On rate-optimal nonparametric wavelet regression with long memory moving average errors

We consider the wavelet-based estimators of mean regression function with long memory moving average errors and investigate their asymptotic rates of convergence based on thresholding of empirical wavelet coefficients. We show that these estimators achieve nearly optimal minimax convergence rates within a logarithmic term over a large range of Besov function classes <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$B^{s}_{p,q}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msubsup> <mrow> <mi>B</mi> </mrow> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msubsup> </mrow> </math> </EquationSource> </InlineEquation>. Therefore, in the presence of long memory non-Gaussian moving average noise, wavelet estimators still achieve nearly optimal convergence rates and provide explicitly the extraordinary local adaptability. The theory is illustrated with some numerical examples. Copyright Springer Science+Business Media Dordrecht 2013