3.2 in Muirhead(1982),we have the following. Analogouslyto Definition DEFINITION 2. A matrixU withdensityfunction(11) is said to have the multivariatebeta distribution withparameters1/2and p/2. Bm(1/2,p/2) A matrixV withdensityfunction(11) forU = Im- V is said to have the multivariate beta distribution Bm(p/2,1/2)withparametersp/2and 1/2(note thatone thenneeds to decomposeIm- V = H1LHI XH1 E Vi,m,L E R). It is clear fromTheorem7 that,forn = 1, Definition1 is a special case of Definition2. Furthermore, by readingTheorem7 "backwards"and switching the roles ofA and B, one obtainsanotherproofforTheorem1 forthe case n = 1.
Acknowledgment. I am gratefulforusefulcommentsfromthe referee.
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ON SINGULARWISHARTAND BETA DISTRIBUTIONS
405
REFERENCES BOLLERSLEV, T., CHOU,R. Y. and KRONER,K F. (1992). ARCH modelingin finance-a reviewof
52 5-59. thetheoryand empiricalevidence.J. Econometrics ofthevaluesofthecorrelation coefficient in samples R. A. (1915). Frequencydistribution froman indefinitely largepopulation. Biometrika 10 507-521. ofmatrixargument. Ann.ofMath.61 474-523. HERZ,C. S. (1955). Bessel functions HoRN,R. A. and JOHNSON, C. R. (1985). MatrixAnalysis.CambridgeUniv.Press. (Reprintedin 1987.) JAMES, A. T. (1954). Normalmultivariate analysisand theorthogonal group.Ann.Math.Statist. 25 40-75. ofcertainstatistics.Ann.Math.Statist.30 KHATRI, C. G. (1959). On the mutualindependence 1258-1262. StatisticalTheory.Wiley,NewYork. MUIRHEAD, R. J. (1982). AspectsofMultivariate OLKIN,I. and Roy,S. N. (1954). On multivariate distribution Ann.Math.Statist.25 329theory. 339. OLKIN, I. and RuBIN,H. (1964). Multivariate beta distributions and independence of properties theWishartdistribution. Ann.Math.Statist.35 261-269. to integrated SHEPHARD, N. G. (1994). Local scale models-statespace alternative GARCH processes.J. Econometrics 60 181-202. UHLIG, H. (1992). Bayesian vector autoregressions with stochasticvolatility.Unpublished manuscript. J.(1928). The generalizedproductmomentdistribution WISHART, in samplesfroma normalmultivariatepopulation.Biometrika 20A 32-43.
FISHER,
DEPARTMENTOF ECONOMICS PRINCETON UNIVERSITY PRINCETON, NEW JERSEY 08544
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The Annals of Statistics
1994,Vol.22,No. 1, 395-405
ON SINGULAR WISHART AND SINGULAR MULTIVARIATE BETA DISTRIBUTIONS BY HARALD UHLIG
PrincetonUniversity This paperextendsthe studyofWishartand multivariate beta distributionsto the singularcase, wherethe rankis belowthe dimensionality. The usual conjugacyis extendedto this case. A volumeelementon the space ofpositivesemidefinite m x m matricesofrankn < m is introduced and some transformation established.The densityfunction properties is foundforall rank-nWishartdistributions as wellas therank-1multivariTo do that,the Jacobianforthe transformation ate beta distribution. to the singularvalue decomposition ofgeneralm x n matricesis calculated. The resultsin thispaperare usefulin particularforupdatinga Bayesian posterior whentrackinga time-varying variance-covariance matrix.
