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.
Model can be seen as prototypical for the more complex situation when several bumps are to be detected. We do not intend to provide novel methodology for this situation in this paper, rather Theorem provides a benchmark for detecting such a bump which then can be used to evaluate the detection power of any method designed for this task. Minimax detection has a long history, see e.g. the seminal series of papers by or the monograph by . More recently, provided a general approach based on convex optimization. In case of independent observations, the problem of detecting a bump has been considered in , and our strategy of proof for the lower bound is adopted from @cite_0 . The additional difficulty we have to treat is due to the dependency structure @math in . We also mention for a model with a simultaneous bump in the variance, and for heterogeneous noise, however still restricted to independent observations.
@cite_0
1972902995
Suppose that one observes a process Y on the unit interval, where dY(t) = n 1 2f(t)dt + dW(t) with an unknown function parameter f, given scale parameter n N > 1 (sample size) and standard Brownian motion W. We propose two classes of tests of qualitative nonparametric hypotheses about f such as monotonicity or concavity. These tests are asymptotically optimal and adaptive in a certain sense. They are constructed via a new class of multiscale statistics and an extension of Levy's modulus of continuity of Brownian Motion.