The Annals of Statistics

Adaptive estimation for Hawkes processes; application to genome analysis

Patricia Reynaud-Bouret and Sophie Schbath

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The aim of this paper is to provide a new method for the detection of either favored or avoided distances between genomic events along DNA sequences. These events are modeled by a Hawkes process. The biological problem is actually complex enough to need a nonasymptotic penalized model selection approach. We provide a theoretical penalty that satisfies an oracle inequality even for quite complex families of models. The consecutive theoretical estimator is shown to be adaptive minimax for Hölderian functions with regularity in (1/2, 1]: those aspects have not yet been studied for the Hawkes’ process. Moreover, we introduce an efficient strategy, named Islands, which is not classically used in model selection, but that happens to be particularly relevant to the biological question we want to answer. Since a multiplicative constant in the theoretical penalty is not computable in practice, we provide extensive simulations to find a data-driven calibration of this constant. The results obtained on real genomic data are coherent with biological knowledge and eventually refine them.

Article information

Ann. Statist., Volume 38, Number 5 (2010), 2781-2822.

First available in Project Euclid: 20 July 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62G20: Asymptotic properties
Secondary: 46N60: Applications in biology and other sciences 65C60: Computational problems in statistics

Hawkes process model selection oracle inequalities data-driven penalty minimax risk adaptive estimation unknown support genome analysis


Reynaud-Bouret, Patricia; Schbath, Sophie. Adaptive estimation for Hawkes processes; application to genome analysis. Ann. Statist. 38 (2010), no. 5, 2781--2822. doi:10.1214/10-AOS806.

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