Adaptive tests of homogeneity for a Poisson process

Magalie Fromont (1) , Béatrice Laurent (2) , Patricia Reynaud-Bouret (3)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 JAD - Laboratoire Jean Alexandre Dieudonné

We propose to test the homogeneity of a Poisson process observed on a finite interval. In this framework, we first provide lower bounds for the uniform separation rates in $\mathbb{L}^2$ norm over classical Besov bodies and weak Besov bodies. Surprisingly, the obtained lower bounds over weak Besov bodies coincide with the minimax estimation rates over such classes. Then we construct non asymptotic and nonparametric testing procedures that are adaptive in the sense that they achieve, up to a possible logarithmic factor, the optimal uniform separation rates over various Besov bodies simultaneously. These procedures are based on model selection and thresholding methods. We finally complete our theoretical study with a Monte Carlo evaluation of the power of our tests under various alternatives.

Mots clés: model selection; thresholding rule; Poisson process; adaptive hypotheses testing; uniform separation rate; minimax separation rate

Domaines: Théorie [stat.TH]

Magalie Fromont, Béatrice Laurent, Patricia Reynaud-Bouret. Adaptive tests of homogeneity for a Poisson process. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2011, 47 (1), pp.176-213. ⟨10.1214/10-aihp367⟩. ⟨hal-00867074⟩