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( @math ) 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.
Higher Criticism is a method for detecting signals that are both sparse and weak. Although rst proposed in cases where the noise variables are independent, Higher Criticism also has reasonable performance in settings where those variables are correlated. In this paper we show that, by exploiting the nature of the correlation, performance can be improved by using a modied approach which exploits the potential advantages that correlation has to oer. Indeed, it turns out that the case of independent noise is the most dicult of all, from a statistical viewpoint, and that more accurate signal detection (for a given level of signal sparsity and strength) can be obtained when correlation is present. We characterize the advantages of correlation by showing how to incorporate them into the denition of an optimal detection boundary. The boundary has particularly attractive properties when correlation decays at a polynomial rate or the correlation matrix is Toeplitz.