Adaptive tests of homogeneity for a Poisson process

Fromont, M.; Laurent, B.; Reynaud-Bouret, P.

Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 1, pp. 176-213.

Nous proposons de tester l'homogénéité d'un processus de Poisson observé sur un intervalle borné. Nous établissons tout d'abord des bornes inférieures pour les vitesses de séparation uniformes relativement à la norme sur des Besov bodies classiques ou faibles. De façon surprenante, nous obtenons des bornes inférieures sur les Besov bodies faibles qui coïncident avec les vitesses minimax d'estimation sur ce type de classe. Ensuite, nous construisons des procédures de tests non asymptotiques et non paramétriques qui sont adaptatives, au sens où elles atteignent, à un facteur logarithmique près dans certains cas, les vitesses de séparation optimales sur plusieurs classes d'alternatives simultanément. Ces procédures sont basées sur des méthodes de sélection de modèles et de seuillage. Enfin, nous complétons cette étude théorique par des simulations afin d'estimer par la méthode de Monte Carlo la puissance de nos tests sous diverses alternatives.

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 -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 non-parametric 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.

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