Fan, Jianqing Local linear regression smoothers and their minimax efficiencies. (English) Zbl 0773.62029 Ann. Stat. 21, No. 1, 196-216 (1993). Summary: We introduce a smooth version of local linear regression estimators and address their advantages. The MSE and MISE of the estimators are computed explicitly. It turns out that the local linear regression smoothers have nice sampling properties and high minimax efficiency — they are not only efficient in rates but also nearly efficient in constant factors.In the nonparametric regression context, the asymptotic minimax lower bound is developed via the heuristic of the “hardest one-dimensional subproblem” of D. L. Donoho and R. C. Liu [ibid. 19, No. 2, 668-701 (1991; Zbl 0754.62029)]. Connections of the minimax risk with the modulus of continuity are made. The lower bound is also applicable for estimating conditional mean (regression) and conditional quantiles for both fixed and random design regression problems. Cited in 11 ReviewsCited in 274 Documents MSC: 62G07 Density estimation 62G20 Asymptotic properties of nonparametric inference 62G05 Nonparametric estimation Keywords:nonparametric regression; hardest one-dimensional subproblem; mean squared error; mean integrated squared error; MSE; MISE; local linear regression smoothers; minimax efficiency; asymptotic minimax lower bound; minimax risk; modulus of continuity; conditional mean; conditional quantiles; fixed and random design regression problems Citations:Zbl 0754.62029 PDFBibTeX XMLCite \textit{J. Fan}, Ann. Stat. 21, No. 1, 196--216 (1993; Zbl 0773.62029) Full Text: DOI