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

url: https://modelslab.com/ai-news/adaptive-estimation-for-the-nonparametric-bivariate-additive-model-in-random-design-with-long-memory-dependent-errors
canonical: https://modelslab.com/ai-news/adaptive-estimation-for-the-nonparametric-bivariate-additive-model-in-random-design-with-long-memory-dependent-errors
type: website
component: Content/AINews/Single
generated_at: 2026-01-20T14:32:31.608893Z
---

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

>   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.



---

*This markdown version is optimized for AI agents and LLMs.*

**Links:**
- [Website](https://modelslab.com)
- [API Documentation](https://docs.modelslab.com)
- [Blog](https://modelslab.com/blog)

---
*Generated by ModelsLab - 2026-01-20*