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Theses

Tests minimax et adaptatifs pour la détection et la localisation d’une rupture dans un processus de Poisson

Abstract : Motivated by applications in cybersecurity and epidemiology, this thesis addresses the problem of detecting and localising an abrupt change in a Poisson process. Firstly, we are interested in the detection of a change in the intensity of a Poisson process characterised by a jump (non transitory change) or a bump (transitory change) from constant. We propose a complete study from the nonasymptotic minimax testing point of view when the constant baseline intensity is known or unknown. The question of minimax adaptation with respect to each parameter (height, location, length) of the change is tackled, leading to a comprehensive overview of the various minimax separation rate regimes. Secondly, we focus on the issue of simultaneously detecting and localising a non transitory change in the intensity of a Poisson process when the baseline intensity is known, and on the construction of change point estimation procedures. This problem, formulated as a multiple testing problem, is studied from the nonasymptotic minimax perspective. The question of minimax adaptation with respect to the change height and location is broached and we identify two regimes in the minimax family-wise separation rate. A correspondence between minimax multiple testing procedures and confidence intervals for the jump localisation is also established. The minimax or minimax adaptive tests and multiple tests are based on simple linear counting statistics derived from Neyman-Pearson tests or quadratic statistics, possibly combined with aggregation strategies.
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Submitted on : Monday, May 30, 2022 - 4:21:22 PM
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  • HAL Id : tel-03682007, version 1

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Fabrice Grela. Tests minimax et adaptatifs pour la détection et la localisation d’une rupture dans un processus de Poisson. Statistiques [math.ST]. Université Rennes 2, 2021. Français. ⟨NNT : 2021REN20052⟩. ⟨tel-03682007⟩

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