Adaptive estimation for Hawkes processes; application to genome analysis. (English) Zbl 1200.62135
Summary: 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 [see A. G. Hawkes and D. Oakes, J. Appl. Probab. 11, 493–503 (1974; Zbl 0305.60021)]. 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 Hölder 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.
MSC:
| 62P10 | Applications of statistics to biology and medical sciences; meta analysis |
| 92C40 | Biochemistry, molecular biology |
| 65C60 | Computational problems in statistics (MSC2010) |
| 62G05 | Nonparametric estimation |
| 62G20 | Asymptotic properties of nonparametric inference |
| 60E15 | Inequalities; stochastic orderings |
| 46N30 | Applications of functional analysis in probability theory and statistics |
Keywords:
Hawkes process; model selection; oracle inequalities; data-driven penalty; minimax risk; adaptive estimation; unknown support; genome analysisCitations:
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