Statistical inference for intrinsic wavelet estimators of SPD matrices in a log-Euclidean manifold

TL;DR

This paper develops a statistical inference framework for wavelet estimators of SPD matrix curves in a log-Euclidean space, ensuring positive-definiteness and permutation-equivariance, with proven asymptotic properties and practical confidence regions.

Contribution

It introduces a novel wavelet estimator for SPD matrices in a non-Euclidean setting with proven asymptotic normality and confidence sets, advancing nonparametric curve estimation in Riemannian geometry.

Findings

Asymptotic normality of the wavelet estimator is established.

Explicit asymptotic variance expressions are derived.

Numerical simulations validate the inference methods.

Abstract

In this paper we treat statistical inference for an intrinsic wavelet estimator of curves of symmetric positive definite (SPD) matrices in a log-Euclidean manifold. This estimator preserves positive-definiteness and enjoys permutation-equivariance, which is particularly relevant for covariance matrices. Our second-generation wavelet estimator is based on average-interpolation and allows the same powerful properties, including fast algorithms, known from nonparametric curve estimation with wavelets in standard Euclidean set-ups. The core of our work is the proposition of confidence sets for our high-level wavelet estimator in a non-Euclidean geometry. We derive asymptotic normality of this estimator, including explicit expressions of its asymptotic variance. This opens the door for constructing asymptotic confidence regions which we compare with our proposed bootstrap scheme for inference. Detailed numerical simulations confirm the appropriateness of our suggested inference schemes.