At the edge of Donsker's Theorem: Asymptotics of multiscale scan statistics

TL;DR

This paper develops a feasible multiscale testing method for nonparametric inference that maintains optimal detection capabilities and extends to complex time series, using a novel weak convergence approach where classical Donsker's theorem fails.

Contribution

It introduces a new theoretical framework for multiscale tests based on multiplicative weighting and weak convergence, enabling broader applicability and improved statistical feasibility.

Findings

Preserves optimal detection properties of multiscale tests

Extends applicability to nonstationary nonlinear time series

Introduces a novel weak convergence framework in H"older spaces

Abstract

For nonparametric inference about a function, multiscale testing procedures resolve the need for bandwidth selection and achieve asymptotically optimal detection performance against a broad range of alternatives. However, critical values strongly depend on the noise distribution, and we argue that existing methods are either statistically infeasible, or asymptotically sub-optimal. To address this methodological challenge, we show how to develop a feasible multiscale test via weak convergence arguments, by replacing the additive multiscale penalty with a multiplicative weighting. 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. Inference for signal discovery, goodness-of-fit testing of regression functions, and multiple changepoint detection is studied in detail, and we apply the new methodology to analyze the April 2025 power blackout on the Iberian peninsula. Our methodology is enabled by a novel functional central limit in H\"older spaces with critical modulus of continuity, where Donsker's theorem fails to hold due to lack of tightness. Probabilistically, we discover a novel form of thresholded weak convergence that holds only in the upper support of the distribution.