At the Edge of Donsker's Theorem: Asymptotics of Multiscale Scan Statistics

Abstract: Bandwidth selection is a critical aspect of statistical tests based on nonparametric regression or density estimation, and multiscale test procedures address this issue by incorporating all bandwidths simultaneously.

Remarkably, this approach avoids a severe multiple testing penalty and instead offers a "free lunch" in statistical inference: the test achieves asymptotically optimal detection power across all scales without prior knowledge of the effect's length scale.

However, constructing valid multiscale procedures is highly dependent on the sampling setting and often relies on bounding unknown and difficult-to-identify quantities, such as exponential tail bounds.

In this work, we demonstrate how to obtain a feasible multiscale test for a broad range of sampling settings via weak convergence arguments, by replacing the additive multiscale penalty with a multiplicative weighting. Specifically, we derive tightness conditions for the functional central limit in Hölder spaces with a critical modulus of continuity, where Donsker's theorem fails to hold. Probabilistically, we discover a novel form of restricted weak convergence that holds only in the tail of the distribution.

This new theoretical foundation preserves the optimal detection properties of multiscale tests and extends their applicability to nonstationary nonlinear time series via a tailored bootstrap scheme. Applications to signal discovery, goodness-of-fit testing of regression functions, and multiple changepoint detection are studied in detail.

By offering a fresh perspective on multiscale statistics, we aim to facilitate their adaptation to a broader range of statistical problems.

Joint work with Johann Köhne (University of Göttingen)