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Register a new userAsymptotic local efficiency of Cram{e}r--von Mises tests for multivariate independence
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Christian Genest
Publication date
2007
fields
Mathematical Statistics
and research's language is
English
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Deheuvels [J. Multivariate Anal. 11 (1981) 102--113] and Genest and R{e}millard [Test 13 (2004) 335--369] have shown that powerful rank tests of multivariate independence can be based on combinations of asymptotically independent Cram{e}r--von Mises statistics derived from a M{o}bius decomposition of the empirical copula process. A result on the large-sample behavior of this process under contiguous sequences of alternatives is used here to give a representation of the limiting distribution of such test statistics and to compute their relative local asymptotic efficiency. Local power curves and asymptotic relative efficiencies are compared under familiar classes of copula alternatives.
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Let $F_N$ and $F$ be the empirical and limiting spectral distributions of an $Ntimes N$ Wigner matrix. The Cram{e}r-von Mises (CvM) statistic is a classical goodness-of-fit statistic that characterizes the distance between $F_N$ and $F$ in $ell^2$-norm. In this paper, we consider a mesoscopic approximation of the CvM statistic for Wigner matrices, and derive its limiting distribution. In the appendix, we also give the limiting distribution of the CvM statistic (without approximation) for the toy model CUE.
Few methods in Bayesian non-parametric statistics/ machine learning have received as much attention as Bayesian Additive Regression Trees (BART). While BART is now routinely performed for prediction tasks, its theoretical properties began to be understood only very recently. In this work, we continue the theoretical investigation of BART initiated by Rockova and van der Pas (2017). In particular, we study the Bernstein-von Mises (BvM) phenomenon (i.e. asymptotic normality) for smooth linear functionals of the regression surface within the framework of non-parametric regression with fixed covariates. As with other adaptive priors, the BvM phenomenon may fail when the regularities of the functional and the truth are not compatible. To overcome the curse of adaptivity under hierarchical priors, we induce a self-similarity assumption to ensure convergence towards a single Gaussian distribution as opposed to a Gaussian mixture. Similar qualitative restrictions on the functional parameter are known to be necessary for adaptive inference. Many machine learning methods lack coherent probabilistic mechanisms for gauging uncertainty. BART readily provides such quantification via posterior credible sets. The BvM theorem implies that the credible sets are also confidence regions with the same asymptotic coverage. This paper presents the first asymptotic normality result for BART priors, providing another piece of evidence that BART is a valid tool from a frequentist point of view.
In this paper, we study the asymptotic posterior distribution of linear functionals of the density. In particular, we give general conditions to obtain a semiparametric version of the Bernstein-Von Mises theorem. We then apply this general result to nonparametric priors based on infinite dimensional exponential families. As a byproduct, we also derive adaptive nonparametric rates of concentration of the posterior distributions under these families of priors on the class of Sobolev and Besov spaces.
The prominent Bernstein -- von Mises (BvM) result claims that the posterior distribution after centering by the efficient estimator and standardizing by the square root of the total Fisher information is nearly standard normal. In particular, the prior completely washes out from the asymptotic posterior distribution. This fact is fundamental and justifies the Bayes approach from the frequentist viewpoint. In the nonparametric setup the situation changes dramatically and the impact of prior becomes essential even for the contraction of the posterior; see [vdV2008], [Bo2011], [CaNi2013,CaNi2014] for different models like Gaussian regression or i.i.d. model in different weak topologies. This paper offers another non-asymptotic approach to studying the behavior of the posterior for a special but rather popular and useful class of statistical models and for Gaussian priors. First we derive tight finite sample bounds on posterior contraction in terms of the so called effective dimension of the parameter space. Our main results describe the accuracy of Gaussian approximation of the posterior. In particular, we show that restricting to the class of all centrally symmetric credible sets around pMLE allows to get Gaussian approximation up to order (n^{-1}). We also show that the posterior distribution mimics well the distribution of the penalized maximum likelihood estimator (pMLE) and reduce the question of reliability of credible sets to consistency of the pMLE-based confidence sets. The obtained results are specified for nonparametric log-density estimation and generalized regression.
In this paper, we analyze the impact of compressed sensing with complex random matrices on Fisher information and the Cram{e}r-Rao Bound (CRB) for estimating unknown parameters in the mean value function of a complex multivariate normal distribution. We consider the class of random compression matrices whose distribution is right-orthogonally invariant. The compression matrix whose elements are i.i.d. standard normal random variables is one such matrix. We show that for all such compression matrices, the Fisher information matrix has a complex matrix beta distribution. We also derive the distribution of CRB. These distributions can be used to quantify the loss in CRB as a function of the Fisher information of the non-compressed data. In our numerical examples, we consider a direction of arrival estimation problem and discuss the use of these distributions as guidelines for choosing compression ratios based on the resulting loss in CRB.
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