
Model Selection Criteria 137
over the set Ω
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