Adaptive goodness-of-fit tests in a density model

Given an i.i.d. sample drawn from a density $f$, we propose to test that $f$ equals some prescribed density $f_0$ or that $f$ belongs to some translation/scale family. We introduce a multiple testing procedure based on an estimation of the $\mathbb{L}_2$-distance between $f$ and $f_0$ or between $f$ and the parametric family that we consider. For each sample size $n$, our test has level of significance $\alpha$. In the case of simple hypotheses, we prove that our test is adaptive: it achieves the optimal rates of testing established by Ingster [J. Math. Sci. 99 (2000) 1110--1119] over various classes of smooth functions simultaneously. As for composite hypotheses, we obtain similar results up to a logarithmic factor. We carry out a simulation study to compare our procedures with the Kolmogorov--Smirnov tests, or with goodness-of-fit tests proposed by Bickel and Ritov [in Nonparametric Statistics and Related Topics (1992) 51--57] and by Kallenberg and Ledwina [Ann. Statist. 23 (1995) 1594--1608].Comment: Published at http://dx.doi.org/10.1214/009053606000000119 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

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

Considering two independent Poisson processes, we address the question of testing equality of their respective intensities. We first propose single tests whose test statistics are U-statistics based on general kernel functions. The corresponding critical values are constructed from a non-asymptotic 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, an aggregation procedure, which allows us to import ideas coming from model selection, thresholding and/or approximation kernels adaptive estimation. The resulting multiple tests are proved to be of level \alpha, and to satisfy non-asymptotic oracle type conditions for the classical L2-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

Bootstrap and permutation tests of independence for point processes

Motivated by a neuroscience question about synchrony detection in spike train analysis, we deal with the independence testing problem for point processes. We introduce non-parametric test statistics, which are rescaled general $U$-statistics, whose corresponding critical values are constructed from bootstrap and randomization/permutation approaches, making as few assumptions as possible on the underlying distribution of the point processes. We derive general consistency results for the bootstrap and for the permutation w.r.t. to Wasserstein's metric, which induce weak convergence as well as convergence of second order moments. The obtained bootstrap or permutation independence tests are thus proved to be asymptotically of the prescribed size, and to be consistent against any reasonable alternative. A simulation study is performed to illustrate the derived theoretical results, and to compare the performance of our new tests with existing ones in the neuroscientific literature

Adaptive tests for periodic signal detection with applications to laser vibrometry

International audienceInitially motivated by a practical issue in target detection via laser vibrometry, we are interested in the problem of periodic signal detection in a Gaussian fixed design regression framework. Assuming that the signal belongs to some periodic Sobolev ball and that the variance of the noise is known, we first consider the problem from a minimax point of view: we evaluate the so-called minimax separation rate which corresponds to the minimal l2−distance between the signal and zero so that the detection is possible with prescribed probabilities of error. Then, we propose a testing procedure which is available when the variance of the noise is unknown and which does not use any prior information about the smoothness degree or the period of the signal. We prove that it is adaptive in the sense that it achieves, up to a possible logarithmic factor, the minimax separation rate over various periodic Sobolev balls simultaneously. The originality of our approach as compared to related works on the topic of signal detection is that our testing procedure is sensitive to the periodicity assumption on the signal. A simulation study is performed in order to evaluate the effect of this prior assumption on the power of the test. We do observe the gains that we could expect from the theory. At last, we turn to the application to target detection by laser vibrometry that we had in view

Surrogate data methods based on a shuffling of the trials for synchrony detection: the centering issue

International audienceWe investigate several distribution-free dependence detection procedures, all based on a shuffling of the trials, from a statistical point of view. The mathematical justification of such procedures lies in the bootstrap principle and its approximation properties. In particular, we show that such a shuffling has mainly to be done on centered quantities-that is, quantities with zero mean under independence-to construct correct p-values, meaning that the corresponding tests control their false positive (FP) rate. Thanks to this study, we introduce a method, named permutation UE, which consists of a multiple testing procedure based on permutation of experimental trials and delayed coincidence count. Each involved single test of this procedure achieves the prescribed level, so that the corresponding multiple testing procedure controls the false discovery rate (FDR), and this with as few assumptions as possible on the underneath distribution, except independence and identical distribution across trials. The mathematical meaning of this assumption is discussed, and it is in particular argued that it does not mean what is commonly referred in neuroscience to as cross-trials stationarity. Some simulations show, moreover, that permutation UE outperforms the trial-shuffling of Pipa and Grün ( 2003 ) and the MTGAUE method of Tuleau-Malot et al. ( 2014 ) in terms of single levels and FDR, for a comparable amount of false negatives. Application to real data is also provided

DMTs and Covid-19 severity in MS: a pooled analysis from Italy and France

We evaluated the effect of DMTs on Covid-19 severity in patients with MS, with a pooled-analysis of two large cohorts from Italy and France. The association of baseline characteristics and DMTs with Covid-19 severity was assessed by multivariate ordinal-logistic models and pooled by a fixed-effect meta-analysis. 1066 patients with MS from Italy and 721 from France were included. In the multivariate model, anti-CD20 therapies were significantly associated (OR = 2.05, 95%CI = 1.39–3.02, p < 0.001) with Covid-19 severity, whereas interferon indicated a decreased risk (OR = 0.42, 95%CI = 0.18–0.99, p = 0.047). This pooled-analysis confirms an increased risk of severe Covid-19 in patients on anti-CD20 therapies and supports the protective role of interferon