Abstract
We present a greedy method for simultaneously performing local bandwidth selection and variable selection in nonparametric regression. The method starts with a local linear estimator with large bandwidths, and incrementally decreases the bandwidth of variables for which the gradient of the estimator with respect to bandwidth is large. The method¯called rodeo (regularization of derivative expectation operator)¯conducts a sequence of hypothesis tests to threshold derivatives, and is easy to implement. Under certain assumptions on the regression function and sampling density, it is shown that the rodeo applied to local linear smoothing avoids the curse of dimensionality, achieving near optimal minimax rates of convergence in the number of relevant variables, as if these variables were isolated in advance.
Highlights
Estimating a high-dimensional regression function is notoriously difficult due to the curse of dimensionality
We focus on local linear regression, it should be noted that very similar results are obtained with kernel regression, which requires no matrix inversion
The stochastic analysis of γ1 from its mean is similar to the analysis of A12 and we have |γ1 − ν2f (x) tr(H HR)|/s0(h) = o(1) uniformly
Summary
We present a greedy method for simultaneously performing local bandwidth selection and variable selection in nonparametric regression. The method starts with a local linear estimator with large bandwidths, and incrementally decreases the bandwidth of variables for which the gradient of the estimator with respect to bandwidth is large. The method—called rodeo (regularization of derivative expectation operator)—conducts a sequence of hypothesis tests to threshold derivatives, and is easy to implement. Under certain assumptions on the regression function and sampling density, it is shown that the rodeo applied to local linear smoothing avoids the curse of dimensionality, achieving near optimal minimax rates of convergence in the number of relevant variables, as if these variables were isolated in advance
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