单指标模型是一类非常重要的半参数回归模型, 不仅可以降低数据维数, 克服多元数据中的``维数祸根''问题, 而且能抓住高维数据的主要特征. 文章研究部分线性单指标模型的M-估计, 利用B-样条近似技术逼近非参数函数,提出了获得模型中未知参数M-估计的方法, 在一些正则条件下, 研究了回归函数以及回归系数的M-估计的渐近性质. 随机模拟结果表明了文中M-估计具有稳健性.

The single-index model is an important tool in multivariate nonparametric regression. Not only it can reduce the dimension of data and avoid the so-called ``curse of dimensionality" in multivariate nonparametric regression, but also can capture the main feature of high-dimensional data. This paper deals with M-estimator for the partial linear single-index model. An M-estimator procedure based on B-spline approximation is proposed in this paper. Under some mild regular conditions, the asymptotic properties of the proposed M-estimator of unknown function and the M-estimator of parameter are investigated. A finite sample simulation study indicates that the M-estimators proposed in this paper are robust.