Minimax rates in sparse, high-dimensional change point detection
Haoyang Liu, Chao Gao, Richard J. Samworth
The Annals of Statistics · 2021 · Volume 49 · Issue 2
Abstract
We study the detection of a sparse change in a high-dimensional mean vector as a minimax testing problem. Our first main contribution is to derive the exact minimax testing rate across all parameter regimes for n independent, p-variate Gaussian observations. This rate exhibits a phase transition when the sparsity level is of order ploglog(8n) and has a very delicate dependence on the sample size: in a certain sparsity regime, it involves a triple iterated logarithmic factor in n. Further, in a dense asymptotic regime, we identify the sharp leading constant, while in the corresponding sparse asymptotic regime, this constant is determined to within a factor of 2. Extensions that cover spatial and temporal dependence, primarily in the dense case, are also provided.
Keywords
62C20, iterated logarithm, Minimax detection boundary, time series
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