Consistency and normality of Huber-Dutter estimators for partial linear model

For partial linear model Y = X ^τ β ₀ + g ₀(T) + ∈ with unknown β ₀ ∈ ȑ ^d and an unknown smooth function g ₀, this paper considers the Huber-Dutter estimators of β ₀, scale σ for the errors and the function g ₀ approximated by the smoothing B-spline functions, respectively. Under some regularity conditions, the Huber-Dutter estimators of β ₀ and σ are shown to be asymptotically normal with the rate of convergence n ^−1/2 and the B-spline Huber-Dutter estimator of g ₀ achieves the optimal rate of convergence in nonparametric regression. A simulation study and two examples demonstrate that the Huber-Dutter estimator of β ₀ is competitive with its M-estimator without scale parameter and the ordinary least square estimator. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10671106, 10771017)