Bump detection in heterogeneous Gaussian regression

TL;DR

This paper investigates how a simultaneous bump in variance affects the difficulty of detecting a mean bump in Gaussian regression, providing explicit detection boundaries and adaptive testing strategies.

Contribution

It introduces a detailed analysis of detection boundaries considering simultaneous mean and variance bumps, including explicit bounds and adaptive methods for unknown bump heights.

Findings

Explicit detection boundaries depending on mean and variance bump behavior

Identification of regimes where variance bumps facilitate detection

Development of adaptive tests for unknown bump heights

Abstract

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.