## Bernoulli

• Bernoulli
• Volume 20, Number 2 (2014), 514-544.

### Testing monotonicity via local least concave majorants

#### Abstract

We propose a new testing procedure for detecting localized departures from monotonicity of a signal embedded in white noise. In fact, we perform simultaneously several tests that aim at detecting departures from concavity for the integrated signal over various intervals of different sizes and localizations. Each of these local tests relies on estimating the distance between the restriction of the integrated signal to some interval and its least concave majorant. Our test can be easily implemented and is proved to achieve the optimal uniform separation rate simultaneously for a wide range of Hölderian alternatives. Moreover, we show how this test can be extended to a Gaussian regression framework with unknown variance. A simulation study confirms the good performance of our procedure in practice.

#### Article information

Source
Bernoulli, Volume 20, Number 2 (2014), 514-544.

Dates
First available in Project Euclid: 28 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.bj/1393593996

Digital Object Identifier
doi:10.3150/12-BEJ496

Mathematical Reviews number (MathSciNet)
MR3178508

Zentralblatt MATH identifier
06291812

#### Citation

Akakpo, Nathalie; Balabdaoui, Fadoua; Durot, Cécile. Testing monotonicity via local least concave majorants. Bernoulli 20 (2014), no. 2, 514--544. doi:10.3150/12-BEJ496. https://projecteuclid.org/euclid.bj/1393593996

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#### Supplemental materials

• Supplementary material: Supplement to “Testing monotonicity via local least concave majorants”. We collect in the supplement [1] the most technical proofs. Specifically, we prove how to reduce to the case $\sigma=1$, we prove (19) and we provide a detailed proof for (17) and all results in Section 4.