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Local linear regression smoothers and their minimax efficiencies. (English) Zbl 0773.62029

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.


MSC:
62G07Density estimation
62G20Nonparametric asymptotic efficiency
62G05Nonparametric estimation