Linear Wavelet Estimation for Regression with Additive-Multiplicative Noise for Ergodic Data

Objectives This paper builds on the linear wavelet estimator for nonparametric regression models incorporating additive and multiplicative noise under i.i.d. conditions. Material and Methods We extend their approach to scenarios where the covariate process is strictly stationary and ergodic. A significant advancement is the establishment of a martingale framework that elucidates the asymptotic properties of the projection estimator without requiring assumptions beyond ergodicity. Results We demonstrate that this estimator achieves the minimax-optimal convergence rate concerning the mean integrated square error (MISE) across Besov spaces. Additionally, we propose a data-driven technique for selecting the truncation parameter, and we validate our theoretical results through numerical simulations conducted on ergodic data. Conclusion This extension broadens the theoretical and practical reach of wavelet-based regression to a wide class of dependent data scenarios and provides a robust estimation framework for ergodic data common in fields like econometrics and signal processing.