"Monte Carlo Estimates of the Log Determinant of ... - Spatial Statistics

52 R.P. Barry, R.K. Pace / Linear Algebra and its Applications 289 (1999) 41±54
x 0 Ax ˆ z 2 1 k A;i ‡ ‡ z 2 n k A;i;
where …z 1 ; . . . ; z n † are N…0; I†. Then
x 0 Ax
x 0 x ˆ z2 1 k A;1 ‡ ‡ z 2 n k A;n
z 2 1 ‡ ‡ ;
z2 n
where …z 1 ; . . . ; z n † is N…0; I†. This fraction can be rewritten
z 2 1
P n k A;1 ‡ z2 2
P n
iˆ1 z2 i
iˆ1 z2 i
k A;2 ‡ ‡
z2 n
P n
iˆ1 z2 i
k A;n :
As each z 2 i
is chi-square with one degree of freedom, and z 2 i
is independent of z 2 j
for i 6ˆ j, the coecients W 1 ; . . . ; W n have the Dirichlet…1=2† distribution
(Johnson and Kotz, 1972, p. 231).
The mean, variance and covariance of n-variate Dirichlet…1=2† random
variables are:
EW i ˆ 1
n ; Var…W …1=2†…n=2 1=2† 2…n 1†
i† ˆ ˆ
…n=2† 2 …n=2 ‡ 1† n 2 …n ‡ 2† ;
2
Cov…W i ; W j † ˆ
n 2 …n ‡ 2†
(Johnson and Kotz, 1972, p. 233). For additional information on the Dirichlet
distribution, see Narayanan (1990). (
Result 2. For any real n n matrix A and x N n …0; I†
E x0 Ax
x 0 x ˆ tr…A†=n:
Proof. For A symmetric we can use Result 1 to get
E x0 Ax
x 0 x ˆ EW 1k A;1 ‡ ‡ EW n k A;n ˆ …k A;1 ‡ ‡ k A;n †=n ˆ tr…A†=n
If A is not symmetric, then consider B ˆ …1=2†…A t ‡ A†. Clearly B is symmetric
and has the same trace as A. However, for any vector c; c 0 Ac ˆ c 0 Bc. Thus
x 0 Ax=x 0 x must have the same distribution as x 0 Bx=x 0 x and therefore is an
unbiased estimator of tr…A†=n. (
Result 3. For real symmetric A,
Var x0 Ax
x 0 x ˆ 2VAR…k† ;
n ‡ 2
where VAR…k† is the population variance of the eigenvalues of A: