Geoscience Reference
In-Depth Information
ˆ
ˆ
T
The estimation of the error variance
is therefore
V
=
E
[(
θ
−
θ
)(
θ
−
θ
)
]
θθ
approximately given by:
+
TT
+
+
+
T
T
VEF vF
~(
)~
FVF
,
where
VE v
νν
=
(
)
θθ
θ
θ
θ
vv
θ
F
by finite difference, by carrying out
n
simulations successively varying each
component:
To be able to use this expression, we determine the Jacobian matrix
θ
ˆ
ˆ
.
According to the ergodicity hypothesis
, the
θ
→
θ
+
δθ
i
i
i
variance matrix
can be calculated using the autocorrelation function of
V
vv
ˆ
the residuals
. This gives:
v
ˆ
=
y
−
f
(
θ
)
1
N
∑
, where
Vij
(, )~
vv
ˆˆ
ˆ
when
v
=
0
(
t
+
i
−
j
)
∉
[
N
]
+
i
−
j
vv
t
t
+−
i
j
N
t
=
1
6.4. Hypothesis test and confidence regions
Given the parametric vector
of dimension
n
, a supposedly normal
θ
ˆ
estimator
θ
and its variance matrix
, the variable
D
(
deviance
)
V
θθ
ˆ
ˆ
2
T
−
1
complies with a law of
χ
to
n
degrees of freedom.
D
=
(
θ
−
θ
)
V
(
θ
−
θ
)
θθ
ˆ
with probability
P
is where
D
is below a
threshold
S
, given by the tables of
The confidence region around
θ
2
χ
. As a result, for
P =
90% and
n
= 6,
tables give
S =
10.65.
This means that we are able to carry out hypothesis tests. Therefore, to
test the assumption
S
2
< 1.62, we will perform the identification under this
constraint, and calculate
D
, where
ˆ
and
θ
result from free identification,
V
θθ
and
from forced identification by the constraint. The hypothesis will be
rejected at likelihood
P
if
D > S
.
θ
We can also compare the relative likelihood of two hypotheses, by
comparing the deviance
D
of two estimations in relation to the same
reference created by free estimation.
ˆ
and
ˆ
If we extract from
θ
a given parameter
and its variance
V
θθ
,
θ
V
i
θθ
i
the confidence levels are as follows:
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