OSCV (version 1.0)

H_I: The family of two-sided cross-validation kernels

Description

The family of two-sided cross-validation kernels \(H_I\) 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 \(H_I\),

sigma

second parameter of the cross-validation kernel \(H_I\).

Value

The value of \(H_I(u;\alpha,\sigma)\).

Details

The family of the two-sided cross-validation kernels \(H_I(u;\alpha,\sigma)=(1+\alpha)\phi(u)-\alpha\phi(u/\sigma)/\sigma\), where \(\phi\) denotes the Gaussian kernel, \(-\infty<\alpha<\infty\) and \(\sigma>0\) 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 \(\alpha=16.8954588\) and \(\sigma=1.01\). 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 \(\phi\) is the special case of \(H_I\) obtained by either setting \(\alpha=0\) or \(\sigma=1\).

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")
## ---------------------------------------------