Estimation non-param\'etrique de la densit\'e spectrale d'un processus gaussien \'echantillonn\'e al\'eatoirement

TL;DR

This paper develops nonparametric wavelet-based estimators for the spectral density of Gaussian processes with stationary increments, addressing both continuous and randomly sampled observations, with theoretical and practical validation.

Contribution

It introduces new wavelet-based estimators for spectral density in Gaussian processes, including cases with random sampling, and provides theoretical CLTs and MISE estimates.

Findings

Established CLTs for both estimators

Provided MISE estimates for the estimators

Validated methods with simulations and real heartbeat data

Abstract

From a wavelet analysis, one derives a nonparametrical estimator for the spectral density of a Gaussian process with stationary increments. First, the idealistic case of a continuous time path of the process is considered. A punctual Central Limit Theorem (CLT) and an estimation of the Mean Integrate Square Error (MISE) are established. Next, to fit the applications, one considers the case where one observes a path at random times. One built a second estimator obtained by replacing the wavelet coefficients by their discretizations. A second CLT and the corresponding estimation of the MISE are provided. Finally, simulation results and an application on the heartbeat time series of marathon runners are presented.