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

Global gauge anomalies in coset models of conformal field theory

We study the occurrence of global gauge anomalies in the coset models of two-dimensional conformal field theory that are based on gauged WZW models. A complete classification of the non-anomalous theories for a wide family of gauged rigid adjoint or twisted-adjoint symmetries of WZW models is achieved with the help of Dynkin's classification of Lie subalgebras of simple Lie algebras.Comment: 25 page

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

Superluminality in the Bi- and Multi- Galileon

We re-explore the Bi- and Multi-Galileon models with trivial asymptotic conditions at infinity and show that propagation of superluminal fluctuations is a common and unavoidable feature of these theories, unlike previously claimed in the literature. We show that all Multi-Galileon theories containing a Cubic Galileon term exhibit superluminalities at large distances from a point source, and that even if the Cubic Galileon is not present one can always find sensible matter distributions in which there are superluminal modes at large distances. In the Bi-Galileon case we explicitly show that there are always superluminal modes around a point source even if the Cubic Galileon is not present. Finally, we briefly comment on the possibility of avoiding superluminalities by modifying the asymptotic conditions at infinity.Comment: 32 pages, minor changes, few references adde

Verbing and nouning in French : toward an ecologically valid approach to sentence processing

La présente thèse utilise la technique des potentiels évoqués afin d’étudier les méchanismes neurocognitifs qui sous-tendent la compréhension de la phrase. Plus particulièrement, cette recherche vise à clarifier l’interaction entre les processus syntaxiques et sémantiques chez les locuteurs natifs et les apprenants d’une deuxième langue (L2). Le modèle “syntaxe en premier” (Friederici, 2002, 2011) prédit que les catégories syntaxiques sont analysées de façon précoce: ce stade est reflété par la composante ELAN (Early anterior negativity, Négativité antérieure gauche), qui est induite par les erreurs de catégorie syntaxique. De plus, ces erreurs semblent empêcher l’apparition de la composante N400 qui reflète les processus lexico-sémantiques. Ce phénomène est défini comme le bloquage sémantique (Friederici et al., 1999). Cependant, la plupart des études qui observent la ELAN utilisent des protocoles expérimentaux problématiques dans lesquels les différences entre les contextes qui précèdent la cible pourraient être à l’origine de résultats fallacieux expliquant à la fois l’apparente “ELAN” et l’absence de N400 (Steinhauer & Drury, 2012). La première étude rééevalue l’approche de la “syntaxe en premier” en adoptant un paradigme expériemental novateur en français qui introduit des erreurs de catégorie syntaxique et les anomalies de sémantique lexicale. Ce dessin expérimental équilibré contrôle à la fois le mot-cible (nom vs. verbe) et le contexte qui le précède. Les résultats récoltés auprès de locuteurs natifs du français québécois ont révélé un complexe N400-P600 en réponse à toutes les anomalies, en contradiction avec les prédictions du modèle de Friederici. Les effets additifs des manipulations syntaxique et sémantique sur la N400 suggèrent la détection d’une incohérence entre la racine du mot qui avait été prédite et la cible, d’une part, et l’activation lexico-sémantique, d’autre part. Les réponses individuelles se sont pas caractérisées par une dominance vers la N400 ou la P600: au contraire, une onde biphasique est présente chez la majorité des participants. Cette activation peut donc être considérée comme un index fiable des mécanismes qui sous-tendent le traitement des structures syntagmatiques. La deuxième étude se concentre sur les même processus chez les apprenants tardifs du français L2. L’hypothèse de la convergence (Green, 2003 ; Steinhauer, 2014) prédit que les apprenants d’une L2, s’ils atteignent un niveau avancé, mettent en place des processus de traitement en ligne similaires aux locuteurs natifs. Cependant, il est difficile de considérer en même temps un grand nombre de facteurs qui se rapportent à leurs compétences linguistiques, à l’exposition à la L2 et à l’âge d’acquisition. Cette étude continue d’explorer les différences inter-individuelles en modélisant les données de potentiels-évoqués avec les Forêts aléatoires, qui ont révélé que le pourcentage d’explosition au français ansi que le niveau de langue sont les prédicteurs les plus fiables pour expliquer les réponses électrophysiologiques des participants. Plus ceux-ci sont élevés, plus l’amplitude des composantes N400 et P600 augmente, ce qui confirme en partie les prédictions faites par l’hypothèse de la convergence. En conclusion, le modèle de la “syntaxe en premier” n’est pas viable et doit être remplacé. Nous suggérons un nouveau paradigme basé sur une approche prédictive, où les informations sémantiques et syntaxiques sont activées en parallèle dans un premier temps, puis intégrées via un recrutement de mécanismes contrôlés. Ces derniers sont modérés par les capacités inter-individuelles reflétées par l’exposition et la performance.The present thesis uses event-related potentials (ERPs) to investigate neurocognitve mechanisms underlying sentence comprehension. In particular, these two experiments seek to clarify the interplay between syntactic and semantic processes in native speakers and second language learners. Friederici’s (2002, 2011) “syntax-first” model predicts that syntactic categories are analyzed at the earliest stages of speech perception reflected by the ELAN (Early left anterior negativity), reported for syntactic category violations. Further, syntactic category violations seem to prevent the appearance of N400s (linked to lexical-semantic processing), a phenomenon known as “semantic blocking” (Friederici et al., 1999). However, a review article by Steinhauer and Drury (2012) argued that most ELAN studies used flawed designs, where pre-target context differences may have caused ELAN-like artifacts as well as the absence of N400s. The first study reevaluates syntax-first approaches to sentence processing by implementing a novel paradigm in French that included correct sentences, pure syntactic category violations, lexical-semantic anomalies, and combined anomalies. This balanced design systematically controlled for target word (noun vs. verb) and the context immediately preceding it. Group results from native speakers of Quebec French revealed an N400-P600 complex in response to all anomalous conditions, providing strong evidence against the syntax-first and semantic blocking hypotheses. Additive effects of syntactic category and lexical-semantic anomalies on the N400 may reflect a mismatch detection between a predicted word-stem and the actual target, in parallel with lexical-semantic retrieval. An interactive rather than additive effect on the P600 reveals that the same neurocognitive resources are recruited for syntactic and semantic integration. Analyses of individual data showed that participants did not rely on one single cognitive mechanism reflected by either the N400 or the P600 effect but on both, suggesting that the biphasic N400-P600 ERP wave can indeed be considered to be an index of phrase-structure violation processing in most individuals. The second study investigates the underlying mechanisms of phrase-structure building in late second language learners of French. The convergence hypothesis (Green, 2003; Steinhauer, 2014) predicts that second language learners can achieve native-like online- processing with sufficient proficiency. However, considering together different factors that relate to proficiency, exposure, and age of acquisition has proven challenging. This study further explores individual data modeling using a Random Forests approach. It revealed that daily usage and proficiency are the most reliable predictors in explaining the ERP responses, with N400 and P600 effects getting larger as these variables increased, partly confirming and extending the convergence hypothesis. This thesis demonstrates that the “syntax-first” model is not viable and should be replaced. A new account is suggested, based on predictive approaches, where semantic and syntactic information are first used in parallel to facilitate retrieval, and then controlled mechanisms are recruited to analyze sentences at the interface of syntax and semantics. Those mechanisms are mediated by inter-individual abilities reflected by language exposure and performance

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