Wavelet Variance for Random Fields: an M-Estimation Framework

TL;DR

This paper introduces a robust M-estimation framework for wavelet variance in multidimensional random fields, providing improved asymptotic properties and finite sample performance over existing methods.

Contribution

It extends wavelet variance estimation to multidimensional fields with a robust M-estimation approach, including joint asymptotic analysis and conditions for high-dimensional cases.

Findings

The estimator demonstrates superior finite sample performance.

Simulation studies confirm robustness and accuracy.

Applications show practical effectiveness.

Abstract

We present a general M-estimation framework for inference on the wavelet variance. This framework generalizes the results on the scale-wise properties of the standard estimator and extends them to deliver the joint asymptotic properties of the estimated wavelet variance vector. Moreover, this is achieved by extending the estimation of the wavelet variance to multidimensional random fields and by stating the necessary conditions for these properties to hold when the size of the wavelet variance vector goes to infinity with the sample size. Finally, these results generally hold when using bounded estimating functions thereby delivering a robust framework for the estimation of this quantity which improves over existing methods both in terms of asymptotic properties and in terms of its finite sample performance. The proposed estimator is investigated in simulation studies and different applications highlighting its good properties.