1. Introduction. Considern drawsYj,j = 1, . . ,n, froma normaldistributionAf(O, E), whereE is m x m and positivedefinite. The randomvariable X = Y271YjYJY has a Wishartdistribution Wm(n, E). Usually,Wishartdistributionsare studiedonlyforn > m- 1. ThispaperextendsthestudyofWishartas well as multivariate beta distributions to the singularcase, where0 < n < m, n an integer, thatis, wheretherankoftherandommatrixis belowits dimensionality.The usual conjugacybetweenWishartand beta distributions [see Muirhead(1982),Theorem3.3.1] is extendedto thiscase (see Theorems1 and 7). A volumeelementon the space ofpositivedefinitem x m matricesofrank n < m is introduced(see Theorem2) and sometransformation propertiesestablished(see Theorems3 and 4-this is necessaryin orderto talk sensibly about densitieson that space. The densityis foundforthe rank-nWishart distribution forall integersn, 0 < n < m (see Theorem6) and therank-1multivariatebeta distribution (see Theorem7). To do that,the Jacobianforthe transformation to the singularvalue decomposition ofgeneralm x n matrices is calculated(see Theorem5). This paperthusextendstheresultsestablished byFisher(1915),Wishart(1928),James(1954),Khatri(1959), Olkinand Roy (1954) and Olkin and Rubin(1964). Theirresultsare presentedclearlyand conciselyin bookformin Muirhead(1982): We willfollowhis terminology and formulations closely. The resultsin thispaperare,in particular, fundamental and usefulforupdatinga Bayesianposterior whentrackinga time-varying variance-covariance matrix:that,in fact,motivatedthisinvestigation [see Uhlig(1992)]. Imagine the followingtime series model,whichis a simplemultivariatestate space alternativeto the popularARCH models.[For an overviewofthe extensive ReceivedJuly1992;revisedJune1993. AMS 1991subjectclassifications. Primary62H10; secondary62E15. Key wordsand phrases.Wishartdistribution, beta distribution, Stiefelmanifold,singular matrixdistributions, conjugacy. 395
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396
ARCH literature,see Bollerslev,Chou and Kroner(1992). For a univariate see Shephard(1994).] Thereis an unobservableprestate space specification, cisionmatrixPt, evolvingovertimeaccordingto (1)
Pt =+u(Pt11)'Qtu(Pt p
l),
whereU(Pt-1) is the upper-triangular Choleskyfactor,that is, that uppertriangularmatrixT withpositivediagonalelementswhichsatisfiesPt-, = T'T. Suppose that the Qt are drawni.i.d. froma multivariatebeta distribua researcherstartswith the priorthat tion 3m(p/2,1/2).Suppose further, Pt-, - Wm(A-', AS7-'), whereA = 1/(p+ 1), so Pt-, is Wishartdistributed, thatE[Pt-1]-1= St_,.If the usual conjugacybetweenWishartand multivariate beta distributions holds,thenthepriorforPt is a WishartWm(p, ST-X1/p). Supposenowthattheresearcherobservesa singledrawYtfroma multivariate normaldistribution withthatprecisionmatrix,Yt - K(O,P7'). The posterior forPtis thengivenbya Wishart Wm(A-1, AS-'), where St = AYtYt+(l- A)St_j so that E[PtI-L = St and the game can begin anew. To findthe parameter p governingthe degreeof timevariation,the explicitlikelihoodfunctionis needed.The problemwiththeseargumentsis thatthe singularmultivariate beta distributions 1/2) have yet to be definedand the "usual conju!3m(p/2, gacy"betweenWishartand this multivariatebeta distribution has yetto be established.To do that,singularWishartdistributions have to be analyzedas well since theyare fundamentalforthe studyof singularmultivariatebeta in orderto statethelikelihoodfunction distributions. Furthermore, explicitly, the densityfunction fora L3m(p/2, randomvariableQ has yet 1/2)-distributed to be found,sinceIm- Q is ofrank1 and thusis singularalmostsurely.Solvingtheseproblemsis thepurposeofthispaper.In thecourseofdoingso,some generallyusefultheoremsforthe analysisofmultivariaterandomvariables are established. 2. Results. Unless statedotherwise, all ournotation,definitions and terfollow minology Muirhead(1982). First,we generalizethe definition ofmultivariatebeta distribution B1m(n/2,p/2) to integers0 < n < m. Let m and n be positiveintegers,p > m - 1 and E be ofsize m x m and positivedefinite.Recall Definition3.1.3 in Muirhead(1982), that a randomvariableA is ifA can be writtenas Wm(n, E)-distributed
A = E,Y)Y,
withYj, A(O,E) i.i.d.
