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θ
k
e
−
θ
(
X
i
)
k!. Such a situation frequently occurs in low-
intensity imaging, e.g., confocal microscopy and positron emission tomography. It
also serves as an approximation of the density model, obtained by a binning proce-
dure.
Y
i
k
X
i
X
i
P
(
=
)=
(
)
Example
4:
color images In color images, Y
i
denotes a vector of values in a three-
dimensionalcolorspaceatpixelcoordinates X
i
.Afourthcomponentmaycodetrans-
parencyinformation. heobserved vectors Y
i
can otenbemodeledasamultivariate
Gaussian,i.e., Y
i
4
withsomeunknowncovariance Σthatmaydepend
on θ. Additionally we will usually observe some spatial correlation.
N
(
θ
(
X
i
)
,Σ
)
Local Modeling
8.1.2
We now formally introduce our model. Let
P=(
P
θ
, θ
Θ
)
be a family of probabil-
where Θ is asubset of the real line R
. We assume that this family
is dominated by a measure P and denote p
ity measures on
Y
(
y, θ
)=
dP
θ
dP
(
y
)
.Wesupposethat
each Y
i
is, conditionally on X
i
=
x, distributed with density p
(ċ
, θ
(
x
))
.hedensity
is parameterized by some unknown function θ
which we aim to estimate.
Aglobalparametric structuresimply means that the parameter θ doesnotdependon
the location; that is, the distribution of every “observation” Y
i
coincides with P
θ
for
some θ
(
x
)
on
X
Θandalli. his assumption reduces the original problem to the classical
parametric situation and the well-developed parametric theory is applied here to es-
timate the underlying parameter θ. In particular, the maximum likelihood estimate
θ
θ
=
(
Y
,...,Y
n
)
of θ, which is defined by the maximization of the log-likelihood
n
i
=
L
(
θ
)=
log p
(
Y
i
, θ
)
(
.
)
is root-n consistent and asymptotically e
cient under rather general conditions.
Sucha global parametric assumption is typically too restrictive. heclassical non-
parametric approach is based on the idea of localization: for every point x,thepara-
metric assumption is only fulfilled locally in a vicinity of x. We therefore use a local
model concentrated in some neighborhood of the point x.
hemostgeneral waytodescribealocal modelisbasedonweights. Let,forafixed
x, a nonnegative weight w
i
=
w
i
(
x
)
be assigned to the observations Y
i
at X
i
, i
=
,...,n.Whenestimatingthelocalparameter θ
(
x
)
,everyobservationY
i
isusedwith
the weight w
i
(
x
)
. his leads to the local (weighted) maximum likelihood estimate
n
i
=
θ
(
x
)=
argsup
θ
Θ
w
i
(
x
)
log p
(
Y
i
, θ
)
(
.
)
Note that this definition is a special case of a more general local linear (polynomial)
likelihood model when the underlying function θ is modelled linearly (polynomi-
