Adaptive estimation for the nonparametric bivariate additive model in random design with long-memory dependent errors

TL;DR

This paper develops adaptive wavelet-based estimators for nonparametric bivariate additive models with long-memory errors, achieving near-optimal convergence rates under various noise structures and extending to higher dimensions.

Contribution

It introduces a novel adaptive estimation method for additive models with long-memory errors, handling both homoskedastic and heteroskedastic cases, and extends to higher dimensions without curse of dimensionality.

Findings

Estimator is fully adaptive in homoskedastic case.

Convergence rates depend on long-memory strength.

Method extends to r-dimensional additive models.

Abstract

We investigate the nonparametric bivariate additive regression estimation in the 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. The proposed approach is extended to the general $r$-dimensional additive case, with $r>2$, and the corresponding convergence rates are free from the curse of dimensionality.