Tow-Stage Estimation For Additive Models By Local Likelihood Adjustment

In this paper, the two-stage estimation for nonparametric additive models by local likelihood adjustment is investigated. Different from Horowitz and Mammen's (2004) two-stage estimation, our first-stage estimators are designed not only for dimension reduction but also as initial approximations to all the additive components, the first-stage estimators used parametric technique and the second-stage estimators are obtained by using one-dimensional nonparametric technique to refine the first-stage ones. From this procedure, we can reveal a relationship between the regression function spaces and convergence rate, and then provide estimators that are of optimality in the sense that one-dimensional meansquared error (MSE) of the O(n^-1), which can be achieved when the underlying models are actually some parametric models, we all know in the sense that one-dimensional nonparametric is O(n^-4/5. This shows that our estimation procedure is good in certain sense. Furthermore, the method can be readily extended to handle additive models with any smooth known links, although in this paper the link is identity. Simulation studies show that our estimator really have some good behavior.The Section 2 of this paper , introduce the two-stage estimators concretely. The asymptotic behavior of the proposed estimators is investigated, and give the bias and variance in Section 3. In Section 4, additive models with a known link and random design models are discussed simply . Simulation and compare the advantage of two-stage estimation, and illustrate the theoretical conclusions in Section 5. The technical proofs are relegated to Section 6.