OSCV (version 1.0)

L_I: The family of one-sided cross-validation kernels

Description

The one-sided counterpart of the kernel H_I. See expressions (15) and (8) of Savchuk and Hart (2017).

Usage

L_I(u, alpha, sigma)

Arguments

u

numerical vector of argument values,

alpha

first parameter of the cross-validation kernel \(L_I\),

sigma

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

Value

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

Details

The family of the one-sided cross-validation kernels \(L_I\) indexed by the parameters \(-\infty<\alpha<\infty\) and \(\sigma>0\). This family is used in the OSCV implementations in both regression context (see Savchuk and Hart (2017)) and density estimation context (see Savchuk (2017)). The special members of the family:

• The one-sided Gaussian kernel \(L_G\) is obtained by either setting \(\alpha=0\) for any \(\sigma>0\) or by setting \(\sigma=1\) for any \(-\infty<\alpha<\infty\).

The bandwidth selected by \(L_I\) should be multiplied by a reascaling constant before it is used in computing the ultimate Gaussian (regression or density) estimate. In the case of a smooth (regression or density) function the rescaling constant is C_smooth.

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. (2017). One-sided cross-validation for nonsmooth density functions, arXiv:1703.05157.

See Also

H_I, C_smooth, OSCV_LI_dens.

Examples

## Not run: ------------------------------------
# # Plotting the robust one-sided kernel from Savchuk and Hart (2017) with 
# # alpha=16.8954588 and sigma=1.01.
# u=seq(-1,5,len=1000)
# rker=L_I(u,16.8954588,1.01)
# Gker=L_I(u,0,1)
# dev.new()
# plot(u,rker,'l',lwd=3,cex.axis=1.7, cex.lab=1.7)
# title(main="One-sided kernels: L_I (robust) and L_G",cex=1.7)
# lines(u,Gker,lty="dashed",lwd=3)
# legend(0.5,2.5,lty=c("solid","dashed"),lwd=c(3,3),legend=c("L_I","L_G"),cex=1.7)
# legend(2,1.5,legend=c("alpha=16.8955","sigma=1.01"),cex=1.5)
## ---------------------------------------------