.
Sanders et al. provide a comprehensive dataset of infrared emissions and other characteristics of objects beyond our galaxy [567]. These data are available from the website for our book. Let
X
denote the log of the variable labeled F12, which is the total 12-μm-band flux measurement on each object.
a.
Fit a normal kernel density estimate for
X, using bandwidths derived from the UCV(h) criterion, Silverman’s rule of thumb, the Sheather–Jones approach, Terrell’s maximal smoothing principle, and any other approaches you wish. Comment on the apparent suitability of each bandwidth for these data.
b.
Fit kernel density estimates for
X
using uniform, normal, Epanechnikov, and triweight kernels, each with bandwidth equivalent to the Sheather–Jones bandwidth for a normal kernel. Comment.
c.
Fit a nearest neighbor density estimate for
X
as in (10.48) with the uniform and normal kernels. Next fit an Abramson adaptive estimate for
X
using a normal kernel and setting
h
equal to the Sheather–Jones bandwidth for a fixed-bandwidth estimator times the geometric mean of the
f˜X(x
i)1/2 values.
d.
If code for logspline density estimation is available, experiment with this approach for estimating the density of
X.
e.
Let
fˆX
denote the normal kernel density estimate for
X
computed using the Sheather–Jones bandwidth. Note the ratio of this bandwidth to the bandwidth given by Silverman’s rule of thumb. Transform the data back to the original scale (i.e.,
Z
= exp{X}), and fit a normal kernel density estimate
fˆZ, using a bandwidth equal to Silverman’s rule of thumb scaled down by the ratio noted previously. (This is an instance where the robust scale measure is far superior to the sample standard deviation.) Next, transform
fˆX
back to the original scale using the change-of-variable formula for densities, and compare the two resulting estimates of density for
Z
on the region between 0 and 8. Experiment further to investigate the relationship between density estimation and nonlinear scale transformations. Comment.