Journal of Multivariate Analysis最新文献

Two-dimensional distributions whose marginal distributions are uniform are called bivariate copulas. Among them, the one that satisfies given constraints on expectation and is closest to being an independent distribution in the sense of Kullback–Leibler divergence is called the minimum information bivariate copula. The density function of the minimum information copula contains a set of functions called the normalizing functions, which are often difficult to compute. Although a number of proper scoring rules for probability distributions having normalizing constants such as exponential families have been proposed, these scores are not applicable to the minimum information copulas due to the normalizing functions. In this paper, we propose the conditional Kullback–Leibler score, which avoids computation of the normalizing functions. The main idea of its construction is to use pairs of observations. We show that the proposed score is strictly proper in the space of copula density functions and therefore the estimator derived from it has asymptotic consistency. Furthermore, the score is convex with respect to the parameters and can be easily optimized by the gradient methods.

Many authors have exploited the fact that the distribution of the multivariate probability integral transformation (PIT) of a continuous random vector $X\in R^d$ with cumulative distribution function $F_X$ is free of the marginal distributions. While most of these methods are based on the cdf of $W=F_X\left(X\right)$, this paper introduces the weighted characteristic function (WCf) of $W$. A sample version of the WCf of $W$ based on pseudo-observations is proposed and its weak limit in a space of complex functions is formally established. This result can be used to define test statistics for multivariate independence and goodness-of-fit in copula models, whose asymptotic behaviour comes from the weak convergence of the empirical WCf process. Simulations show the good sampling properties of these new tests, and an illustration is given on the multivariate Cook and Johnson dataset.

The Gaussian copula is a powerful tool that has been widely used to model spatial and/or temporal correlated data with arbitrary marginal distributions. However, this kind of model can potentially be too restrictive since it expresses a reflection symmetric dependence. In this paper, we propose a new spatial copula model that makes it possible to obtain random fields with arbitrary marginal distributions with a type of dependence that can be reflection symmetric or not.

Particularly, we propose a new random field with uniform marginal distributions that can be viewed as a spatial generalization of the classical Clayton copula model. It is obtained through a power transformation of a specific instance of a beta random field which in turn is obtained using a transformation of two independent Gamma random fields.

For the proposed random field, we study the second-order properties and we provide analytic expressions for the bivariate distribution and its correlation. Finally, in the reflection symmetric case, we study the associated geometrical properties. As an application of the proposed model we focus on spatial modeling of data with bounded support. Specifically, we focus on spatial regression models with marginal distribution of the beta type. In a simulation study, we investigate the use of the weighted pairwise composite likelihood method for the estimation of this model. Finally, the effectiveness of our methodology is illustrated by analyzing point-referenced vegetation index data using the Gaussian copula as benchmark. Our developments have been implemented in an open-source package for the R statistical environment.

Studying the multivariate extension of copula correlation yields a dimension reduction principle, which turns out to be strongly related with the ‘simple measure of conditional dependence’ $T$ recently introduced by Azadkia and Chatterjee (2021). In the present paper, we identify and investigate the dependence structure underlying this dimension reduction principle, provide a strongly consistent estimator for it, and demonstrate its broad applicability. For that purpose, we define a bivariate copula capturing the scale-invariant extent of dependence of an endogenous random variable $Y$ on a set of $d\ge 1$ exogenous random variables $X=(X_1,\dots ,X_d)$, and containing the information whether $Y$ is completely dependent on $X$, and whether $Y$ and $X$ are independent. The dimension reduction principle becomes apparent insofar as the introduced bivariate copula can be viewed as the distribution function of two random variables $Y$ and $Y^{\prime }$ sharing the same conditional distribution and being conditionally independent given $X$. Evaluating this copula uniformly along the diagonal, i.e., calculating Spearman’s footrule, leads to an unconditional version of Azadkia and Chatterjee’s ‘simple measure of conditional dependence’ $T$. On the other hand, evaluating this copula uniformly over the unit square, i.e., calculating Spearman’s rho, leads to a distribution-free coefficient of determination (also known as ‘copula correlation’). Several real data examples illustrate the importance of the introduced methodology.

We propose an approach to construct a new family of generalized Farlie–Gumbel–Morgenstern (GFGM) copulas that naturally scales to high dimensions. A GFGM copula can model moderate positive and negative dependence, cover different types of asymmetries, and admits exact expressions for many quantities of interest such as measures of association or risk measures in actuarial science or quantitative risk management. More importantly, this paper presents a new method to construct high-dimensional copulas based on mixtures of power functions and may be adapted to more general contexts to construct broader families of copulas. We construct a family of copulas through a stochastic representation based on multivariate Bernoulli distributions and Coxian-2 distributions. This paper will cover the construction of a GFGM copula and study its measures of multivariate association and dependence properties. We explain how to sample random vectors from the new family of copulas in high dimensions. Then, we study the bivariate case in detail and find that our construction leads to an asymmetric modified Huang–Kotz FGM copula. Finally, we study the exchangeable case and provide insights into the most negative dependence structure within this new class of high-dimensional copulas.

We study the asymptotic normality of the posterior distribution of canonical parameter in the exponential family under the Diaconis–Ylvisaker prior which is a conjugate prior when the dimension of parameter space increases with the sample size. We prove under mild conditions on the true parameter value ${\theta }_0$ and hyperparameters of priors, the difference between the posterior distribution and a normal distribution centered at the maximum likelihood estimator, and variance equal to the inverse of the Fisher information matrix goes to 0 in the expected total variation distance. The proof assumes dimension of parameter space $d$ grows linearly with sample size $n$ only requiring $d=o\left(n\right)$. En route, we derive a concentration inequality of the quadratic form of the maximum likelihood estimator without any specific assumption such as sub-Gaussianity. A specific illustration is provided for the Multinomial-Dirichlet model with an extension to the density estimation and Normal mean estimation problems.

This paper provides comparison results for general factor models with respect to the supermodular and directionally convex order. These results extend and strengthen previous ordering results from the literature concerning certain classes of mixture models as mixtures of multivariate normals, multivariate elliptic and exchangeable models to general factor models. For the main results, we first strengthen some known orthant ordering results for the multivariate

In this paper we study $L_p$-norm spherical copulas for arbitrary $p\in [1,\infty ]$ and arbitrary dimensions. The study is motivated by a conjecture that these distributions lead to a sharp bound for the value of a certain generalized mean difference. We fully characterize conditions for existence and uniqueness of $L_p$-norm spherical copulas. Explicit formulas for their densities and correlation coefficients are derived and the distribution of the radial part is determined. Moreover, statistical inference and efficient simulation are considered.

This paper provides a structured overview of the contents of the Special Issue of the Journal of Multivariate Analysis on “Copula modeling from Abe Sklar to the present day,” along with a brief history of the development of the field.

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.