Statistical inference for wavelet curve estimators of symmetric positive definite matrices

In this paper we treat statistical inference for a wavelet estimator of curves of symmetric positive definite (SPD) using the log-Euclidean distance. 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 (AI) 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 AI 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.