
58 Modeling and In verse Problems in the Presence of Uncertainty
We can obtain the standard errors SE
k
(
ˆ
θ
OLS
) (discussed in more detail below)
for the kth element of
ˆ
θ
OLS
by calculating SE
k
(
ˆ
θ
OLS
) ≈
q
ˆ
Σ
N
kk
.
We note that (3.18) represents the estimate for σ
2
0
of (3.13) with the factor
1
N
replaced by the factor
1
N − κ
θ
. In the linear case the estimate with
1
N
can
be shown to be biased downward (i.e., biased too low) a nd the same behavior
can be observed in the ge ne ral nonlinear case – se e Chapter 12 of [35] and p.
28 of [22]. The subtraction of κ
θ
degrees of freedom reflects the fact that
ˆ
θ
has been computed to satisfy the κ
θ
normal equations (3.11). We also ...