On asymptotically optimal wavelet estimation of trend functions under long-range dependence

TL;DR

This paper develops an adaptive wavelet estimation method for trend functions in long-range dependent Gaussian time series, providing asymptotic optimality and demonstrating strong performance through simulations.

Contribution

It introduces a data-adaptive wavelet estimator that achieves asymptotic optimality under long-range dependence, balancing smoothness and discontinuities.

Findings

Optimal mean integrated squared error derived

Estimator performs well for smooth and discontinuous trends

Simulation results confirm theoretical asymptotic behavior

Abstract

We consider data-adaptive wavelet estimation of a trend function in a time series model with strongly dependent Gaussian residuals. Asymptotic expressions for the optimal mean integrated squared error and corresponding optimal smoothing and resolution parameters are derived. Due to adaptation to the properties of the underlying trend function, the approach shows very good performance for smooth trend functions while remaining competitive with minimax wavelet estimation for functions with discontinuities. Simulations illustrate the asymptotic results and finite-sample behavior.