L_I {OSCV}

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

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.

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 robust kernel used in Savchuk and Hart (2017) and Savchuk (2017) is obtained by setting \alpha=16.8954588 and \sigma=1.01;

• 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.

Value

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

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)

## End(Not run)