Graphics Reference
In-Depth Information
regions. he kernel estimate for Kansas has a dominant mode consistent with the
eastern plains of Colorado.
Kernel Variants
5.1.4
Minor adjustments allow for estimation of density derivatives as well as the function
itself. he derivative of f is a reasonable estimate of the derivative of f and can be
estimated directly as
n
i
=
nh
x
−
x
i
f
′
K
′
(
x
)=
(
.
)
h
he variance of this estimate is inherently greater than that of f ,andtheoptimal
choice of h is correspondingly larger.
Returning to estimates of f , a variety of approaches are available for reducing the
bias of a kernel estimate. See Jones and Signorini (
)for a survey and comparison
of some of the most common higher-order methods. As a rule, most of these will
have fairly minimal effect on the visual impact of the estimate.
he oldest and best-known higher-order methods replace the p.d.f.kernel K with
one involving carefully computed negative regions so as to have second, and possi-
bly higher, moments equal to
(Bartlett
). For example, starting with standard
(second-order) kernel K,let
∫
t
m
K
s
m
=
(
t
)
dt.
(
.
)
hen the function
s
t
s
−
K
()
(
t
)=
K
(
t
)
(
.
)
s
−
s
will have a
second moment and be a fourth-order kernel. Using K
()
t
in place
(
)
of K
will reduce the bias of the estimate. Unfortunately, such an estimate will fre-
quentlyincludeundesirablevisualartifacts,includingextramodesandnegativelobes
inthetails ofthe density estimate,soisnotrecommendedforvisualization purposes.
Bias-reduction methodsinvolving “variable location” approaches(Samiuddinand
El-Sayyad,
;HallandMinnotte,
)providebettersolutions. Insteadofreplac-
ing the kernel K,these methodsuseoneofourgood second-orderkernels butcenter
them on transformed values, γ
(
t
)
, rather than on the raw x
i
sthemselves.hese
transformations depend upon pilot estimates of f and its derivatives in a way that
will reduce the bias. For example, replacing x
i
with
(
x
i
)
f
′
h
s
x
i
(
)
γ
(
x
i
)=
x
i
+
(
.
)
f
(
x
i
)
provides fourth-order bias and improved estimation near local extrema, but the use
of the standard kernel guarantees positivity and fewer extraneous bumps than found
with the fourth-order kernel.
