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In-Depth Information
Yetanother approachtobiasreduction reliesonvariable bandwidths. Bychoosing
proportional to f
−
h
i
, Abramson (
) demonstrated that this is
yet another way to improve the order of the bias; see also Terrell and Scott (
)
and Sain and Scott (
). his approach also has much to recommend it for density
visualization. Byusingsmallbandwidths nearmodeswesharpentheestimateswhere
bias is greatest. At the same time, wide kernels for isolated points in the tails of the
density lead to smoother estimates with fewer spurious modes there.
A compromise between the single bandwidth of a standard kernel estimate and
the n bandwidths of a variable bandwidth estimate may be found in the filtered ker-
nel technique of Marchette et al. (
). he authors use an initial normal mixture
distribution estimate with a small number of components to select bandwidths pro-
portional to the component standard deviations. Kernels with each bandwidth are
averaged for each x
i
with weights proportional to their component densities at x
i
.
In this way, the smoothness of each region may be adjusted individually to empha-
size large, important features, whiledeemphasizing unimportant features such asthe
many minor modes that may oten be observed in the tails of densities.
Figure
.
shows these variants for the data of Fig.
.
. For the fourth-order and
variable-location estimates, the bandwidth is h
=
h
(
x
i
)
(
x
i
)
.
, while the variable-bandwidth
and filtered estimates have the same value for the geometric averages of their band-
widths. Eachpanel also includes the standard kernel estimate in greyforcomparison
(asinthemiddlepanelofFig.
.
).Allofthevariantsstrengthenandsharpenthelarge
mode on the right. he fourth-order and variable-location estimates also strengthen
the smaller bumps and modes in the let tail, while the variable-bandwidth and fil-
tered estimates deemphasize the same.
=
Figure
.
.
Variant kernel density estimates for minimum temperature data
