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ally) in x; see e.g., Fan et al. (
). However, our approach focuses on the choice of
localizing weights in a data-driven way rather than on the method of local approxi-
mation of the function θ.
Acommonwaytochoosetheweights w
i
(
x
)
istodefinethemintheform w
i
(
x
)=
where h is a bandwidth, ρ
K
loc
is the Euclidean
distance between x and the design point X
i
,andK
loc
is a location kernel.hisap-
proachisintrinsically based ontheassumption that thefunction θ issmooth. Itleads
to a local approximation of θ
(
l
i
)
with
l
i
=
ρ
(
x, X
i
)
h
(
x, X
i
)
within a ball with some small radius h centered on
the point x, see e.g., Tibshirani and Hastie (
); Hastie and Tibshirani (
); Fan
et al. (
); Carroll et al. (
); Cai et al. (
).
An alternative approach is termed localization by a window. his simply restricts
the model to a subset (window) U
(
x
)
=
U
(
x
)
of the design space which depends on
x;thatis,w
i
(
x
)=
(
X
i
U
(
x
))
.ObservationsY
i
with X
i
outside the region U
(
x
)
are not used to estimate the value θ
. his kind of localization arises, for example,
in the regression tree approach, in change point estimation (see e.g., Müller,
;
Spokoiny,
),and inimagedenoising (seeQiu,
;Polzehl andSpokoiny,
),
among many other situations.
In our procedure we do not assume any special structure for the weights w
i
(
x
)
;
that is, any configuration of weights is allowed. he weights are computed in an it-
erative way from the data. In what follows we identify the set W
(
x
)
(
x
)=
w
(
x
)
,...,
w
n
(
x
)
and the local model in x described by these weights and use the notation
n
i
=
L
(
W
(
x
)
, θ
)=
w
i
(
x
)
log p
(
Y
i
, θ
)
.
hen θ
(
x
)=
argsup
θ
L
(
W
(
x
)
, θ
)
. For simplicity we will assume the case where
θ
(
x
)
describestheconditional expectation
E
(
Y
x
)
and the local estimate isobtained
explicitly as
θ
(
x
)=
i
w
i
(
x
)
Y
i
i
w
i
(
x
)
(
.
)
he quality of the estimation heavily depends on the localizing scheme we se-
lected.Weillustrate thisissuebyconsidering kernelweights w
i
(
x
)=
K
loc
(
ρ
(
x, X
i
)
h
are
concentrated within the ball of radius h at the point x.Asmallbandwidthh leads to
a very strong localization. In particular, if the bandwidth h is smaller than the dis-
tance from x to the nearest neighbor, then the resulting estimate coincides with the
observation at x. Increasing the bandwidth amplifies the noise reduction that can be
achieved. However, the choice of a large bandwidth may lead to estimation bias if
the local parametric assumption of a homogeneous structure is not fulfilled in the
selected neighborhood.
he classical approach to solving this problem is based on a model selection idea.
Oneassumesagiven setofbandwidth candidates
)
wherethe kernel K
loc
is supportedon
[
,
]
. hen the positive weights w
i
(
x
)
,andoneofthemisselectedin
adata-driven waytoprovidetheoptimal quality ofestimation. heglobalbandwidth
selection problemassumesthesame kernel structureoflocalizing schemes w
i
h
k
(
x
)
for
