The two-sample problem for Poisson processes: adaptive tests with a non-asymptotic wild bootstrap approach

Patricia Reynaud-Bouret (1) , Magalie Fromont (2) , Béatrice Laurent (3, 4)

1 JAD - Laboratoire Jean Alexandre Dieudonné
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 IMT - Institut de Mathématiques de Toulouse UMR5219
4 INSA Toulouse - Institut National des Sciences Appliquées - Toulouse

Considering two independent Poisson processes, we address the question of testing equality of their respective intensities. We construct multiple testing procedures from the aggregation of single tests whose testing statistics come from model selection, thresholding and/or kernel estimation methods. The corresponding critical values are computed through a non-asymptotic wild bootstrap approach. The obtained tests are proved to be exactly of level $\alpha$, and to satisfy non-asymptotic oracle type inequalities. From these oracle type inequalities, we deduce that our tests are adaptive in the minimax sense over a large variety of classes of alternatives based on classical and weak Besov bodies in the univariate case, but also Sobolev and anisotropic Nikol'skii-Besov balls in the multivariate case. A simulation study furthermore shows that they strongly perform in practice.

Mots clés: two-sample problem; Poisson process; bootstrap; adaptive tests; minimax separation rates; kernel methods; aggregation methods

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Patricia Reynaud-Bouret, Magalie Fromont, Béatrice Laurent. The two-sample problem for Poisson processes: adaptive tests with a non-asymptotic wild bootstrap approach. 2012. ⟨hal-00679102v1⟩