Explore the vast range of titles available.

We investigate the problem of testing whether d possibly multivariate random variables, which may or may not be continuous, are jointly (or mutually) independent. Our method builds on ideas of the two-variable Hilbert–Schmidt independence criterion but allows for an arbitrary number of variables. We embed the joint distribution and the product of the marginals in a reproducing kernel Hilbert space and define the d-variable Hilbert–Schmidt independence criterion dHSIC as the squared distance between the embeddings. In the population case, the value of dHSIC is 0 if and only if the d variables are jointly independent, as long as the kernel is characteristic. On the basis of an empirical estimate of dHSIC, we investigate three non-parametric hypothesis tests: a permutation test, a bootstrap analogue and a procedure based on a gamma approximation. We apply non-parametric independence testing to a problem in causal discovery and illustrate the new methods on simulated and real data sets.

Testing mutual independence for high-dimensional observations is a fundamental statistical challenge. Popular tests based on linear and simple rank correlations are known to be incapable of detecting nonlinear, nonmonotone relationships, calling for methods that can account for such dependences. To address this challenge, we propose a family of tests that are constructed using maxima of pairwise rank correlations that permit consistent assessment of pairwise independence. Built upon a newly developed Cramér-type moderate deviation theorem for degenerate U-statistics, our results cover a variety of rank correlations including Hoeffding’s D, Blum–Kiefer–Rosenblatt’s R and Bergsma–Dassios–Yanagimoto’s τ∗. The proposed tests are distribution-free in the class of multivariate distributions with continuous margins, implementable without the need for permutation, and are shown to be rate-optimal against sparse alternatives under the Gaussian copula model. As a by-product of the study, we reveal an identity between the aforementioned three rank correlation statistics, and hence make a step towards proving a conjecture of Bergsma and Dassios.

In this paper, we study distance covariance, Hilbert–Schmidt covariance (aka Hilbert–Schmidt independence criterion [In Advances in Neural Information Processing Systems (2008) 585–592]) and related independence tests under the high dimensional scenario. We show that the sample distance/Hilbert–Schmidt covariance between two random vectors can be approximated by the sum of squared componentwise sample cross-covariances up to an asymptotically constant factor, which indicates that the standard distance/Hilbert–Schmidt covariance based test can only capture linear dependence in high dimension. Under the assumption that the components within each high dimensional vector are weakly dependent, the distance correlation based t test developed by Székely and Rizzo (J. Multivariate Anal. 117 (2013) 193–213) for independence is shown to have trivial limiting power when the two random vectors are nonlinearly dependent but component-wisely uncorrelated. This new and surprising phenomenon, which seems to be discovered and carefully studied for the first time, is further confirmed in our simulation study. As a remedy, we propose tests based on an aggregation of marginal sample distance/Hilbert–Schmidt covariances and show their superior power behavior against their joint counterparts in simulations. We further extend the distance correlation based t test to those based on Hilbert–Schmidt covariance and marginal distance/Hilbert–Schmidt covariance. A novel unified approach is developed to analyze the studentized sample distance/Hilbert–Schmidt covariance as well as the studentized sample marginal distance covariance under both null and alternative hypothesis. Our theoretical and simulation results shed light on the limitation of distance/Hilbert–Schmidt covariance when used jointly in the high dimensional setting and suggest the aggregation of marginal distance/Hilbert–Schmidt covariance as a useful alternative.

We propose a test of independence of two multivariate random vectors, given a sample from the underlying population. Our approach is based on the estimation of mutual information, whose decomposition into joint and marginal entropies facilitates the use of recently developed efficient entropy estimators derived from nearest neighbour distances. The proposed critical values may be obtained by simulation in the case where an approximation to one marginal is available or by permuting the data otherwise. This facilitates size guarantees, and we provide local power analyses, uniformly over classes of densities whose mutual information satisfies a lower bound. Our ideas may be extended to provide new goodness-of-fit tests for normal linear models based on assessing the independence of our vector of covariates and an appropriately defined notion of an error vector. The theory is supported by numerical studies on both simulated and real data.

The experiences of victimisation and living conditions of LGBTI (lesbian, gay, bisexual, transgender and intersex) people are not homogeneous and vary by geographical location. There have been comparisons between the heterogeneous experiences of these population groups in different countries, regions and cities, which have attracted both academic and media attention. This article uses data for Spain from a European Union survey to examine how participants' responses to victimisation and living conditions vary according to where they live along the rural-urban continuum. After applying chi-squared tests of independence and standardised residuals, it was found that the experiences of LGBTI people were not homogeneous, and the specific problems faced by each group of participants could be identified.

