Wavelet estimation for the nonparametric additive model in random design and long-memory dependent errors

Rida Benhaddou and Qing Liu

Journal of Nonparametric Statistics, 2024, vol. 36, issue 4, 1088-1113

Abstract: We investigate the nonparametric additive regression estimation in random design and long-memory errors and construct adaptive thresholding estimators based on wavelet series. The proposed approach achieves asymptotically near-optimal convergence rates when the unknown function and its univariate additive components belong to Besov space. We consider the problem under two noise structures; (1) homoskedastic Gaussian long memory errors and (2) heteroskedastic Gaussian long memory errors. In the homoskedastic long-memory error case, the estimator is completely adaptive with respect to the long-memory parameter. In the heteroskedastic long-memory case, the estimator may not be adaptive with respect to the long-memory parameter unless the heteroskedasticity is of polynomial form. In either case, the convergence rates depend on the long-memory parameter only when long-memory is strong enough, otherwise, the rates are identical to those under i.i.d. errors. In addition, convergence rates are free from the curse of dimensionality.

Date: 2024
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DOI: 10.1080/10485252.2023.2296523

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