Minimax-rate adaptive nonparametric regression has been intensively studied under the assumption of independent or uncorrelated errors in the literature. In many applications, however, the errors are dependent, including both short- and long-range dependent situations. In such a case, adaptation with respect to the unknown dependence is important. We present a general result in this direction under Gaussian errors. It is assumed that the covariance matrix of the errors is known to be in a list of specifications possibly including independence, short-range dependence and long-range dependence as well. The regression function is known to be in a countable (or uncountable but well-structured) collection of function classes. Adaptive estimators are constructed to attain the minimax rate of convergence automatically for each function class under each correlation specification in the corresponding lists.
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Acknowledgements This work was supported by National Natural Science Foundation of China (Grant No. 61572109). This paper is dedicated to Professor Lincheng Zhao, in celebration of his 75th birthday. Both authors were inspired by Professor Lincheng Zhao as a leading scholar in statistical sciences. The second author had the honor of being his MS student (co-advised by Professor Baiqi Miao), and received the solid initial training of minimax philosophy and tools. The authors greatly appreciate the helpful comments by two referees on improving their work.
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- adaptive estimation
- long-range dependence
- nonparametric regression
- rate of convergence