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

44 R.P. Barry, R.K. Pace / Linear Algebra and its Applications 289 (1999) 41±54
where
s
na m‡1
F ˆ
…m ‡ 1†…1 a† ‡ 1:96 s 2 …V 1 ; . . . ; V p †
:
p
The interval …V F ; V ‡ F † is an asymptotic 95% con®dence interval for
log det…I aD†. We select the ``tuning constants'' m and p to give the desired
degree of approximation …F †.
Proof. By the triangle inequality:
jV log det…I aD†j 6 jV EV j ‡ jEV log det…I aD†j:
The sampling distribution of a mean of independent, ®nite variance random
variables gives us
s !
s
P jV EV j 6 1:96
2 …V 1 ; . . . ; V p †
0:95:
p
The bound for the term jEV log det…I aD†j is given in the next theorem.
(
Theorem
na m‡1
jEV log det…I aD†j 6
…m ‡ 1†…1 a† :
Proof. Start with the power series expansion of the matrix function
log…I aD†:
log…I aD† ˆ X1 D k a k
ˆ Xm D k a k
‡ X1 D k a k
:
k k
k
kˆ1
Now, the trace of log…I aD† is
X n
iˆ1
kˆ1
log…1 ak D;i † ˆ log det…I aD†:
kˆm‡1
The nice property tr…cA† ‡ tr…dB† ˆ tr…cA ‡ dB† has the consequence:
log det…I aD† ˆ tr…log…I aD†† ˆ Xm
6 Xm
kˆ1
tr…D k †a k
k
‡ X1
kˆm‡1
kˆ1
tr…D k †a k
k
tr…D k †a k
m ‡ 1 :
‡ X1
kˆm‡1
tr…D k †a k
k
The expansion of log det…I aD† in terms of the trace of D k
expansion (Martin, 1993).
is the Martin