Journal of Multivariate Analysis最新文献

Consider a multivariate distribution of $(X,Y)$, where $X$ is a vector of predictor variables and $Y$ is a response variable. Results are obtained for comparing the conditional and marginal tail indices, ${\xi }_{Y|X}\left(x\right)$ and ${\xi }_Y$, based on conditional distributions $\left\{F_{Y|X}\left(\cdot \right|x)\right\}$ and marginal distribution $F_Y$, respectively. For a multivariate distribution based on a copula, the conditional tail index can be decomposed into a product of copula-based conditional tail indices and the marginal tail index. In some applications, one may want ${\xi }_{Y|X}\left(x\right)$ to be non-constant, and some new copula families are derived to facilitate this.

A novel positive dependence property is introduced, called positive measure inducing (PMI for short), being fulfilled by numerous copula classes, including Gaussian, Student $t$, Fréchet, Farlie–Gumbel–Morgenstern and Frank copulas; it is conjectured that even all positive quadrant dependent Archimedean copulas meet this property. From a geometric viewpoint, a PMI copula concentrates more mass near the main diagonal than in the opposite diagonal. A striking feature of PMI copulas is that they impose an ordering on a certain class of copula-induced measures of concordance, the latter originating in Edwards et al. (2004) and including Spearman’s rho $\rho$ and Gini’s gamma $\gamma$, leading to numerous new inequalities such as $3\gamma \ge 2\rho$. The measures of concordance within this class are estimated using (classical) empirical copulas and the intrinsic construction via empirical checkerboard copulas, and the estimators’ asymptotic behavior is determined. Building upon the presented inequalities, asymptotic tests are constructed having the potential of being used for detecting whether the underlying dependence structure of a given sample is PMI, which in turn can be used for excluding certain copula families from model building. The excellent performance of the tests is demonstrated in a simulation study and by means of a real-data example.

Multivariate random fields are commonly used in spatial statistics and natural science to model coregionalized variables. In this context, the matrix-valued covariance function plays a central role in capturing their spatial continuity and interdependence. This study aims to contribute to the literature on covariance modeling by proposing new parametric families of isotropic matrix-valued functions exhibiting non-monotonic behaviors, namely hole effects and cross-dimples. The benefit of the proposed models is shown on a bivariate data set consisting of concentrations of airborne particulate matter.

In this study, three asymptotic behaviors of hierarchical clustering are defined and studied with strict conditions under several asymptotic settings, from large samples to high dimensionality, when having two independent populations. We proceed with the current comprehension of the asymptotic properties of hierarchical clustering in high-dimensional, low-sample-size (HDLSS) settings. For high-dimensional data, the asymptotic properties of hierarchical clustering are demonstrated under mild and practical settings, and we present simulation studies and hierarchical clustering performance discussions. Furthermore, hierarchical clustering was theoretically investigated when both the dimension and sample size approach infinity, and we generalized a latent number of populations considering hierarchical clustering in multiclass HDLSS settings.

In this paper, we consider high-dimensional quantile tensor regression using a general convex decomposable regularizer and analyze the statistical performances of the estimator. The rates are stated in terms of the intrinsic dimension of the estimation problem, which is, roughly speaking, the dimension of the smallest subspace that contains the true coefficient. Previously, convex regularized tensor regression has been studied with a least squares loss, Gaussian tensorial predictors and Gaussian errors, with rates that depend on the Gaussian width of a convex set. Our results extend the previous work to nonsmooth quantile loss. To deal with the non-Gaussian setting, we use the concept of Rademacher complexity with appropriate concentration inequalities instead of the Gaussian width. For the multi-linear nuclear norm penalty, our Orlicz norm bound for the operator norm of a random matrix may be of independent interest. We validate the theoretical guarantees in numerical experiments. We also demonstrate advantage of quantile regression over mean regression, and compare the performance of convex regularization method and nonconvex decomposition method in solving quantile tensor regression problem in simulation studies.

