h.mlcv: Maximum-Likelihood Cross-validation for Bandwidth Selection

In kedd: Kernel Estimator and Bandwidth Selection for Density and Its Derivatives

Description

The (S3) generic function h.mlcv computes the maximum likelihood cross-validation (Kullback-Leibler information) bandwidth selector of a one-dimensional kernel density estimate.

Usage

h.mlcv(x, ...)
## Default S3 method:
h.mlcv(x, lower = 0.1, upper = 5, tol = 0.1 * lower, 
         kernel = c("gaussian", "epanechnikov", "uniform", "triangular", 
         "triweight", "tricube", "biweight", "cosine"), ...)

Arguments

x	vector of data values.
lower, upper	range over which to maximize. The default is almost always satisfactory.
tol	the convergence tolerance for optimize.
kernel	a character string giving the smoothing kernel to be used, with default "gaussian".
...	further arguments for (non-default) methods.

Details

h.mlcv maximum-likelihood cross-validation implements for choosing the optimal bandwidth h of kernel density estimator.

This method was proposed by Habbema, Hermans, and Van den Broeck (1971) and by Duin (1976). The maximum-likelihood cross-validation (MLCV) function is defined by:

MLCV(h) = n^-1 sum( log(hat(f(h))),i=1...n)

the estimate hat(f)(x) on the subset (X_j)_(j != i) denoting the leave-one-out estimator, can be written:

hat(f)(X_i) = 1/(n-1) h sum(K(x(j)-x(i)/h), j != i)

Define that h(mlcv) as good which approaches the finite maximum of MLCV(h):

h(mlcv)= argmax MLCV(h) = argmax n^-1 sum(log(sum(K(x(j)-x(i)/h),j != i)),i=1...n) - log((n-1)h)

Value

x	data points - same as input.
data.name	the deparsed name of the x argument.
n	the sample size after elimination of missing values.
kernel	name of kernel to use
h	value of bandwidth parameter.
mlcv	the maximal likelihood CV value.

Author(s)

Arsalane Chouaib Guidoum [e-mail]

References

Habbema, J. D. F., Hermans, J., and Van den Broek, K. (1974) A stepwise discrimination analysis program using density estimation. Compstat 1974: Proceedings in Computational Statistics. Physica Verlag, Vienna.

Duin, R. P. W. (1976). On the choice of smoothing parameters of Parzen estimators of probability density functions. IEEE Transactions on Computers, C-25, 1175–1179.

See Also

plot.h.mlcv, see lcv in package locfit.

Examples

h.mlcv(bimodal)
h.mlcv(bimodal, kernel ="epanechnikov")

Example output

Call:		Maximum-Likelihood Cross-Validation

Data: bimodal (200 obs.);	Kernel: gaussian
Max CV = -1.419;	Bandwidth 'h' = 0.2303


Call:		Maximum-Likelihood Cross-Validation

Data: bimodal (200 obs.);	Kernel: epanechnikov
Max CV = -1.414;	Bandwidth 'h' = 0.448

kedd documentation built on May 2, 2019, 7:32 a.m.