In this talk we treat statistical inference for an intrinsic wavelet estimator of curves of symmetric positive definite (SPD) matrices in a log-Euclidean manifold. Examples for these arise in Diffusion Tensor Imaging or related medical imaging problems as well as in computer vision and for neuroscience problems.

Our proposed linear wavelet (kernel) 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.

At the heart of this talk is the proposition of confidence sets based on our 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. Numerical simulations confirm the appropriateness of our suggested inference schemes.

This is joint work with D. Rademacher (U Heidelberg) and J. Krebs (KU Eichstätt) [J. Stat. Plan. Inf. (2023)]. We also refer to work in progress with G. Bailly (UCLouvain) on the more adaptive non-linear wavelet threshold estimator version of this approach.

In this talk, we tackle the problem of estimating time-varying covariance matrices (TVCM; i.e. covariance matrices with entries being time-dependent curves) characterized by either pronounced local peaks or inhomogeneous smoothness. To address this challenge, wavelet denoising estimators are particularly appropriate. Specifically, we model TVCM using a signal-noise model within the Riemannian manifold of symmetric positive definite matrices (endowed with the log-Euclidean metric) and use the intrinsic wavelet transform, designed for curves in Riemannian manifolds. Within this non-Euclidean framework, the proposed estimators preserve positive definiteness.

Although linear wavelet estimators for smooth TVCM achieve good results in various scenarios, they are less suitable if the underlying curve features singularities. Consequently, our estimator is designed around a nonlinear thresholding scheme, tailored to the characteristics of the noise in covariance matrix regression models. The effectiveness of this novel nonlinear scheme is assessed by deriving mean-squared error consistency and by numerical simulations, and its practical application is demonstrated using electroencephalography data. Joint work with R. von Sachs (UCLouvain).

Comparative evaluation of forecasts of statistical functionals relies on comparing averaged losses of competing forecasts after the realization of the quantity $Y$, on which the functional is based, has been observed. Motivated by high-frequency finance, in this paper we investigate how proxies $tilde Y$ for $Y$ - say volatility proxies - which are observed together with $Y$ can be utilized to improve forecast comparisons.
We extend previous results on robustness of loss functions for the mean to general moments and ratios of moments, and show in terms of the variance of differences of losses that using proxies will increase the power in comparative forecast tests. These results apply both to testing conditional as well as unconditional dominance. Finally, we numerically illustrate the theoretical results, both for simulated high-frequency data as well as for high-frequency log returns of several cryptocurrencies.

The study of the function-indexed empirical process and its weak convergence under dependence has a substantial history. A great share of the literature deals with empirical processes constructed from stationary time series, and recently, some results concerning nonstationary settings have also been published. In contrast, much less is known about the sequential empirical process indexed in function classes, and the literature so far seems to focus on the stationary case. In an attempt to partially close this gap, its weak convergence in a nonstationary setting is studied, and it is shown to be asymptotically equicontinuous provided suitable uniform moment bounds are available for the increments of the empirical process. Restrictions on the dependence are imposed solely in terms of these bounds and their assumed properties in order to achieve a certain degree of generality. Limitations, possible extensions and statistical applications are discussed.