Bandwidth selectors for Gaussian kernels in `density`

.

`bw.nrd0(x)`bw.nrd(x)

bw.ucv(x, nb = 1000, lower = 0.1 * hmax, upper = hmax,
tol = 0.1 * lower)

bw.bcv(x, nb = 1000, lower = 0.1 * hmax, upper = hmax,
tol = 0.1 * lower)

bw.SJ(x, nb = 1000, lower = 0.1 * hmax, upper = hmax,
method = c("ste", "dpi"), tol = 0.1 * lower)

x

numeric vector.

nb

number of bins to use.

lower, upper

range over which to minimize. The default is
almost always satisfactory. `hmax`

is calculated internally
from a normal reference bandwidth.

method

either `"ste"`

("solve-the-equation") or
`"dpi"`

("direct plug-in"). Can be abbreviated.

tol

for method `"ste"`

, the convergence tolerance for
`uniroot`

. The default leads to bandwidth estimates
with only slightly more than one digit accuracy, which is sufficient
for practical density estimation, but possibly not for theoretical
simulation studies.

A bandwidth on a scale suitable for the `bw`

argument
of `density`

.

`bw.nrd0`

implements a rule-of-thumb for
choosing the bandwidth of a Gaussian kernel density estimator.
It defaults to 0.9 times the
minimum of the standard deviation and the interquartile range divided by
1.34 times the sample size to the negative one-fifth power
(= Silverman's ‘rule of thumb’, Silverman (1986, page 48, eqn (3.31)))
*unless* the quartiles coincide when a positive result
will be guaranteed.

`bw.nrd`

is the more common variation given by Scott (1992),
using factor 1.06.

`bw.ucv`

and `bw.bcv`

implement unbiased and
biased cross-validation respectively.

`bw.SJ`

implements the methods of Sheather & Jones (1991)
to select the bandwidth using pilot estimation of derivatives.
The algorithm for method `"ste"`

solves an equation (via
`uniroot`

) and because of that, enlarges the interval
`c(lower, upper)`

when the boundaries were not user-specified and
do not bracket the root.

The last three methods use all pairwise binned distances: they are of
complexity \(O(n^2)\) up to `n = nb/2`

and \(O(n)\)
thereafter. Because of the binning, the results differ slightly when
`x`

is translated or sign-flipped.

Scott, D. W. (1992)
*Multivariate Density Estimation: Theory, Practice, and
Visualization.*
New York: Wiley.

Sheather, S. J. and Jones, M. C. (1991).
A reliable data-based bandwidth selection method for kernel density
estimation.
*Journal of the Royal Statistical Society series B*,
**53**, 683--690.
http://www.jstor.org/stable/2345597.

Silverman, B. W. (1986).
*Density Estimation*.
London: Chapman and Hall.

Venables, W. N. and Ripley, B. D. (2002).
*Modern Applied Statistics with S*.
Springer.

`bandwidth.nrd`

, `ucv`

,
`bcv`

and `width.SJ`

in
package MASS, which are all scaled to the `width`

argument
of `density`

and so give answers four times as large.

# NOT RUN { require(graphics) plot(density(precip, n = 1000)) rug(precip) lines(density(precip, bw = "nrd"), col = 2) lines(density(precip, bw = "ucv"), col = 3) lines(density(precip, bw = "bcv"), col = 4) lines(density(precip, bw = "SJ-ste"), col = 5) lines(density(precip, bw = "SJ-dpi"), col = 6) legend(55, 0.035, legend = c("nrd0", "nrd", "ucv", "bcv", "SJ-ste", "SJ-dpi"), col = 1:6, lty = 1) # }