h.ccv: Complete Cross-Validation for Bandwidth Selection

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

Description

The (S3) generic function h.ccv computes the complete cross-validation bandwidth selector of r'th derivative of kernel density estimator one-dimensional.

Usage

h.ccv(x, ...)
## Default S3 method:
h.ccv(x, deriv.order = 0, lower = 0.1 * hos, upper = hos, 
         tol = 0.1 * lower, kernel = c("gaussian", "triweight", 
         "tricube", "biweight", "cosine"), ...)

Arguments

x	vector of data values.
deriv.order	derivative order (scalar).
lower, upper	range over which to minimize. The default is almost always satisfactory. hos (Over-smoothing) is calculated internally from an kernel, see details.
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.ccv complete cross-validation implements for choosing the bandwidth h of a r'th derivative kernel density estimator.

Jones and Kappenman (1991) proposed a so-called complete cross-validation (CCV) in kernel density estimator. This method can be extended to the estimation of derivative of the density, basing our estimate of integrated squared density derivative (Peter and Marron 1987) on the bar(theta)(h;r)'s, we get the following, start from R(hat(f)(h;r)) - bar(theta)(h;r) as an estimate of MISE. Thus, h(r)_(CCV), say, is the h that minimises:

CCV(h;r)= R(K(x;r))/ n h^(2r+1) + R(hat(f)(h;r))- bar(theta)(h;r) + 0.5 mu(K(x)) h^2 bar(theta)(h;r+1) + 1/24 (6 mu(K(x))^2 - delta(K(x))) h^4 bar(theta)(h;r+2)

with

R(hat(f)(h;r)) = int (hat(f)(x;r))^2 dx =R(k(x;r))/n h^(2r+1) + (-1)^r / n (n-1) h^(2r+1) sum(sum(K(.;r)*K(.;r)(x(j)-x(i)/h)), i=1...n, j=1...n, j != i)

and

bar(theta)(h;r) = (-1)^r / n(n-1) h^(2r+1) sum(sum(K((x(j)-x(i)/h);2r)), i=1...n, j=1...n, j != i)

and K(x;r)*K(x;r) is the convolution of the r'th derivative kernel function K(x;r) (see kernel.conv and kernel.fun); R(K(x;r)) = int K(x;r)^2 dx and mu(K(x)) = int x^2 K(x) dx, delta(K(x)) = int x^4 K(x) dx.

The range over which to minimize is hos Oversmoothing bandwidth, the default is almost always satisfactory. See George and Scott (1985), George (1990), Scott (1992, pp 165), Wand and Jones (1995, pp 61).

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
deriv.order	the derivative order to use.
h	value of bandwidth parameter.
min.ccv	the minimal CCV value.

Author(s)

Arsalane Chouaib Guidoum [e-mail]

References

Jones, M. C. and Kappenman, R. F. (1991). On a class of kernel density estimate bandwidth selectors. Scandinavian Journal of Statistics, 19, 337–349.

Peter, H. and Marron, J.S. (1987). Estimation of integrated squared density derivatives. Statistics and Probability Letters, 6, 109–115.

See Also

plot.h.ccv.

Examples

## Derivative order = 0

h.ccv(kurtotic,deriv.order = 0)

## Derivative order = 1

h.ccv(kurtotic,deriv.order = 1)

Example output

Call:		Complete Cross-Validation

Derivative order = 0
Data: kurtotic (200 obs.);	Kernel: gaussian
Min CCV = 0.04780886;	Bandwidth 'h' = 0.3508383


Call:		Complete Cross-Validation

Derivative order = 1
Data: kurtotic (200 obs.);	Kernel: gaussian
Min CCV = 0.7207028;	Bandwidth 'h' = 0.4856272

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