Local CQR Smoothing: An Efficient and Safe Alternative to Local Polynomial Regression

Summary.  Local polynomial regression is a useful non-parametric regression tool to explore fine data structures and has been widely used in practice. We propose a new non-parametric regression technique called local composite quantile regression smoothing to improve local polynomial regression further. Sampling properties of the estimation procedure proposed are studied. We derive the asymptotic bias, variance and normality of the estimate proposed. The asymptotic relative efficiency of the estimate with respect to local polynomial regression is investigated. It is shown that the estimate can be much more efficient than the local polynomial regression estimate for various non-normal errors, while being almost as efficient as the local polynomial regression estimate for normal errors. Simulation is conducted to examine the performance of the estimates proposed. The simulation results are consistent with our theoretical findings. A real data example is used to illustrate the method proposed.     

具有Riemann函数核的奇积分方程解的讨论

本文借助于偏微分方程的Riemann方法,讨论一类具有Riemann函数核的奇异积分方程的解的存在性。     

Nonparametric testing of closeness between two unknown distribution functions

Based on the kernel integrated square difference and applying a central limit theorem for degenerate V-statistic proposed by Hall (1984), this paper proposes a consistent nonparametric test of closeness between two unknown density functions under quite mild conditions. We only require the unknown density functions to be bounded and continuous. Monte Carlo simulations show that the proposed tests perform well for moderate sample sizes.     

Asymptotic behaviour of the mean integrated squared error of kernel density estimators for dependent observations

Let X₁, X₂, … be a strictly stationary sequence of observations, and g be the joint density of (X₁, …, X_d) for some fixed d ? 1. We consider kernel estimators of the density g. The asymptotic behaviour of the mean integrated squared error of the kernel estimators is obtained under an assumption of weak dependence between the observations.     

Estimation of a symmetric density

Kraft, Lepage, and van Eeden (1985) have suggested using a symmetrized version of the kernel estimator when the true density f of the observation is known to be symmetric around a possibly unknown point θ. The effect of this symmetrization device depends on the smoothness of f * f(x) = f f(x+t)f(t) dt at zero. We show that if θ has to be estimated and if f is not absolutely continuous, symmetrization may deteriorate the estimate.     

A note on bias reduction in variable-kernel density estimates

It is well known that the inverse-square-root rule of Abramson (1982) for the bandwidth h of a variable-kernel density estimator achieves a reduction in bias from the fixed-bandwidth estimator, even when a nonnegative kernel is used. Without some form of “clipping” device similar to that of Abramson, the asymptotic bias can be much greater than O(h⁴) for target densities like the normal (Terrell and Scott 1992) or even compactly supported densities. However, Abramson used a nonsmooth clipping procedure intended for pointwise estimation. Instead, we propose a smoothly clipped estimator and establish a globally valid, uniformly convergent bias expansion for densities with uniformly continuous fourth derivatives. The main result extends Hall's (1990) formula (see also Terrell and Scott 1992) to several dimensions, and actually to a very general class of estimators. By allowing a clipping parameter to vary with the bandwidth, the usual O(h⁴) bias expression holds uniformly on any set where the target density is bounded away from zero.