Nonparametric adaptive time-dependent multivariate function estimation

TL;DR

This paper develops minimax optimal wavelet estimators for nonparametric, time-dependent multivariate functions observed in Gaussian noise, with theoretical bounds and practical demonstrations.

Contribution

It introduces adaptive nonlinear wavelet estimators for inhomogeneous time-dependent functions, achieving asymptotic minimax optimality.

Findings

Derived minimax lower bounds for $L^2$-risk as noise intensity decreases.

Proposed adaptive nonlinear wavelet estimators are asymptotically optimal.

Validated estimators with simulated and real data examples.

Abstract

We consider the nonparametric estimation problem of time-dependent multivariate functions observed in a presence of additive cylindrical Gaussian white noise of a small intensity. We derive minimax lower bounds for the $L^2$-risk in the proposed spatio-temporal model as the intensity goes to zero, when the underlying unknown response function is assumed to belong to a ball of appropriately constructed inhomogeneous time-dependent multivariate functions, motivated by practical applications. Furthermore, we propose both non-adaptive linear and adaptive non-linear wavelet estimators that are asymptotically optimal (in the minimax sense) in a wide range of the so-constructed balls of inhomogeneous time-dependent multivariate functions. The usefulness of the suggested adaptive nonlinear wavelet estimator is illustrated with the help of simulated and real-data examples.