Bandwidth selectors for Gaussian kernels in `density`

.

1 2 3 4 5 6 7 8 9 10 11 12 |

`x` |
numeric vector. |

`nb` |
number of bins to use. |

`lower, upper` |
range over which to minimize. The default is
almost always satisfactory. |

`method` |
either |

`tol` |
for method |

`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.

A bandwidth on a scale suitable for the `bw`

argument
of `density`

.

Long vectors `x`

are not supported, but neither are they by
`density`

and kernel density estimation and for more than
a few thousand points a histogram would be preferred.

B. D. Ripley, taken from early versions of package MASS.

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

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

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

`density`

.

`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.

1 2 3 4 5 6 7 8 9 10 11 12 | ```
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)
``` |

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