## Bernoulli

- Bernoulli
- Volume 24, Number 2 (2018), 1266-1306.

### Bump detection in heterogeneous Gaussian regression

Farida Enikeeva, Axel Munk, and Frank Werner

#### 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.

#### Article information

**Source**

Bernoulli, Volume 24, Number 2 (2018), 1266-1306.

**Dates**

Received: May 2015

Revised: May 2016

First available in Project Euclid: 21 September 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1505980896

**Digital Object Identifier**

doi:10.3150/16-BEJ899

**Mathematical Reviews number (MathSciNet)**

MR3706794

**Zentralblatt MATH identifier**

06778365

**Keywords**

change point detection heterogeneous Gaussian regression minimax testing theory

#### Citation

Enikeeva, Farida; Munk, Axel; Werner, Frank. Bump detection in heterogeneous Gaussian regression. Bernoulli 24 (2018), no. 2, 1266--1306. doi:10.3150/16-BEJ899. https://projecteuclid.org/euclid.bj/1505980896