Abstract
Let \(\{X_n: n\in {{\mathbb {N}}}\}\) be a linear process with density function \(f(x)\in L^2({{\mathbb {R}}})\). We study wavelet density estimation of f(x). Under some regular conditions on the characteristic function of innovations, we achieve, based on the number of nonzero coefficients in the linear process, the minimax optimal convergence rate of the integrated mean squared error of density estimation. Considered wavelets have compact support and are twice continuously differentiable. The number of vanishing moments of mother wavelet is proportional to the number of nonzero coefficients in the linear process and to the rate of decay of characteristic function of innovations. Theoretical results are illustrated by simulation studies with innovations following Gaussian, Cauchy and chi-squared distributions.
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Acknowledgements
We thank Johannes Schmidt-Hieber and Fangjun Xu for many helpful comments. We are also grateful to the reviewers for numerous constructive remarks and comments that helped to improve the manuscript. The research of Aleksandr Beknazaryan and Peter Adamic is partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant RGPIN-2017-05595. The research of Hailin Sang is partially supported by the Simons Foundation Grant 586789, USA.
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Beknazaryan, A., Sang, H. & Adamic, P. On the integrated mean squared error of wavelet density estimation for linear processes. Stat Inference Stoch Process 26, 235–254 (2023). https://doi.org/10.1007/s11203-022-09281-9
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DOI: https://doi.org/10.1007/s11203-022-09281-9