Variance function estimation in nonparametric regression is considered. We derived the minimax rate of convergence for both the univariate case and the multivariate case through difference based method. It was shown that the optimal order of differences depends on the number of dimensions. We are particularly interested in the effect of the unknown mean function on the estimation of the variance function. Our results indicate that, 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. In addition the results also correct the optimal rate claimed in Hall and Carroll (1989, JRSSB).
Also, a wavelet thresholding approach to adaptive variance function estimation in heteroscedastic nonparametric regression is introduced for the univariate case. The estimator is constructed by applying wavelet thresholding to the squared first-order differences of the observations. We show that the variance function estimator is nearly optimally adaptive to the smoothness of both the mean and variance functions. The estimator is shown to achieve the optimal adaptive rate of convergence under the pointwise mean squared error simultaneously over a range of smoothness classes. The estimator is also adaptively within a logarithmic factor of the minimax risk under the global integrated mean squared error. Numerical implementation and simulation results are also discussed.
|School:||University of Pennsylvania|
|School Location:||United States -- Pennsylvania|
|Source:||DAI-B 69/09, Dissertation Abstracts International|
|Keywords:||Adaptive estimation, Nonparametric regression, Variance function estimation, Wavelets|
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