Journal of Multivariate Analysis最新文献

Multivariate functional data are often observed in many scientific fields. This paper considers a multi-sample equal-covariance matrix function testing problem for multivariate functional data. Two new tests are proposed and studied. The asymptotic properties of the two tests under the null hypothesis and a local alternative are investigated. Two methods for approximating the null distributions of the test statistics are described. It is shown that the two tests are root-$n$ consistent. Two simulation studies are conducted to evaluate the finite sample performance of the proposed tests. Finally, the two tests are illustrated via applications to three real multivariate functional data sets.

Mean–variance mixtures of normal distributions are very flexible: they model many nonnormal features, such as skewness, kurtosis and multimodality. Special cases include generalized asymmetric Laplace distributions, mixtures of two normal distributions with proportional covariance matrices, scale mixtures of normal distributions and normal distributions. This paper investigates the skewness of multivariate mean–variance normal mixtures. The special case of mixtures of two normal distributions with proportional covariance matrices is treated in greater detail. The paper derives the analytical forms of prominent measures of multivariate skewness and applies them to model-based clustering, normalizing linear transformations, projection pursuit and normality testing. The practical relevance of the theoretical results is assessed with both real and simulated data.

This work is concerned with testing conditional independence under a factor model setting. We propose a novel multivariate test for non-Gaussian data based on the Hilbert–Schmidt independence criterion (HSIC). Theoretically, we investigate the convergence of our test statistic under both the null and the alternative hypotheses, and devise a bootstrap scheme to approximate its null distribution, showing that its consistency is justified. Methodologically, we generalize the HSIC-based independence test approach to a situation where data follow a factor model structure. Our test requires no nonparametric smoothing estimation of functional forms including conditional probability density functions, conditional cumulative distribution functions and conditional characteristic functions under the null or alternative, is computationally efficient and is dimension-free in the sense that the dimension of the conditioning variable is allowed to be large but finite. Further extension to nonlinear, non-Gaussian structure equation models is also described in detail and asymptotic properties are rigorously justified. Numerical studies demonstrate the effectiveness of our proposed test relative to that of several existing tests.

The family of multivariate skew-normal distributions has many interesting properties. It is shown here that these hold for a general class of skew-elliptical distributions. For this class, several stochastic representations are established and then their probabilistic properties, such as characteristic function, moments, quadratic forms as well as transformation properties, are investigated.

Standard high-dimensional factor models assume that the comovements in a large set of variables could be modeled using a small number of latent factors that affect all variables. In many relevant applications in economics and finance, heterogeneous comovements specific to some known groups of variables naturally arise, and reflect distinct cyclical movements within those groups. This paper develops two new statistical tests that can be used to investigate whether there is evidence supporting group-specific heterogeneity in the data. The paper also proposes and proves the validity of a permutation approach for approximating the asymptotic distributions of the two test statistics.

In the present paper, we tackle the problem of testing $H_{0q}:{\lambda }_q>{\lambda }_{q+1}=\cdots ={\lambda }_p$, where ${\lambda }_1,\dots ,{\lambda }_p$ are the scatter matrix eigenvalues of an elliptical distribution on $R^p$. This is a classical problem in multivariate analysis which is very useful in dimension reduction. We analyse the problem using the Le Cam asymptotic theory of experiments and show that contrary to the testing problems on eigenvalues and eigenvectors of a scatter matrix tackled in Hallin et al. (2010), the non-specification of nuisance parameters has an asymptotic cost for testing $H_{0q}$. We moreover derive signed-rank tests for the problem that enjoy the property of being asymptotically distribution-free under ellipticity. The van der Waerden rank test uniformly dominates the classical pseudo-Gaussian procedure for the problem. Numerical illustrations show the nice finite-sample properties of our tests.

In this paper, we study the quantile regression (QR) estimation for the partially functional linear model with the responses being right-censored and the censoring indicators being missing at random (MAR). Firstly, we construct the weighted QR estimators for both the infinite-dimensional slope function and the finite scalar parameters of the model by combining the methods of calibration, imputation and inverse probability weighting. Then, some asymptotic properties such as the convergence rate of the estimator for the slope function and the asymptotic distribution of the estimator for the finite scalar parameters are obtained respectively. Moreover, to select the scalar covariates in the model, we also give a variable selection procedure by the method of adaptive LASSO penalty and establish the oracle property of the proposed weighted penalized estimators simultaneously. Finally, some simulation studies and a real data analysis are carried out to show the performances of the proposed methods.

We study the limiting behavior of singular values of a lag-$\tau$ sample auto-correlation matrix $R_{\tau }^{ϵ}$ of large dimensional vector white noise process, the error term $ϵ$ in the high-dimensional factor model. We establish the limiting spectral distribution (LSD) that characterizes the global spectrum of $R_{\tau }^{ϵ}$, and derive the limit of its largest singular value. All the asymptotic results are derived under the high-dimensional asymptotic regime where the data dimension and sample size go to infinity proportionally. Under mild assumptions, we show that the LSD of $R_{\tau }^{ϵ}$ is the same as that of the lag-$\tau$ sample auto-covariance matrix. Based on this asymptotic equivalence, we additionally show that the largest singular value of $R_{\tau }^{ϵ}$ converges almost surely to the right end point of the support of its LSD. Based on these results, we further propose two estimators of total number of factors with lag-$\tau$ sample auto-correlation matrices in a factor model. Our theoretical results are fully supported by numerical experiments as well.

We introduce projection divergence in the reproducing kernel Hilbert space to test for statistical independence and measure the degree of nonlinear dependence. We suggest a slicing procedure to estimate the kernel projection divergence, which divides a random sample of size $n$ into $H$ slices, each of size $c$. The entire procedure has the complexity of $O\left(n^2\right)$, which is prohibitive if $n$ is extremely large. To alleviate computational complexity, we implement this slicing procedure together with a block-wise estimation, which divides the whole sample into $B$ blocks, each of size $d$. This block-wise and slicing estimation has the complexity of $O\left\{n(c+d+logn)\right\}$, which reduces the computational complexity substantially if $c$ and $d$ are relatively small. The resultant estimation is asymptotically normal and has the convergence rate of ${\{n\left(cd\right)/(c+d)\}}^{-1/2}$. More importantly, this block-wise implementation has the same asymptotic properties as the naive slicing estimation, if $c$ is relatively small, indicating that the block-wise implementation does not result in power loss in independence tests. We demonstrate the computational efficiencies and theoretical properties of this block-wise and slicing estimation through simulations and an application to psychological datasets.

This paper assumes a context in which cause–effect relationships between random variables can be represented by a Gaussian linear structural equation model and the corresponding directed acyclic graph. We consider the situation where we observe a set of random variables satisfying the so-called back-door criterion. When the ordinary least squares method is utilized to estimate the total effect, we formulate the unbiased estimator of the causal effect (the estimated causal effect) on the variance of the outcome variable with external intervention in which a treatment variable is set to a specified constant value. In addition, we provide the variance formula for the estimated causal effect on the variance. The variance formula proposed in this paper is exact, in contrast to those in most previous studies on estimating causal effects.