OSCV (version 1.0)

C_smooth: The OSCV smooth rescaling constant.

Description

Computing the OSCV smooth rescaling constant that corresponds to using the two-sided kernel H_I for the cross-validation purposes and the Gaussian kernel in the estimation stage. The constant is applicable for the OSCV versions in the regression and kernel density estimation contexts.

Usage

C_smooth(alpha, sigma)

Arguments

alpha

first parameter of the two-sided cross-validation kernel H_I,

sigma

second parameter of the two-sided cross-validation kernel H_I.

Value

The OSCV smooth rescaling constant \(C\) for the given values of the parameters \(\alpha\) and \(\sigma\).

Details

Computation of the OSCV rescaling constant \(C\) (see (10) in Savchuk and Hart (2017) or (3) in Savchuk (2017)). The constant is a function of the parameters \((\alpha,\sigma)\) of the two-sided cross-validation kernel H_I defined by expression (15) in Savchuk and Hart (2017). The Gaussian kernel is used for computing the ultimate (regression or density) estimate. The constant is used in the OSCV versions for kernel regression and density estimation. Notice that in the cases \(\alpha=0\), \(\sigma>0\) and \(\sigma=1\), \(-\infty<\alpha<\infty\) the kernel H_I reduces to the Gaussian kernel.

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

L_I, H_I, OSCV_reg, h_OSCV_reg, OSCV_LI_dens, OSCV_Gauss_dens, h_OSCV_dens, loclin.

Examples

# OSCV rescaling constant for the robust cross-validation kernel with 
# (alpha,sigma)=(16.8954588,1.01).
C_smooth(16.8954588,1.01)
# OSCV smooth rescaling constant in the case when the kernel H_I is Gaussian.
C_smooth(1,1)