Graphics Reference
In-Depth Information
Local estimation:
Now compute new local MLE estimates θ
(
k
)
i
of θ
(
X
i
)
as
θ
(
k
)
i
(
k
)
i
N
(
k
)
i
N
(
k
)
i
(
k
)
ij
, S
(
k
)
i
(
k
)
ij
=
S
with
=
j
w
=
j
w
Y
j
.
(
k
)
i
Adaptive control:
Compute the memory parameter as η
i
=
K
mem
(
m
)
. Define
θ
(
k
)
i
η
i
θ
(
k
)
i
θ
(
k
−)
i
=
+(
−
η
i
)
and
(
k
)
i
η
i
N
(
k
)
i
(
k
−)
i
N
η
i
N
=
+(
−
)
Stopping:
Stop if h
(
k
)
h
max
, otherwise set h
(
k
)
c
h
h
(
k
−)
,increasek by
, and
=
continue with the adaptation step.
Choice of Parameters: Propagation Condition
8.2.2
he proposed procedure involves several parameters. he most important one is the
scale parameter λ in the statistical penalty
s
ij
. he special case λ
simply leads
to a kernel estimate with bandwidth h
max
. We propose to choose λ as the smallest
value satisfying a propagation condition. his condition requires that, if the local
assumption is valid globally (i.e., θ
=
(
x
)
θ does not depend on x), then with high
probability the final estimate for h
max
coincides at every point with the global
estimate. More formally we request that, in this case, for each iteration k,
=
θ
(
k
)
θ
(
k
)
θ
(
k
)
E
(
X
)−
(
X
) <
α
E
(
X
)−
θ
(
.
)
for a specified constant α
. Here
θ
(
k
)
(
k
)
ij
(
k
)
ij
(
X
i
)=
j
K
loc
(
l
)
Y
j
j
K
loc
(
l
)
(
.
)
denotes the nonadaptive kernel estimate employing the bandwidth h
(
k
)
from step
k.hevalueλ provided by this condition does not depend on the unknown model
parameter θ and can therefore be approximately found by simulation. his allows
us to select default values for λ depending on the specified family of the probability
distribution
P=(
P
θ
, θ
Θ
)
. Default values for λ in the examples below are selected
for a value of α
.
.
he second parameter of interest is the maximal bandwidth h
max
, which controls
both the numerical complexity of the algorithm and the smoothness within homo-
geneous regions.
he scale parameter τ in the memory penalty
m
i
canalsobechosentomeetthe
propagation condition (
.
). he special case τ
=
=
turns off the adaptive control
step.
Additionally we specify a number of parameters and kernel functions that have
less influence on the resulting estimates. As a default, the kernel functions are cho-
sen as K
loc
x
e
−
x
I
x
<
.Ifthedesignison
(
x
)=
K
mem
(
x
)=(
−
)
+
and K
stat
(
x
)=
