Variance Function Estimation In Multivariate Nonparametric Regression
Tony Cai, Michael Levine, and Lie Wang
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Abstract:
Variance function estimation in multivariate nonparametric regression
is considered and the minimax rate of convergence is established. Our
work uses the approach that generalizes the one used in Munk,
Bissantz, Wagner and Freitag (2005, JRSSB) for the constant variance
case. As
is the case when the number of dimensions d = 1,
and very much contrary to the common practice, it is often not
desirable to base the estimator of the variance function on the
residuals from an optimal estimator of the mean. Instead it is
desirable to use estimators of the mean with minimal bias. Another
important conclusion is that the first order difference-based
estimator that achieves minimax rate of convergence in one-dimensional
case does not do the same in the high dimensional case.
Instead, the optimal order of differences depends on the number of dimensions.
- Paper:
pdf file.
- Other related papers:
Wang, L., Brown, L.D., Cai, T. & Levine, M. (2008).
Effect of mean on variance function estimation in nonparametric regression.
The Annals of Statistics 36, 646-664.Cai, T. & Wang, L. (2008).
Adaptive variance function estimation in heteroscedastic nonparametric regression.
The Annals of Statistics 36, 2025-2054.