H_I {OSCV} | R Documentation |
The family of two-sided cross-validation kernels
.
Description
The family of two-sided cross-validation kernels defined by equation (15) of Savchuk and Hart (2017).
Usage
H_I(u, alpha, sigma)
Arguments
u |
numerical vector of argument values, |
alpha |
first parameter of the cross-validation kernel |
sigma |
second parameter of the cross-validation kernel |
Details
The family of the two-sided cross-validation kernels , where
denotes the Gaussian kernel,
and
are the parameters of the kernel. See expression (15) of Savchuk and Hart (2017). The robust kernel plotted in Figure 1 of Savchuk and Hart (2017) is obtained by setting
and
. Note that the kernels
are also used for the bandwidth selection purposes in the indirect cross-validation (ICV) method (see expression (4) of Savchuk, Hart, and Sheather (2010)). The kernel
is a two-sided analog of the one-sided kernel
L_I
. The Gaussian kernel is the special case of
obtained by either setting
or
.
Value
The value of .
References
Savchuk, O.Y., Hart, J.D. (2017). Fully robust one-sided cross-validation for regression functions. Computational Statistics, doi:10.1007/s00180-017-0713-7.
Savchuk, O.Y., Hart, J.D., Sheather, S.J. (2010). Indirect cross-validation for density estimation. Journal of the American Statistical Association, 105(489), 415-423.
See Also
L_I
, C_smooth
, OSCV_reg
, loclin
.
Examples
## Not run:
# Plotting the robust kernel from Savchuk and Hart (2017) with alpha=16.8954588 and sigma=1.01.
u=seq(-5,5,len=1000)
ker=H_I(u,16.8954588,1.01)
dev.new()
plot(u,ker,'l',lwd=3,cex.axis=1.7, cex.lab=1.7)
title(main="Robust kernel H_I along with the Gaussian kernel (phi)",cex=1.7)
lines(u,dnorm(u),lty="dashed",lwd=3)
legend(-4.85,0.3,lty=c("solid","dashed"),lwd=c(3,3),legend=c("H_I","phi"),cex=1.5)
legend(1,0.4,legend=c("alpha=16.8955","sigma=1.01"),cex=1.5,bty="n")
## End(Not run)