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Modeling and Inverse Problems in the Presence of Uncertainty
book

Modeling and Inverse Problems in the Presence of Uncertainty

by H. T. Banks, Shuhua Hu, W. Clayton Thompson
April 2014
Intermediate to advanced content levelIntermediate to advanced
405 pages
13h
English
Chapman and Hall/CRC
Content preview from Modeling and Inverse Problems in the Presence of Uncertainty
Model Selection Criteria 137
over the set
q
to obtain the ordinary le ast squares e stimate
ˆ
q
OLS
. Hence, we
have
ˆ
q
OLS
=
ˆ
q
MLE
.
Note that
ln(L(θ|y))
∂σ
=
P
ν
i=1
N
i
σ
+
P
ν
i=1
P
N
i
j=1
(y
ij
f
i
(t
j
; q))
2
σ
3
.
Hence, by the above equation we see that the maximum likelihood estimate
ˆσ
MLE
of σ is given by
ˆσ
2
MLE
=
P
ν
i=1
P
N
i
j=1
(y
ij
f
i
(t
j
;
ˆ
q
MLE
))
2
P
ν
i=1
N
i
.
Substituting
ˆ
q
MLE
for q and ˆσ
MLE
for σ in (4.29) yields
ln(L(
ˆ
θ
MLE
|y)) =
P
ν
i=1
N
i
2
ln (2π)
P
ν
i=1
N
i
2
ν
X
i=1
N
i
2
!
ln
P
ν
i=1
P
N
i
j=1
(y
ij
f
i
(t
j
;
ˆ
q
MLE
))
2
P
ν
i=1
N
i
!
.
Then by the above equation and (4.12) we find that the AIC value is given by
AIC =
ν
X
i=1
N
i
!
ln
P
ν
i=1
P
N
i
j=1
(y
ij
f
i
(t
j
;
ˆ
q
MLE
))
2
P
ν
i=1
N
i
!
+
ν
X
i=1
N
i
[1 + ln (2π)] + 2(κ
q
+ 1).
Note
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Publisher Resources

ISBN: 9781482206432