Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Adaptive tests of homogeneity for a Poisson process

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

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Abstract

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 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.

Résumé

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 $\mathbb {L}^{2}$ 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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 1 (2011), 176-213.

Dates
First available in Project Euclid: 4 January 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1294170235

Digital Object Identifier
doi:10.1214/10-AIHP367

Mathematical Reviews number (MathSciNet)
MR2779402

Zentralblatt MATH identifier
1207.62161

Subjects
Primary: 62G10: Hypothesis testing
Secondary: 62G20: Asymptotic properties

Keywords
Poisson process adaptive hypotheses testing uniform separation rate minimax separation rate model selection thresholding rule

Citation

Fromont, M.; Laurent, B.; Reynaud-Bouret, P. Adaptive tests of homogeneity for a Poisson process. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 1, 176--213. doi:10.1214/10-AIHP367. https://projecteuclid.org/euclid.aihp/1294170235


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