Local Walsh-average regression

Local polynomial regression is widely used for nonparametric regression. However, the efficiency of least squares (LS) based methods is adversely affected by outlying observations and heavy tailed distributions. On the other hand, the least absolute deviation (LAD) estimator is more robust, but may be inefficient for many distributions of interest. Kai et al. (2010) [13] propose a nonparametric regression technique called local composite quantile regression (LCQR) smoothing to improve local polynomial regression further. However, the performance of LCQR depends on the choice of the number of quantiles to combine, a meta parameter which plays a vital role in balancing the performance of LS and LAD based methods. To overcome this issue, we propose a novel method termed the local Walsh-average regression (LWAR) estimator by minimizing a locally Walsh-average based loss function. Under the same assumptions in Kai et al. (2010) [13], we theoretically show that the proposed estimator is highly efficient across a wide spectrum of distributions. Its asymptotic relative efficiency with respect to the LS based method is closely related to that of the signed-rank Wilcoxon test in comparison with the t-test. Both of the theoretical and numerical results demonstrate that the performance of the new approach and LCQR is at least comparable in estimating the nonparametric regression function or its derivatives and in some cases the new approach performs better than the LCQR with commonly recommended number of quantiles, especially for estimating the regression function.