h.mcv {kedd}

Modified Cross-Validation for Bandwidth Selection

Description

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

Usage

h.mcv(x, ...)
## Default S3 method:
h.mcv(x, deriv.order = 0, lower = 0.1 * hos, upper = 2 * hos, 
         tol = 0.1 * lower, kernel = c("gaussian", "epanechnikov", "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.mcv modified cross-validation implements for choosing the bandwidth h of a r'th derivative kernel density estimator.

Stute (1992) proposed a so-called modified cross-validation (MCV) in kernel density estimator. This method can be extended to the estimation of derivative of a density, the essential idea based on approximated the problematic term by the aid of the Hajek projection (see Stute 1992). The minimization criterion is defined by:

MCV(h;r) = \frac{R\left(K^{(r)}\right)}{nh^{2r+1}} + \frac{(-1)^{r}}{n(n-1)h^{2r+1}}\sum_{i=1}^{n} \sum_{j=1;j \neq i}^{n} \varphi^{(r)} \left(\frac{X_{j}-X_{i}}{h}\right)

whit

\varphi^{(r)}(c) = \left(K^{(r)} \ast K^{(r)} - K^{(2r)} - \frac{\mu_{2}(K)}{2}K^{(2r+2)} \right)(c)

and K^{(r)} \ast K^{(r)} (x) is the convolution of the r'th derivative kernel function K^{(r)}(x) (see kernel.conv and kernel.fun); R\left(K^{(r)}\right) = \int_{R} K^{(r)}(x)^{2} dx and \mu_{2}(K) = \int_{R}x^{2} 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.mcv

the minimal MCV value.

Author(s)

Arsalane Chouaib Guidoum [e-mail]

References

Heidenreich, N. B., Schindler, A. and Sperlich, S. (2013). Bandwidth selection for kernel density estimation: a review of fully automatic selectors. Advances in Statistical Analysis.

Stute, W. (1992). Modified cross validation in density estimation. Journal of Statistical Planning and Inference, 30, 293–305.

See Also

plot.h.mcv.

Examples

## Derivative order = 0

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

## Derivative order = 1

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