Adaptive tests of homogeneity for a Poisson process

作者：Fromont M^*; Laurent B; Reynaud Bouret P

来源：Annales de l Institut Henri Poincare-Probabilites et Statistiques, 2011, 47(1): 176-213.

DOI：10.1214/10-AIHP367

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

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 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.

• 出版日期2011-2