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June 2013 The two-sample problem for Poisson processes: Adaptive tests with a nonasymptotic wild bootstrap approach
Magalie Fromont, Béatrice Laurent, Patricia Reynaud-Bouret
Ann. Statist. 41(3): 1431-1461 (June 2013). DOI: 10.1214/13-AOS1114

Abstract

Considering two independent Poisson processes, we address the question of testing equality of their respective intensities. We first propose testing procedures whose test statistics are $U$-statistics based on single kernel functions. The corresponding critical values are constructed from a nonasymptotic wild bootstrap approach, leading to level $\alpha$ tests. Various choices for the kernel functions are possible, including projection, approximation or reproducing kernels. In this last case, we obtain a parametric rate of testing for a weak metric defined in the RKHS associated with the considered reproducing kernel. Then we introduce, in the other cases, aggregated or multiple kernel testing procedures, which allow us to import ideas coming from model selection, thresholding and/or approximation kernels adaptive estimation. These multiple kernel tests are proved to be of level $\alpha$, and to satisfy nonasymptotic oracle-type conditions for the classical $\mathbb{L} _{2}$-norm. From these conditions, we deduce that they 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.

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Magalie Fromont. Béatrice Laurent. Patricia Reynaud-Bouret. "The two-sample problem for Poisson processes: Adaptive tests with a nonasymptotic wild bootstrap approach." Ann. Statist. 41 (3) 1431 - 1461, June 2013. https://doi.org/10.1214/13-AOS1114

Information

Published: June 2013
First available in Project Euclid: 1 August 2013

zbMATH: 1273.62102
MathSciNet: MR3113817
Digital Object Identifier: 10.1214/13-AOS1114

Subjects:
Primary: 62G09 , 62G10 , 62G55
Secondary: 62G20

Keywords: adaptive tests , aggregation methods , bootstrap , kernel methods , minimax separation rates , multiple kernel , Poisson process , two-sample problem

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 3 • June 2013
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