H_I: The family of two-sided cross-validation kernels H_I.
In OSCV: One-Sided Cross-Validation

Description Usage Arguments Details Value References See Also Examples

The family of two-sided cross-validation kernels H_I defined by equation (15) of Savchuk and Hart (2017).

1	H_I(u, alpha, sigma)

`u`	numerical vector of argument values,
`alpha`	first parameter of the cross-validation kernel H_I,
`sigma`	second parameter of the cross-validation kernel H_I.

The family of the two-sided cross-validation kernels H_I(u;α,σ)=(1+α)φ(u)-αφ(u/σ)/σ, where φ denotes the Gaussian kernel, -∞<α<∞ and σ>0 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 α=16.8954588 and σ=1.01. Note that the kernels H_I 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 H_I is a two-sided analog of the one-sided kernel L_I. The Gaussian kernel φ is the special case of H_I obtained by either setting α=0 or σ=1.

The value of H_I(u;α,σ).

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.

L_I, C_smooth, OSCV_reg, loclin.

## 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)