For a positivedefinite matrixS, letU(S) denotetheupper-triangular Cholesky factor,thatis, thatupper-triangular matrixT withpositivediagonalelements whichsatisfiesS = T'T. DEFINITION 1. A randomvariableX is 3m(n/2,p/2)-distributed, if it can
be writtenas
X = U(A +B)'-'AU(A +B)-
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ON SINGULAR WISHART ANDBETADISTRIBUTIONS
397
whereA Wm(nf,Im) and B Wm(p,Im)withA represented as aboveforE = I and the Yj,j = 1, .. ., n, independentfromB. X is Bm(n/2,p/2)-distributed if A Wm( (p,Im,)and B - Wm(n, Im,)instead. This definition is moldedafterTheorem3.3.1 in Muirhead(1982) and is therefore containedas a special case in Definition3.3.2 in Muirhead(1982), forn > m - 1. However,forn < m - 1, this definition is new and Theorem 3.3.1 in Muirhead(1982) needsto be establishedfortheseparametersas well. This is done in the following theorem,whichis stated"backwards"fromthe versionin Muirhead(1982) to makeit particularly suitableforthepurposeof posteriorupdatingalludedto in the Introduction. THEOREM1. Let m and n be positiveintegersand letp > m - 1. Let H Then Wm(p+ n,E) and Q Bmf(p/2, n/2)be independent. '
G=-U(H)'QU(H) -Wm(P,2E) PROOF. The theoremfollowsfromthe following somewhatbroaderclaim: CLAIM. Let A
withYj Wm(pjIm),B = Ejn=YjYjl,
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 proofmimicsthe proofof Muirhead[(1982), Theorem3.3.1]. Define Zi = U(H)'U(C)'-'Yj and note that D = Ej lZjZj. It followsfromMuirhead [(1982),Theorem2.1.4] 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.Writingout the densities,it now followsthat (2mP/2rm(p/2))-l
etr(-A/2)(detA)(P-m-
? (2m(n+p)/2 rm ((n+p)/2)(detE)(n+p)/2)
1)/2
-1
etr(-E-'H/2)
(detH)(n+p-m-1)/2
? (27r)-mn/2 etr(-B/2) (dA) A (dH) A (dY1) A ... A (dYn) = (2m(n+P)/2Irm((n +p)/2))
-1
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 (detE) -n/2 etr(-D/2) (dC) A (dG) A (dZj) A ... A (dZn),
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398
exploiting detA= detC detQ, H = G + D and detH= detG/detQ. Inspecting thelatterdensity finishes theproof.0 ofa likelihood Forthecomputation Let m > n > 0 be integers. function, ona spaceofappropriate is needed.Densitiesdo say,a density dimensionality mx m notexistforV Wm(ln, E) orX B13m(p, n/2)onthespaceofsymmetric matrices, sinceV andIm- X are singularand ofrankn almostsurely[see 3.1.4].As shownbelow,however, densities doexist Muirhead (1982),Theorem manifold ofrank-n onthe(mn- n(n- 1)/2)-dimensional semidefinite positive S withn distinct denotethatmanifold m x m matrices positive eigenvalues; A coordinate natural for this manifold is to use the bySm+, global system ,n = x = L diag(1,... , ln)with S H1LH',whereL is n n,diagonal, decomposition 11> 12 > ... > 0 and whereH1 E Vn,m, the(mn- n(n+ 1)/2)-dimensional H1withorthonormal Stiefelmanifold ofm x n matrices columns, H'H1 = I, This parameterization is uniqueup to theassignment ofn arbitrary signs to the columnsofH1. The taskis to definethevolumeelement(dS): with the chosenparameterization, (dS) needsto be defined as somefunction of H1 and L multiplied with(Hl dH1) A Ai=1 dli. [We followMuirhead(1982), page56,inignoring signsoftheoveralldifferential anddefining onlypositive integrals. Forthedefinition of(H' dH1),seeMuirhead (1982),page63 andthe discussion following page67.]NotethatdS = dHj LH' + H1dLH' + HL dH'. Findan m x (m- n) matrix H2,so thatH _ [H1tH2] E O(m), thatis, so that H'H = Im,andlethi be thei-thcolumnofH. LetR = H' dSH andcalculate that [
R=H'dSH=
o]b
where
Ra H= dH1L + dL + (H dH,L)' di,
(11 -12)h/
(1 - 12)k2 dhl
(11-
n)hn dhi
dhl
d12
..
...
(I, -ln)hndhl-
(12- ln) h dh2
(12-1n)hn dh2
dln
[exploitingthe skew symmetry of H, dHj, see Muirhead(1982), bottomof page 64] and Rb =H dH,L llhn'+1dhl
...
