A0712
Title: Robust mean change point testing in high-dimensional data with heavy tails
Authors: Mengchu Li - University of Birmingham (United Kingdom) [presenting]
Abstract: The purpose is to study mean change point testing problems for high-dimensional data with exponentially- or polynomially-decaying tails. In each case, depending on the $\ell_0$-norm of the mean change vector, dense and sparse regimes are separately considered. The boundary between the dense and sparse regimes is characterized under the above two tail conditions for the first time in the change point literature, and novel testing procedures that attain optimal rates are proposed in each of the four regimes up to a poly-iterated logarithmic factor. When the error distributions possess exponentially decaying tails, a near-optimal CUSUM-type statistic is considered. As for polynomially-decaying tails, admitting bounded $\alpha$-th moments for some $\alpha \geq 4$, a median-of-means-type test statistic that achieves a near-optimal testing rate is introduced in both dense and sparse regimes. The investigation in the even more challenging case of $2 \leq \alpha < 4$, unveils a new phenomenon that the minimax testing rate has no sparse regime, i.e.\ testing sparse changes is information-theoretically as hard as testing dense changes. Finally, various extensions are considered where near-optimal performances are also obtained, including testing against multiple change points, allowing temporal dependence as well as fewer than two finite moments in the data-generating mechanisms.
