h.ucv {kedd}

Unbiased (Least-Squares) Cross-Validation for Bandwidth Selection

Description

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

Usage

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

Rudemo (1982) and Bowman (1984) proposed a so-called unbiased (least-squares) cross-validation (UCV) in kernel density estimator. An adaptation of unbiased cross-validation is proposed by Wolfgang et al. (1990) for bandwidth choice in the r'th derivative of kernel density estimator. The essential idea of this methods, for the estimation of f^{(r)}(x) (r is derivative order), is to use the bandwidth h which minimizes the function:

UCV(h;r) = \int \left(\hat{f}_{h}^{(r)}(x)\right)^{2} - 2n^{-1}(-1)^{r}\sum_{i=1}^{n} \hat{f}_{h,i}^{(2r)}(X_{i})

The bandwidth minimizing this function is:

\hat{h}^{(r)}_{ucv} = \arg \min_{h^{(r)}} UCV(h;r)

for r = 0, 1, 2, \dots
where

\int \left(\hat{f}_{h}^{(r)}(x)\right)^{2} = \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} K^{(r)} \ast K^{(r)} \left(\frac{X_{j}-X_{i}}{h}\right)

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).
The estimate \hat{f}_{h,i}^{(2r)}(x) on the subset \{X_{j}\}_{j \neq i} denoting the leave-one-out estimator, can be written:

\hat{f}_{h,i}^{(2r)}(X_{i}) = \frac{1}{(n-1) h^{2r+1}} \sum_{j \neq i} K^{(2r)} \left(\frac{X_{j}-X_{i}}{h}\right)

The function UCV(h;r) is unbiased cross-validation in the sense that E[UCV]=MISE[\hat{f}_{h}^{(r)}(x)]-R(f^{(r)}(x)) (see, Scott and George 1987). Can be simplified to give the computationally:

UCV(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} \left(K^{(r)} \ast K^{(r)} -2K^{(2r)}\right) \left(\frac{X_{j}-X_{i}}{h}\right)

where R\left(K^{(r)}\right) = \int_{R} K^{(r)}(x)^{2} 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.ucv

the minimal UCV value.

Author(s)

Arsalane Chouaib Guidoum [e-mail]

References

Bowman, A. (1984). An alternative method of cross-validation for the smoothing of kernel density estimates. Biometrika, 71, 353–360.

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

Jones, M. C., Marron, J. S. and Sheather,S. J. (1996). A brief survey of bandwidth selection for density estimation. Journal of the American Statistical Association, 91, 401–407.

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

Rudemo, M. (1982). Empirical choice of histograms and kernel density estimators. Scandinavian Journal of Statistics, 9, 65–78.

Scott, D.W. and George, R. T. (1987). Biased and unbiased cross-validation in density estimation. Journal of the American Statistical Association, 82, 1131–1146.

Sheather, S. J. (2004). Density estimation. Statistical Science, 19, 588–597.

Tarn, D. (2007). ks: Kernel density estimation and kernel discriminant analysis for multivariate data in R. Journal of Statistical Software, 21(7), 1–16.

Wand, M. P. and Jones, M. C. (1995). Kernel Smoothing. Chapman and Hall, London.

Wolfgang, H. (1991). Smoothing Techniques, With Implementation in S. Springer-Verlag, New York.

Wolfgang, H., Marron, J. S. and Wand, M. P. (1990). Bandwidth choice for density derivatives. Journal of the Royal Statistical Society, Series B, 223–232.

See Also

plot.h.ucv, see bw.ucv in package "stats" and ucv in package MASS for Gaussian kernel only if deriv.order = 0, hlscv in package ks for Gaussian kernel only if 0 <= deriv.order <= 5, kdeb in package locfit if deriv.order = 0.

Examples

## Derivative order = 0

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

## Derivative order = 1

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