lnh+1 Clhn
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ON SINGULARWISHARTAND BETA DISTRIBUTIONS
399
Appealingtotheanalogythat(dS) wouldbe theexterior productofall distinct, nonzeroentriesin R by Theorem2.1.6 in Muirhead(1982), ifS were a fullranksymmetric matrix,we define(dS) to be theexteriorproductofall entries on and belowthemaindiagonalofRa and ofall entriesinRbtimesa factor2-n to correctforthe factthat each matrixS is the image of2n decompositions H1LH' due to the arbitraryassignmentof signs to the columnsofH1. We therefore get the following theorem. THEOREM2. Let m > n > 0 be integers.On the space S., n ofpositive m x m matricesS ofrankn withn distinctpositiveeigenvalues, semidefinite thevolumeelement(dS) is
(2)
n
n
n
i=l
i 12>
We aim at expressingthe densitieswithrespectto this volumeelement subjectto the restriction that the densityis the same whenevertwo pairs (H1,L) and (H1,L) differ onlyin the assignmentofsignsto the columnsofH1 and H1. The following usefultheoremalso justifiesdefining (dS) in termsof the nonzeroand distinctentriesofR. THEOREM 3. Let X, Y E S.,n be relatedbyX = QYQ', whereQ E O(m). Then(dX) = (dY), where(dX) and (dY) are thevolumeelements 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 decompositionforX. Since (H' dHi) is invariantto multiplication on the leftwithan orthonormal m x m matrix,thatis, since (G'1dG1) = (H' dH,) [see Muirhead (1982),bottomofpage 69], the resultfollows.0
The following theoremis a versionofMuirhead'sTheorem2.1.6 [Muirhead (1982)] forthe case n = 1. The generalrank-ncase is an openproblem. THEOREM4. Let X, Y E Sm, 1 be relatedbyX = BYB', whereB is m x m and offull rank.Form the representations X = 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 normofthe m x 1 vectorconstituting G1),
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400
we have G1 = BH/ IIBHiII and K = IIBH1I2L,and thus
(3)
BH1 II=(K/L)1/2 = 1/j G'BH, =11
G
enablingexplicitcalculationofthe expressionin the theorem.We conjecture thatthe formulaforthe generalrank-ncase is givenby (dX) = det(G/BHi)m+l-n det(B)n(dY). PROOF OF THEOREM 4.
We firstshowthisforthe 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,thatis, G,H E 0(m). Let gi and hi denotethe ith columnsofG and H. Let E = G'DH and notethat G'dXG =EH'dYHE', where,forexample,
(H2'dhlL)' [dL 0 dh L
H'dYH= HH
Thus,the firstcolumnofE H'dYH E' has m
e1leildL + (elleij + eile11)hj dhl L j=2
as its ith entry.Takingthe exteriorproductoverall these entries,ignoring the overallsign fornow and using the abbreviation fil = ellei,, fij = elleij+ eilelj yields m
m\
(dx)=A 1dL +E i=l
=
j=2
E
aEI(m) =(det
fijhjdhiL) m
sgn(a)
Eia(i)
i=l
\m
Lm-ldLA
AhJdhl
j=2
F) (dY),
ofthe operatorA, where11(m)is the set ofall perusingthe skew symmetry mutations of (1, ... , m) [cf.Horn and Johnson(1985), page 8] and where F is
the matrix[fi=l,m The firstcolumnofF is thefirstcolumnofE, multipliedwithe1l = GIDH1, 0. Thejth columnofF is thesumofthejthcolumnofE multipliedwithell and the firstcolumnofF multipliedwithejl/el1.By the rules about calculating withdeterminants, it followsthat det F = det(eulE) = eTmdet E = eTmdet D.
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ON SINGULAR WISHART ANDBETADISTRIBUTIONS
401
Since (dX) has a positivesign,the absolutevalue ofell needs to be taken, demonstrating the claimforB = 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 previoustheoremand the proof above fordiagonalmatricesD, it followsthat (dX) = (dk) = (b'Dfi))m(det D)(dY) = (GIBHi)m(detB)(dY), as claimed. O The following theoremis an extremely usefulcousinofMuirhead'sTheorem 2.1.13 [Muirhead(1982), page 63]. The proofis not a straightforward generalizationofthe proofofthattheorem,but it proceedsalongsimilarlines. Let Z be an m x n (m > n) matrixof rank n and withdistincteigenvaluesof Z'Z. Using the nonsingularpart of the singularvalue decomposition, write Z = H1DP', whereH1 E Vn,m, D is diagonalwithD1l > D22 > ... > Dnn> 0 and P e 0(n): this decomposition is unique up to the arbitrary assignmentof signsto columnsofP as can be seen upon examinationof,forexample,Theorem7.3.5 and its proofin Hornand Johnson[(1985),page 414, withA = Z' there]. THEOREM5. Let Z be an m x n matrixand Z = H1DP' the nonsingular part ofthesingularvalue decomposition, whereH1 e Vn,m, D is diagonal with 0 Dui >D22 > * > Dnn> and P E0(n). Then
