Finite-Sample Variance of Local Polynomials: Analysis and Solutions
分析了非参数回归中局部多项式在有限样本下的方差问题，发现稀疏设计区域会导致方差无界，并提出了局部带宽增加和局部多项式岭回归两种改进方法。
Abstract Fitting local polynomials in nonparametric regression has a number of advantages. The attractive theoretical features are in a partial contradiction to variance properties for random design and to practical experience over a broad range of situations. No upper bound can be given for the conditional variance. The unconditional variance is infinite when using optimal weights with compact support. Properties are better for Gaussian weights. We analyze local polynomials for finite sample size, both theoretically and numerically. It turns out that difficulties arise in sparse regions in the realization of the design, when the realization has locally a small variance and/or a skew empirical distribution. Two small-sample modifications of local polynomials are presented: local increase of bandwidth in sparse regions of the design, and local polynomial ridge regression. Both modifications combine a good finite-sample behavior with the asymptotic advantages of local polynomials.