TY - JOUR
T1 - An adaptive two-stage estimation method for additive models
AU - Lin, Lu
AU - Cui, Xia
AU - ZHU, Lixing
N1 - This research was supported by a NNSF project of China, a NBRP (973 Program) of China, an RFDP of China, two NSF projects of Shandong Province of China and a grant from Research Grants Council of Hong Kong, Hong Kong, China.
PY - 2009/6
Y1 - 2009/6
N2 - In this paper, a two-stage estimation method for non-parametric additive models is investigated. Differing from Horowitz and Mammen's two-stage estimation, our first-stage estimators are designed not only for dimension reduction but also as initial approximations to all of the additive components. The second-stage estimators are obtained by using one-dimensional non-parametric techniques 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 optimal in the sense that, better than the usual one-dimensional mean-squared error (MSE) of the order n-4/5, the MSE of the order n-1; can be achieved when the underlying models are actually parametric. This shows that our estimation procedure is adaptive in a certain sense. Also it is proved that the bandwidth that is selected by cross-validation depends only on one-dimensional kernel estimation and maintains the asymptotic optimality. Simulation studies show that the new estimators of the regression function and all components outperform the existing estimators, and their behaviours are often similar to that of the oracle estimator.
AB - In this paper, a two-stage estimation method for non-parametric additive models is investigated. Differing from Horowitz and Mammen's two-stage estimation, our first-stage estimators are designed not only for dimension reduction but also as initial approximations to all of the additive components. The second-stage estimators are obtained by using one-dimensional non-parametric techniques 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 optimal in the sense that, better than the usual one-dimensional mean-squared error (MSE) of the order n-4/5, the MSE of the order n-1; can be achieved when the underlying models are actually parametric. This shows that our estimation procedure is adaptive in a certain sense. Also it is proved that the bandwidth that is selected by cross-validation depends only on one-dimensional kernel estimation and maintains the asymptotic optimality. Simulation studies show that the new estimators of the regression function and all components outperform the existing estimators, and their behaviours are often similar to that of the oracle estimator.
KW - Adjustment
KW - Kernel estimation
KW - Local fitting
KW - Non-parametric additive model
KW - Non-parametric adjustment
KW - Semiparametric method
UR - http://www.scopus.com/inward/record.url?scp=67949085075&partnerID=8YFLogxK
U2 - 10.1111/j.1467-9469.2008.00629.x
DO - 10.1111/j.1467-9469.2008.00629.x
M3 - Journal article
AN - SCOPUS:67949085075
SN - 0303-6898
VL - 36
SP - 248
EP - 269
JO - Scandinavian Journal of Statistics
JF - Scandinavian Journal of Statistics
IS - 2
ER -