Fully robust one-sided cross-validation for regression functions

Abstract

Fully robust OSCV is a modification of the OSCV method that produces consistent bandwidths in the cases of smooth and nonsmooth regression functions. We propose the practical implementation of the method based on the robust cross-validation kernel $H_I$ in the case when the Gaussian kernel $\phi$ is used in computing the resulting regression estimate. The kernel $H_I$ produces practically unbiased bandwidths in the smooth and nonsmooth cases and performs adequately in the data examples. Negative tails of $H_I$ occasionally result in unacceptably wiggly OSCV curves in the neighborhood of zero. This problem can be resolved by selecting the bandwidth from the largest local minimum of the curve. Further search for the robust kernels with desired properties brought us to consider the quartic kernel for the cross-validation purposes. The quartic kernel is almost robust in the sense that in the nonsmooth case it substantially reduces the asymptotic relative bandwidth bias compared to $\phi$. However, the quartic kernel is found to produce more variable bandwidths compared to $\phi$. Nevertheless, the quartic kernel has an advantage of producing smoother OSCV curves compared to $H_I$. A simplified scale-free version of the OSCV method based on a rescaled one-sided kernel is proposed.

Appendix

Notation For an arbitrary function g, define the following functionals:

$$\begin{aligned}&\begin{array}{l} X_g(u)=\int _{-\infty }^\infty g(x)\cos (2\pi ux)\,dx,\\ Y_g(u)=\int _{-\infty }^\infty g(x)\sin (2\pi ux)\,dx, \end{array}\nonumber \\&I_g=\int _0^{\infty } u^2\left[ X_g^{\prime }(u)(X_g(u)-1)+Y_g^{\prime }(u)Y_g(u)\right] ^2\,du, \end{aligned}$$

(18)

$$\begin{aligned}&B_g=\int _0^\infty \left\{ z\bigl (1-D_g(z)\bigr )+G_g(z)\right\} ^2\,dz+\int _0^\infty \left\{ zD_g(-z)+G_g(-z)\right\} ^2\,dz,\nonumber \\ \end{aligned}$$

(19)

where for all z,

$$\begin{aligned} \begin{array}{l} \displaystyle {D_g(z)=\int _{-\infty }^z g(u)\,du},\\ \displaystyle {G_g(z)=\int _{-\infty }^z ug(u)\,du}. \end{array} \end{aligned}$$

Regression functions Regression functions $r_1$, $r_2$, and $r_3$ are defined below. For each function, $0\le x\le 1$.

$$\begin{aligned} \begin{array}{l} \displaystyle {r_1(x)=5x^{10}(1-x)^2+2.5x^2(1-x)^{10},}\\ \\ \displaystyle {r_2(x)={\left\{ \begin{array}{ll} 0.0125-0.05|x-0.25|,\qquad 0\le x\le 0.5,\\ 0.05|x-0.75|-0.0125,\qquad 0.5<x\le 1. \end{array}\right. }}\\ \\ \displaystyle { r_3(x)=\left\{ \begin{array}{ll} 0.047619\sqrt{x},&{}\quad 0\le x<0.1,\\ 0.035186e^{-20x}+0.010297,&{}\quad 0.1\le x<0.3,\\ 0.142857x-0.032473,&{}\quad 0.3\le x<0.35,\\ 0.142857(x-0.35)(x-0.45)+0.017527,&{}\quad 0.35\le x<0.6,\\ 0.151455-0.214286x,&{}\quad 0.6\le x<0.7,\\ 0.001455-0.214286(x-0.7)^3(x-0.4),&{}\quad 0.7\le x<0.8,\\ 0.004762\ln (10x-7.9)+0.012334,&{}\quad 0.8\le x\le 1. \end{array}\right. } \end{array} \end{aligned}$$