基于Walsh平均的非参数回归模型的稳健估计

Robust Walsh-Average-Based Spline Estimator in Nonparametric Regression Model

由于非参数回归模型复杂灵活,被广泛应用。在众多估计方法中,最小二乘法最为常用,一般情况下具有良好的性质,但在处理厚尾分布及异常点时表现的不够稳健。本文针对此,提出了基于Walsh平均的稳健样条估计。我们理论地推导了估计结果的相合性和渐近正态性;并与多项式样条回归做比较。计算得Walsh平均的样条估计相对于多项式样条回归的渐近相对效率与Wilcoxon符号秩检验相对于t-检验的渐近相对效率是一样的。在正态情形下我们的方法与多项式样条回归差不多,在非正态情形下,我们的方法表现更为稳健,效率明显优于多项式样条回归。

Nonparametric regression model is widely used for its flexibility and complexity.The LS method is always efficient in nonparametric regression model fitting,but it may suffer when the error follows a heavy-tailed distribution or in the presence of outliers.To overcome that issue,we the propose Robust Walsh-Average-Based Spline Estimator.We theoretically proof that our new approach is consistent and asymptotic normal,and compare with Polynomial spline estimation.Its asymptotic relative efficiency with respect to the Polynomial spline estimation is the same as that of the signed-rank Wilcoxon test in comparison with the t-test.Its efficiency compared to Polynomial spline estimation is high.Even for normal,its AREs are merely slightly smaller than 1.