bw.abram.default {spatstat.univar}

Abramson's Adaptive Bandwidths For Numeric Data

Description

Computes adaptive smoothing bandwidths for numeric data, according to the inverse-square-root rule of Abramson (1982).

Usage

 ## Default S3 method:
bw.abram(X, h0, ...,
    at = c("data", "grid"),
    pilot = NULL, hp = h0, trim = 5, smoother = density.default)

Arguments

X

Data for which bandwidths should be calculated. A numeric vector.

h0

A scalar value giving the global smoothing bandwidth in the same units as X. The default is h0=bw.nrd0(X).

...

Arguments passed to density.default (or to smoother) controlling the range of values x at which the density will be estimated, when at="grid".

at

Character string (partially matched) specifying whether to compute bandwidth values only at the data points of X (at = 'data', the default) or on a grid of x values (at = 'grid').

pilot

Optional. Specification of a pilot density (possibly unnormalised). Either a numeric vector giving the pilot density for each data point of X, a function in the R language, or a probability density estimate (object of class "density"). If pilot=NULL the pilot density is computed by applying fixed-bandwidth density estimation to X using bandwidth hp.

hp

Optional. A scalar pilot bandwidth, used for estimation of the pilot density, if pilot is not given.

trim

A trimming value required to curb excessively large bandwidths. See Details. The default is sensible in most cases.

smoother

Smoother for the pilot. A function or character string, specifying the function to be used to compute the pilot estimate when pilot is NULL.

Details

This function computes adaptive smoothing bandwidths using the methods of Abramson (1982) and Hall and Marron (1988).

The function bw.abram is generic. The function bw.abram.default documented here is the default method which is designed for numeric data.

If at="data" (the default) a smoothing bandwidth is computed for each data point in X. Alternatively if at="grid" a smoothing bandwidth is computed for a grid of x values.

Under the Abramson-Hall-Marron rule, the bandwidth at location u is

h(u) = \mbox{\texttt{h0}} * \mbox{min}[ \frac{\tilde{f}(u)^{-1/2}}{\gamma}, \mbox{\texttt{trim}} ]

where \tilde{f}(u) is a pilot estimate of the probability density. The variable bandwidths are rescaled by \gamma, the geometric mean of the \tilde{f}(u)^{-1/2} terms evaluated at the data; this allows the global bandwidth h0 to be considered on the same scale as a corresponding fixed bandwidth. The trimming value trim has the same interpretation as the required ‘clipping’ of the pilot density at some small nominal value (see Hall and Marron, 1988), to necessarily prevent extreme bandwidths (which can occur at very isolated observations).

The pilot density or intensity is determined as follows:

• If pilot is a function in the R language, this is taken as the pilot density.

• If pilot is a probability density estimate (object of class "density" produced by density.default) then this is taken as the pilot density.

• If pilot is NULL, then the pilot intensity is computed as a fixed-bandwidth kernel intensity estimate using density.default applied to the data X using the pilot bandwidth hp.

In each case the pilot density is renormalised to become a probability density, and then the Abramson rule is applied.

Instead of calculating the pilot as a fixed-bandwidth density estimate, the user can specify another density estimation procedure using the argument smoother. This should be either a function or the character string name of a function. It will replace density.default as the function used to calculate the pilot estimate. The pilot estimate will be computed as smoother(X, sigma=hp, ...) if pilot is NULL, or smoother(pilot, sigma=hp, ...) if pilot is a point pattern. If smoother does not recognise the argument name sigma for the smoothing bandwidth, then hp is effectively ignored.

Value

Either a numeric vector of the same length as X giving the Abramson bandwidth for each point (when at = "data", the default), or a function giving the Abramson bandwidths as a function of location.

Author(s)

Tilman Davies [e-mail]. Adapted by Adrian Baddeley [e-mail].

References

Abramson, I. (1982) On bandwidth variation in kernel estimates — a square root law. Annals of Statistics, 10(4), 1217-1223.

Hall, P. and Marron, J.S. (1988) Variable window width kernel density estimates of probability densities. Probability Theory and Related Fields, 80, 37-49.

Silverman, B.W. (1986) Density Estimation for Statistics and Data Analysis. Chapman and Hall, New York.

See Also

bw.abram, bw.nrd0.

Examples

  xx <- rexp(20)
  bw.abram(xx)