## Kernel Density Estimates

You can use the KERNEL option
to superimpose kernel density estimates on histograms.
Smoothing the data distribution with a kernel density estimate
can be more effective than using a histogram
to examine features that might be
obscured by the choice of histogram bins or sampling
variation.
A kernel density estimate can also be more
effective than a parametric curve fit when the
process distribution is multimodal.
See Example 4.5.
The general form of the kernel density estimator is

where *K*_{0}(·) is a kernel function, is the
bandwidth, *n* is the sample size, and *x*_{i} is the *i*^{ th}
observation.

The KERNEL option provides three kernel functions
(*K*_{0}): normal, quadratic, and triangular. You can specify
the function with the K= *kernel-option* in parentheses
after the KERNEL option.
Values for the K= option are
NORMAL, QUADRATIC, and TRIANGULAR (with aliases of N, Q, and T,
respectively).
By default, a normal kernel is used.
The formulas for the kernel functions are

The value of , referred to as the bandwidth parameter,
determines the degree of smoothness in the estimated
density function. You specify indirectly by specifying a standardized bandwidth *c*
with the C= *kernel-option*. If *Q* is the
interquartile range, and *n* is the sample size, then
*c* is related to by the formula

For a specific kernel function, the discrepancy between the
density estimator
and the true density *f*(*x*) is measured
by the mean integrated square error (MISE):

The MISE is the sum of the integrated squared bias and the
variance. An approximate mean integrated square error (AMISE) is

A bandwidth that minimizes AMISE can be derived by
treating *f*(*x*) as the normal density having parameters
and estimated by the sample mean and
standard deviation. If you do not specify a bandwidth
parameter or if you specify C=MISE, the bandwidth that
minimizes AMISE is used. The value of AMISE can be
used to compare different density estimates. For each
estimate, the bandwidth parameter *c*, the kernel function
type, and the value of AMISE are reported in the SAS log.

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.