B0302
Title: Robust mean change point testing in high-dimensional data with heavy tails
Authors: Mengchu Li - University of Warwick (United Kingdom) [presenting]
Yi Yu - University of Warwick (United Kingdom)
Tengyao Wang - London School of Economics (United Kingdom)
Yudong Chen - University of Cambridge (United Kingdom)
Abstract: A mean change point testing problem for high-dimensional data is studied, 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 characterised under the above two tail conditions for the first time in the change point literature and proposes novel testing procedures that attain optimal rates in each of the four regimes up to a poly-iterated logarithmic factor. By comparing with previous results under Gaussian assumptions, the results quantify the costs of heavy-tailedness on the fundamental difficulty of change point testing problems for high-dimensional data. Specifically, when the error vectors follow sub-Weibull distributions, a CUSUM-type statistic is shown to achieve a minimax testing rate up to $\sqrt{\log\log(8n)}$. When the error distributions have 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. Surprisingly, 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.