The experiences of victimisation and living conditions of LGBTI (lesbian, gay, bisexual, transgender and intersex) people are not homogeneous and vary by geographical location. There have been comparisons between the heterogeneous experiences of these population groups in different countries, regions and cities, which have attracted both academic and media attention. This article uses data for Spain from a European Union survey to examine how participants' responses to victimisation and living conditions vary according to where they live along the rural-urban continuum. After applying chi-squared tests of independence and standardised residuals, it was found that the experiences of LGBTI people were not homogeneous, and the specific problems faced by each group of participants could be identified.

Statistical inference on conditional dependence is essential in many fields including genetic association studies and graphical models. The classic measures focus on linear conditional correlations and are incapable of characterizing nonlinear conditional relationship including nonmonotonic relationship. To overcome this limitation, we introduce a nonparametric measure of conditional dependence for multivariate random variables with arbitrary dimensions. Our measure possesses the necessary and intuitive properties as a correlation index. Briefly, it is zero almost surely if and only if two multivariate random variables are conditionally independent given a third random variable. More importantly, the sample version of this measure can be expressed elegantly as the root of a V or U-process with random kernels and has desirable theoretical properties. Based on the sample version, we propose a test for conditional independence, which is proven to be more powerful than some recently developed tests through our numerical simulations. The advantage of our test is even greater when the relationship between the multivariate random variables given the third random variable cannot be expressed in a linear or monotonic function of one random variable versus the other. We also show that the sample measure is consistent and weakly convergent, and the test statistic is asymptotically normal. By applying our test in a real data analysis, we are able to identify two conditionally associated gene expressions, which otherwise cannot be revealed. Thus, our measure of conditional dependence is not only an ideal concept, but also has important practical utility. Supplementary materials for this article are available online.

The authors consider the linear model $Y_{n}\\ =\\ \\Psi X_{n}+\\varepsilon _{n}$ relating a functional response with explanatory variables. They propose a simple test of the nullity of ψ based on the principal component decomposition. The limiting distribution of their test statistic is chi-squared, but this distribution is also an excellent approximation in finite samples. The authors illustrate their method using data from terrestrial magnetic observatories. /// Les auteurs s'intéressent au modèle linéaire $Y_{n}\\ =\\ \\Psi X_{n}+\\varepsilon _{n}$ liant une variable réponse fonctionnelle à des variables explicatives. Ils proposent un test simple de nullité de ψ fondé sur la décomposition en composantes principales. La loi limite de leur statistique est une khi-deux, mais cette loi fournit aussi une excellente approximation à taille finie. Les auteurs illustrent leur méthode au moyen de données provenant d'observatoires du champ magnétique terrestre.

In multi-criteria decision-making and model evaluation, determining the weight of criteria is crucial. With the rapid development of information technology and the advent of the big data era, the need for complex problem analysis and decision-making has intensified. Traditional CRiteria Importance Through Intercriteria Correlation (CRITIC) methods rely on Pearson correlation, which may not adequately address nonlinearity in some scenarios. This study aims to refine the CRITIC method to better accommodate nonlinear relationships and enhance its robustness. We have developed a novel method named CRiteria Importance Through Intercriteria Dependence (CRITID), which leverages cutting-edge independence testing methods such as distance correlation among others. This approach enhances the assessment of intercriteria relationships. Upon application across diverse data distributions, the CRITID method has demonstrated enhanced rationality and robustness relative to the traditional CRITIC method. These improvements significantly benefit multi-criteria decision-making and model evaluation, providing a more accurate and dependable framework for analyzing complex datasets.

This article introduces a novel statistical framework for independent component analysis (ICA) of multivariate data. We propose methodology for estimating mutually independent components, and a versatile resampling-based procedure for inference, including misspecification testing. Independent components are estimated by combining a nonparametric probability integral transformation with a generalized nonparametric whitening method based on distance covariance that simultaneously minimizes all forms of dependence among the components. We prove the consistency of our estimator under minimal regularity conditions and detail conditions for consistency under model misspecification, all while placing assumptions on the observations directly, not on the latent components. U statistics of certain Euclidean distances between sample elements are combined to construct a test statistic for mutually independent components. The proposed measures and tests are based on both necessary and sufficient conditions for mutual independence. We demonstrate the improvements of the proposed method over several competing methods in simulation studies, and we apply the proposed ICA approach to two real examples and contrast it with principal component analysis.