## Asymptotically optimal difference-based estimation of variance in nonparametric regression.(English)Zbl 1377.62102

Summary: We define and compute asymptotically optimal difference sequences for estimating error variance in homoscedastic nonparametric regression. Our optimal difference sequences do not depend on unknowns, such as the mean function, and provide substantial improvements over the suboptimal sequences commonly used in practice. For example, in the case of normal data the usual variance estimator based on symmetric second-order differences is only 64% efficient relative to the estimator based on optimal second-order differences. The efficiency of an optimal $$m$$th-order difference estimator relative to the error sample variance is $$2m/(2m+1)$$. Again this is for normal data, and increases as the tails of the error distribution become heavier.

### MSC:

 62G05 Nonparametric estimation
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