VerifytheWishartevolutiondistributiontheoryofSection10.4.8inthe followinggeneralsetting.

Theq qprecisionmatrix hastheWishartdistribution W(hA)

for somedegreesof freedomh=n+q 1wheren>0sothath>

q 1 andwhereAisthe inversesum-of-squaresmatrix.TheBartlett decomposition,oftenusedforsimulationofWishartmatricesaswellas theoreticaldevelopments (e.g.,Odell andFeiveson1966), is that =

PUUPwherePistheuppertriangularCholeskycomponentofAso thatA=PP,and U=

u11 u12 u13 u1q 0 u22 u23 u2q 0 0 u33 u3q 0 0 0 uqq wherethenonzeroentriesareindependentrandomquantitieswithuij N(01) for 1 i0andUis anupper triangular matrixwiththesameo-diagonalelementsasU,anddiagonalelements uii= iwhere i= i iwith i Be( i(h i+1) 2 (1 i)(h i+

1) 2) forsomeconstants i (01) for i=1:q The i aremutually independent,andindependentofthenormaluij Showthat, for i=1 : q i 2 i(h i+1) independentlyover iand independentlyoftheupper,o-diagonalelementsofU Deducethat W(k b 1A)when i=(k+i 1) (h+i 1) for eachi=1 q andwhere0

Identifythevaluesof(k b)andthe i ineachofthesespecialcases.