Journal of Multivariate Analysis最新文献

As a fundamental concept in information theory, mutual information ($MI$) has been commonly applied to quantify association between random vectors. Most existing nonparametric estimators of $MI$ have unstable statistical performance since they involve parameter tuning. We develop a consistent and powerful estimator, called fastMI, that does not incur any parameter tuning. Based on a copula formulation, fastMI estimates $MI$ by leveraging Fast Fourier transform-based estimation of the underlying density. Extensive simulation studies reveal that fastMI outperforms state-of-the-art estimators with improved estimation accuracy and reduced run time for large data sets. fastMI provides a powerful test for independence that exhibits satisfactory type I error control. Anticipating that it will be a powerful tool in estimating mutual information in a broad range of data, we develop an R package fastMI for broader dissemination.

This manuscript discusses a novel estimation approach for parametric hierarchical Archimedean copula. The parameters and structure of this copula are simultaneously estimated while imposing a non-concave penalty on differences between parameters which coincides with an implicit penalty on the copula’s structure. The asymptotic properties of the resulting penalized estimator are studied and small sample properties are illustrated using simulations.

We introduce a new family of multivariate distributions by taking the component-wise Tukey-$h$ transformation of a random vector following a skew-normal distribution with an alternative parameterization. The proposed distribution is named the skew-normal-Tukey-$h$ distribution and is an extension of the skew-normal distribution for handling heavy-tailed data. We compare this proposed distribution to the skew-$t$ distribution, which is another extension of the skew-normal distribution for modeling tail-thickness, and demonstrate that when there are substantial differences in marginal kurtosis, the proposed distribution is more appropriate. Moreover, we derive many appealing stochastic properties of the proposed distribution and provide a methodology for the estimation of the parameters that can be applied to large dimensions. Using simulations, as well as a wine and a wind speed data application, we illustrate how to draw inferences based on the multivariate skew-normal-Tukey-$h$ distribution.

Testing for homogeneity of two random vectors is a fundamental problem in statistics. In the past two decades, numerous efforts have been made to detect heterogeneity when the random vectors are multivariate or even high dimensional. Due to the “curse of dimensionality”, existing tests based on Euclidean distance may fail to capture the overall homogeneity in high-dimensional settings while can only capture the moment discrepancy. To address this issue, we propose a fully nonparametric test for homogeneity of two random vectors. Our method involves randomly selecting two subspaces consisting of components of the vectors, projecting the subspaces onto one-dimensional spaces, respectively, and constructing the test statistic using the Cramér–von Mises distance of the projections. To enhance the performance, we repeatedly implement this procedure to construct the final test statistic. Theoretically, if the replication time tends to infinity, we can avoid potential power loss caused by lousy directions. Owing to the $U$-statistic theory, the asymptotic null distribution of our proposed test is standard normal, regardless of the parent distributions of the random samples and the relationship between data dimensions and sample sizes. As a result, no re-sampling procedure is needed to determine critical values. The empirical size and power of the proposed test are demonstrated through numerical simulations.

Factor models are a parsimonious way to explain the dependence of variables using several latent variables. In Gaussian 1-factor and structural factor models (such as bi-factor and oblique factor) and their factor copula counterparts, factor scores or proxies are defined as conditional expectations of latent variables given the observed variables. With mild assumptions, the proxies are consistent for corresponding latent variables as the sample size and the number of observed variables linked to each latent variable go to infinity. When the bivariate copulas linking observed variables to latent variables are not assumed in advance, sequential procedures are used for latent variables estimation, copula family selection and parameter estimation. The use of proxy variables for factor copulas means that approximate log-likelihoods can be used to estimate copula parameters with less computational effort for numerical integration.

A broad class of smooth, possibly data-adaptive nonparametric copula estimators that contains empirical Bernstein copulas introduced by Sancetta and Satchell (and thus the empirical beta copula proposed by Segers, Sibuya and Tsukahara) is studied. Within this class, a subclass of estimators that depend on a scalar parameter determining the amount of marginal smoothing and a functional parameter controlling the shape of the smoothing region is specifically considered. Empirical investigations of the influence of these parameters suggest to focus on two particular data-adaptive smooth copula estimators that were found to be uniformly better than the empirical beta copula in all of the considered Monte Carlo experiments. Finally, with future applications to change-point detection in mind, conditions under which related sequential empirical copula processes converge weakly are provided.

We compare measures of concordance that arise as Pearson’s linear correlation coefficient between two random variables transformed so that they follow the so-called concordance-inducing distributions. The class of such transformed rank correlations includes Spearman’s rho, Blomqvist’s beta and van der Waerden’s coefficient. When only the standard axioms of measures of concordance are required, it is not always clear which transformed rank correlation is most suitable to use. To address this question, we compare measures of concordance in terms of their best and worst asymptotic variances of some canonical estimators over a certain set of dependence structures. A simple criterion derived from this approach is that concordance-inducing distributions with smaller fourth moment are more preferable. In particular, we show that Blomqvist’s beta is the optimal transformed rank correlation in this sense, and Spearman’s rho outperforms van der Waerden’s coefficient. Moreover, we find that Kendall’s tau, although it is not a transformed rank correlation of that nature, shares a certain optimal structure with Blomqvist’s beta.

Given two multivariate copulas with corresponding tail dependence functions, we investigate the relation between a natural tail dependence ordering and the order of local stochastic dominance. We show that, although the two orderings are not equivalent in general, they coincide for various important classes of copulas, among them all multivariate Archimedean and bivariate lower extreme value copulas. We illustrate the relevance of our results by an implication to risk management.

This paper considers a dependent competing risks model with the distribution of one risk being a semiparametric proportional hazards model, whereas the model for the other risks and the degree of risk dependence of an Archimedean copula are unknown. Identifiability is shown when there is at least one covariate with at least two values. Estimation is done by means of a $\sqrt{n}$-consistent semiparametric two-step procedure. Applicability and attractive finite sample performance are demonstrated with the help of simulations. An application to unemployment duration confirms the importance of estimating rather than assuming risk dependence.

In this article, we study tests of independence for data with arbitrary distributions in the non-serial case, i.e., for independent and identically distributed random vectors, as well as in the serial case, i.e., for time series. These tests are derived from copula-based covariances and their multivariate extensions using Möbius transforms. We find the asymptotic distributions of these statistics under the null hypothesis of independence or randomness, as well as under contiguous alternatives. This enables us to find out locally most powerful test statistics for some alternatives, whatever the margins. Numerical experiments are performed for Wald’s type combinations of these statistics to assess the finite sample performance.