We study a mean change point testing problem for high-dimensional data, with
exponentially- or polynomially-decaying tails. In each case, depending on the
ℓ0-norm of the mean change vector, we separately consider dense and
sparse regimes. We characterise the boundary between the dense and sparse
regimes under the above two tail conditions for the first time in the change
point literature and propose novel testing procedures that attain optimal rates
in each of the four regimes up to a poly-iterated logarithmic factor. Our
results quantify the costs of heavy-tailedness on the fundamental difficulty of
change point testing problems for high-dimensional data by comparing to the
previous results under Gaussian assumptions.
To be specific, when the error vectors follow sub-Weibull distributions, a
CUSUM-type statistic is shown to achieve a minimax testing rate up to
loglog(8n). When the error distributions have polynomially-decaying
tails, admitting bounded α-th moments for some α≥4, we
introduce a median-of-means-type test statistic that achieves a near-optimal
testing rate in both dense and sparse regimes. In particular, in the sparse
regime, we further propose a computationally-efficient test to achieve the
exact optimality. Surprisingly, our investigation in the even more challenging
case of 2≤α<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. This phenomenon
implies a phase transition of the minimax testing rates at α=4.Comment: 50 pages, 1 figur