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May 2018 Bump detection in heterogeneous Gaussian regression
Farida Enikeeva, Axel Munk, Frank Werner
Bernoulli 24(2): 1266-1306 (May 2018). DOI: 10.3150/16-BEJ899


We analyze the effect of a heterogeneous variance on bump detection in a Gaussian regression model. To this end, we allow for a simultaneous bump in the variance and specify its impact on the difficulty to detect the null signal against a single bump with known signal strength. This is done by calculating lower and upper bounds, both based on the likelihood ratio.

Lower and upper bounds together lead to explicit characterizations of the detection boundary in several subregimes depending on the asymptotic behavior of the bump heights in mean and variance. In particular, we explicitly identify those regimes, where the additional information about a simultaneous bump in variance eases the detection problem for the signal. This effect is made explicit in the constant and/or the rate, appearing in the detection boundary.

We also discuss the case of an unknown bump height and provide an adaptive test and some upper bounds in that case.


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Farida Enikeeva. Axel Munk. Frank Werner. "Bump detection in heterogeneous Gaussian regression." Bernoulli 24 (2) 1266 - 1306, May 2018.


Received: 1 May 2015; Revised: 1 May 2016; Published: May 2018
First available in Project Euclid: 21 September 2017

zbMATH: 06778365
MathSciNet: MR3706794
Digital Object Identifier: 10.3150/16-BEJ899

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability


Vol.24 • No. 2 • May 2018
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