Bump detection in the presence of dependency: Does it ease or does it load?
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We provide the asymptotic minimax detection boundary for a bump, i.e. an abrupt change, in the mean function of a dependent Gaussian process. This will be characterized in terms of the asymptotic behaviour of the bump length and height as well as the dependency structure of the process. A major finding is that for stationary processes the asymptotic minimax detection boundary is generically determined by the value of is spectral density at zero. Hence, a positive long-run variance makes detection harder whereas a negative long-run variance eases bump detection. Finally, our asymptotic analysis is complemented by non-asymptotic results for the subclass of AR(p) processes and confirmed to serve as a good proxy for finite sample scenarios in a simulation study. Our proofs are based on laws of large numbers for non-independent and non-identically distributed arrays of random variables and the asymptotically sharp analysis of the precision matrix of the process.
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