Open Access
November 2020 Bump detection in the presence of dependency: Does it ease or does it load?
Farida Enikeeva, Axel Munk, Markus Pohlmann, Frank Werner
Bernoulli 26(4): 3280-3310 (November 2020). DOI: 10.3150/20-BEJ1226


We provide the asymptotic minimax detection boundary for a bump, i.e. an abrupt change, in the mean function of a stationary Gaussian process. This will be characterized in terms of the asymptotic behavior of the bump length and height as well as the dependency structure of the process. A major finding is that the asymptotic minimax detection boundary is generically determined by the value of its spectral density at zero. Finally, our asymptotic analysis is complemented by non-asymptotic results for 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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Farida Enikeeva. Axel Munk. Markus Pohlmann. Frank Werner. "Bump detection in the presence of dependency: Does it ease or does it load?." Bernoulli 26 (4) 3280 - 3310, November 2020.


Received: 1 June 2019; Revised: 1 April 2020; Published: November 2020
First available in Project Euclid: 27 August 2020

zbMATH: 07256176
MathSciNet: MR4140545
Digital Object Identifier: 10.3150/20-BEJ1226

Keywords: ARMA processes , change point detection , minimax testing , time series , Toeplitz matrices , weak laws of large numbers

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

Vol.26 • No. 4 • November 2020
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