On the Choice of Difference Sequence in a Unified Framework for Variance Estimation in Nonparametric Regression
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AbstractDifference-based methods do not require estimating the mean function in nonparametric regression and are therefore popular in practice. In this paper, we propose a unified framework for variance estimation that combines the linear regression method with the higher-order difference estimators systematically. The unified framework has greatly enriched the existing literature on variance estimation that includes most existing estimators as special cases. More importantly, the unified framework has also provided a smart way to solve the challenging difference sequence selection problem that remains a long-standing controversial issue in nonparametric regression for several decades. Using both theory and simulations, we recommend to use the ordinary difference sequence in the unified framework, no matter if the sample size is small or if the signal-to-noise ratio is large. Finally, to cater for the demands of the application, we have developed a unified R package, named VarED, that integrates the existing difference-based estimators and the unified estimators in nonparametric regression and have made it freely available in the R statistical program http://cran.r-project.org/web/packages/.
CitationDai W, Tong T, Zhu L (2017) On the Choice of Difference Sequence in a Unified Framework for Variance Estimation in Nonparametric Regression. Statistical Science 32: 455–468. Available: http://dx.doi.org/10.1214/17-STS613.
SponsorsTiejun Tong's research was supported by the National Natural Science Foundation of China grant (No. 11671338), and the Hong Kong Baptist University grants FRG1/14-15/044, FRG2/15-16/019 and FRG2/15-16/038. Lixing Zhu's research was supported by the Hong Kong Research Grants Council grant (No. HKBU202810). The authors thank the editor, the associate editor and two reviewers for their constructive comments that have led to a substantial improvement of the paper.
PublisherInstitute of Mathematical Statistics