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residuals. The estimated values are usually called
most likelihood parameter
values
or the
least-squares parameter values
.
The maximum likelihood method applied to the ARMA(
p
,
q
) process with the
sampled values arranged as the components of the vector
y
and with the
]
T
[
yy
,
...,
1 ,
n
non-zero mean
µ
starts with the extended model
p
q
Y
PD P
(
Y
)
z
E
z
¦
¦
i
i
t
i
t
j
t
j
i
1
j
1
with
p+q
+2 parameters
, and
A matrix V(
Į
,
ȕ
)
DEV
,
2
VAR( )
z
P H
{}.
Y
i
j
,
should now be defined so that the relation
2
VAR(
Y
) =
V
VDE,
,
holds, where
T
DDDD
[, , ]
12
p
T
EEEE
[ , ,..., ]
[, , ]
[
12
q
yyy y
YYY Y
T
12
n
T
,
,...,
]
12
n
with the elements of
V
(
Į
,
ȕ
) being proportional to the autocorrelation coefficients
of {
Y
}.
Supposing now that
i
z
values are normally distributed, so will
Y
also be
normally distributed, so that the log-likelihood function will be defined by
1
2
2
T
1
2
L
(, ,, )
PVDE
[log
n
V
log (, ) (
V
D E
y
P
I
)[(, ](
V
D E
y
P V
I
)/
]
2
[1, 1, ..., 1]
T
where
I
is the identity vector
I
.
Given initial values of
Į
and
ȕ
, the maximum estimate of
µ
and
V
are
PDE
ˆ (, ) { [(, )] }/{ [(, ] }
1
IV
T
DE
1
y IV
T
DE
1
I
2
ˆ
T
1
ˆ
VDE
(, )
[
y
PDE
(, )]/{[(, ][
I
V
DE
y
PDE
(, )]
I
n
This, after substituting in the above likelihood equation, gives
1
2
L
(, )
DE
[log
n
VDE
(, ) log (, ]
V
DE
.
0
2
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