The maximum depth estimator (aka depth median) (${\beta }_{RD}^{\ast }$) induced from regression depth (RD) of Rousseeuw and Hubert (1999) is one of the most prevailing estimators in regression. It possesses outstanding robustness similar to the univariate location counterpart. Indeed, ${\beta }_{RD}^{\ast }$ can, asymptotically, resist up to 33% contamination without breakdown, in contrast to the 0% for the traditional (least squares and least absolute deviations) estimators (see Van Aelst and Rousseeuw (2000)). The results from Van Aelst and Rousseeuw (2000) are pioneering, yet they are limited to regression-symmetric populations (with a strictly positive density), the $ϵ$-contamination, maximum-bias model, and in asymptotical sense. With a fixed finite-sample size practice, the most prevailing measure of robustness for estimators is the finite-sample breakdown point (FSBP) (Donoho and Huber, 1983). Despite many attempts made in the literature, only sporadic partial results on FSBP for ${\beta }_{RD}^{\ast }$ were obtained whereas an exact FSBP for ${\beta }_{RD}^{\ast }$ remained open in the last twenty-plus years. Furthermore, is the asymptotic breakdown value $1/3$ (the limit of an increasing sequence of finite-sample breakdown values) relevant in the finite-sample practice? (Or what is the difference between the finite-sample and the limit breakdown values?). Such discussions are yet to be given in the literature. This article addresses the above issues, revealing an intrinsic connection between the regression depth of ${\beta }_{RD}^{\ast }$ and the newly obtained exact FSBP. It justifies the employment of ${\beta }_{RD}^{\ast }$ as a robust alternative to the traditional estimators and demonstrates the necessity and the merit of using the FSBP in finite-sample real practice.

Multivariate normal/independent (MNI) distributions contain many renowned heavy-tailed distributions such as the multivariate $t$, multivariate slash, multivariate contaminated normal, multivariate variance-gamma, and multivariate double exponential distributions. A frequent problem encountered in statistical analysis is the occurrence of truncated observations and non-normality such that theoretical moments are required for the estimation of the truncated multivariate normal/independent (TMNI) distributions. This paper is dedicated to deriving explicit expressions for the moments of the TMNI distributions with supports confined within a hyper-rectangle. A Monte Carlo experiment is undertaken to validate to the correctness of the proposed formulae for five selected members of the TMNI distributions. R scripts and data to reproduce the results are available in the GitHub repository.

Multivariate analysis-of-variance (MANOVA) is a well established tool to examine multivariate endpoints. While classical approaches depend on restrictive assumptions like normality and homogeneity, there is a recent trend to more general and flexible procedures. In this paper, we proceed on this path, but do not follow the typical mean-focused perspective. Instead we consider general quantiles, in particular the median, for a more robust multivariate analysis. The resulting methodology is applicable for all kind of factorial designs and shown to be asymptotically valid. Our theoretical results are complemented by an extensive simulation study for small and moderate sample sizes. An illustrative data analysis is also presented.

This paper studies a flexible approach to analyze high-dimensional nonlinear time series of unconstrained dimension based on linear statistics calculated from spectral average statistics of bilinear forms and nonlinear transformations of lag-window (i.e. band-regularized) spectral density matrix estimators. That class of statistics includes, among others, smoothed periodograms, nonlinear statistics such as coherency, long-run-variance estimators and contrast statistics related to factorial effects as special cases. Especially, we introduce the class of nonlinear spectral averages of the spectral density matrix. Having in mind big data settings, we study a sampling design which includes a sparse sampling scheme. Gaussian approximations with optimal rate are derived for nonlinear time series of growing dimension for these frequency domain statistics and the underlying lag-window (cross-) spectral estimator under non-stationarity. For change-testing (self-standardized) CUSUM statistics are examined. Further, a specific wild bootstrap procedure is proposed to estimate critical values. Simulation studies and an application to SP500 financial returns are provided in a supplement to this paper.

This paper provides a comprehensive estimation framework via nuclear norm plus ${\ell }_1$ norm penalization for high-dimensional approximate factor models with a sparse residual covariance. The underlying assumptions allow for non-pervasive latent eigenvalues and a prominent residual covariance pattern. In that context, existing approaches based on principal components may lead to misestimate the latent rank. On the contrary, the proposed optimization strategy recovers with high probability both the covariance matrix components and the latent rank and the residual sparsity pattern. Conditioning on the recovered low rank and sparse matrix varieties, we derive the finite sample covariance matrix estimators with the tightest error bound in minimax sense and we prove that the ensuing estimators of factor loadings and scores via Bartlett’s and Thomson’s methods have the same property. The asymptotic rates for those estimators of factor loadings and scores are also provided.