Trimmed Cross-Validation for Bandwidth Selection

Description

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

Usage

h.tcv(x, ...)
## Default S3 method:
h.tcv(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

Details

h.tcv trimmed cross-validation implements for choosing the bandwidth h of a r'th derivative kernel density estimator.

Feluch and Koronacki (1992) proposed a so-called trimmed cross-validation (TCV) in kernel density estimator, a simple modification of the unbiased (least-squares) cross-validation criterion. We consider the following "trimmed" version of "unbiased", to be minimized with respect to h:

int (hat(f)(x;r))^2 - 2 (-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) chi(|X(i)-X(j)| > c(n))

where chi(.) denotes the indicator function and c(n) is a sequence of positive constants, c(n)/h^(2r+1) --> 0 as n --> Inf, and

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)

the trimmed cross-validation function is defined by:

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

whit

varphi(c;r)= K(c;r)*K(c;r) - 2 K(c;2r) chi( |c| > c(n)/h^(2r+1) )

here we take c(n) = 1/n, for assure the convergence. Where 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).

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

Author(s)

Arsalane Chouaib Guidoum acguidoum@usthb.dz

References

Feluch, W. and Koronacki, J. (1992). A note on modified cross-validation in density estimation. Computational Statistics and Data Analysis, 13, 143–151.

See Also

plot.h.tcv.

Examples

## Derivative order = 0

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

## Derivative order = 1

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