(4)
(dZ)=2
n(det D)m-n
J(Ds -D
)(HIdHj) A (dD) A (P dP),
i n > 0 be integers. m x m matriceswith of rank-npositivesemidefinite tionon the space Sm+,n distinctpositiveeigenvalueswithrespectto the volumeelement(dS) defined above is givenby
( )
~2mn/2rn (n/2)(detE)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 checkthatS has n distinctpositiveeigenvaluesalmost surely,and we will assume so fromhere onward.The densityforY is given by (10)
(27r mn/2 (det E)-n/2
etr(-E-S-1S2) (dY)
DecomposingY = H1DP' as in theprevioustheoremresultsin S = H1LH', the desiredparameterization, whereL =_D2 and li _ Lsi.Notethat n
n
n
Adli =2n]7Dii AdDii i=l
i=1
i=1
side of(4) and and that detD = (detL)1/2. Replacing(dY) by the right-hand over(P'dP) withCorollary2.1.16 in Muirhead(1982), one thereintegrating foreobtainsthe densitystatedin the theorem.O theoremis a versionofMuirhead'sTheorem3.3.1 [Muirhead The following (1982)] forthe case n = 1. The generalrank-ncase is an open problem.[The generalapproach.Show that U in Theorem7 refereesuggestedthe following + B)-1/2 and thatits momentgenhas thesame distribution as (A + BY'1/2A(A funchypergeometric eratingfunctioncan be writtenin termsofa confluent tionofmatrixargument;see Muirhead(1982). The matrixargumentwillbe a matrixofthe same (and thusreduced)rankas A. Reductionformulasin Herz ofU as a (1955) can thenbe used to rewritethe momentgeneratingfunction momentgeneratingfunction oflowerdimensionality.] THEOREM 7. Let m > 1 be an integerand letp > m - 1. Let A and B be whereA is Wm(1,E) and B is Wm(p,E). Put A + B = T'T, where independent, T is an upper-triangular m x m matrixwithpositivediagonalelements.Let U be them x m symmetric matrixdefinedbyA = T'UT. ThenA + B and U are
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H. UHLIG
404
A + B is Wm(p + 1,E) and thedensityfunction independent; of U on thespace S+ 1 withrespectto thevolumeelement(dU) on thisspace definedaboveis (11)
7r(-m+l)/2 Fm((p +
1)/2) L-m/2 det(Im
r7(1/2)rm(p/2) whereU = H,LH', H1 E Vl,mL E R.
-U)(p-m-)/2
We conjecturethatthe densityin the generalrank-ncase is givenby 7r(-mn+n2)/2
rm((p + n)/2)deL)nmU2dtIm-
rn(n/2)rm(p/2)
(pm12
(
(L)
-
PROOFOF THEOREM 7. The proofis almosta verbatimcopyofthe proofof Muirhead'sTheorem3.3.1 [Muirhead(1982)] and is statedhereforreasonsof A = G,KG', whereG1 E Vl,m,KE R. completeness.Find the representation The joint densityofA and B is 7r(-m+1)/22-m(p+1)/2 (detE)-(p+l)/2 etrt r(1/2)Fm(p/2)
K
-1(A +B) 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 uppertriangularwithpositivediagonalelements,and A = T'UT. Find the representation U = H,LH', H1 E Vl,m,L E DR.Theorem4 impliesthat (dA) A (dC) = (K/L)m/2(det T) (dU) A (dC), that T is a functionof C alone. Substitutinginto the density remembering above and collectingtermsyieldsthe desiredconclusion.O 3.3.2 in Muirhead(1982),we have the following. Analogouslyto Definition DEFINITION 2. A matrixU withdensityfunction(11) is said to have the multivariatebeta distribution withparameters1/2and p/2. Bm(1/2,p/2) A matrixV withdensityfunction(11) forU = Im- V is said to have the multivariate beta distribution Bm(p/2,1/2)withparametersp/2and 1/2(note thatone thenneeds to decomposeIm- V = H1LHI XH1 E Vi,m,L E R). It is clear fromTheorem7 that,forn = 1, Definition1 is a special case of Definition2. Furthermore, by readingTheorem7 "backwards"and switching the roles ofA and B, one obtainsanotherproofforTheorem1 forthe case n = 1.
Acknowledgment. I am gratefulforusefulcommentsfromthe referee.
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ON SINGULARWISHARTAND BETA DISTRIBUTIONS
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DEPARTMENTOF ECONOMICS PRINCETON UNIVERSITY PRINCETON, NEW JERSEY 08544
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