Our article addresses the problem of flexibly estimating a multivariate density while also attempting to estimate its marginals correctly. We do so by proposing two new estimators that try to capture the best features of mixture of normals and copula estimators while avoiding some of their weaknesses. The first estimator we propose is a mixture of normals copula model that is a flexible alternative to parametric copula models such as the normal and t copula. The second is a marginally adapted mixture of normals estimator that improves on the standard mixture of normals by using information contained in univariate estimates of the marginal densities. We show empirically that copula based approaches can behave much better or much worse than estimators based on mixture of normals depending on the properties of the data. We provide fast and reliable implementations of the estimators and illustrate the methodology on simulated and real data.
A Linear Relationship between Correlation and Cohen's Kappa for Binary Data and Simulating Multivariate Nominal and Ordinal Data with Specified Kappa Matrix
We consider the asymptotic distribution of a cell in a 2 x ... x 2 contingency table as the fixed marginal totals tend to infinity. The asymptotic order of the cell variance is derived and a useful diagnostic is given for determining whether the cell has a Poisson limit or a Gaussian limit. There are three forms of Poisson convergence. The exact form is shown to be determined by the growth rates of the two smallest marginal totals. The results are generalized to contingency tables with arbitrary sizes and are further complemented with concrete examples.
Copulas are popular as models for multivariate dependence because they allow the marginal densities and the joint dependence to be modeled separately. However, they usually require that the transformation from uniform marginals to the marginals of the joint dependence structure is known. This can only be done for a restricted set of copulas, e.g. a normal copula. Our article introduces copula-type estimators for flexible multivariate density estimation which also allow the marginal densities to be modeled separately from the joint dependence, as in copula modeling, but overcomes the lack of fl…
Studies that collect multi-outcome data such as tobacco and alcohol use are becoming increasingly common. In principle, multi-outcomes studies investigate the correlations between outcomes, including, causal links and/or joint distributions. Although there are many methods for studying multivariate outcomes, significant limitations regarding scale and interpretation persist. Here we introduce a model based on the exponential-family for discrete binary outcomes that provides a flexible framework for hypothesis testing of multiple binary outcomes in a computationally efficient fashion.
Tests for break points detection in the law of random vectors have been proposed in several papers. Nevertheless, they have often little powers for alternatives involving a change in the dependence between components of vectors. Specific tests for detection of a change in the copula of random vectors have been proposed in recent papers, but they do not allow to conclude of a change in the dependence structure without condition that the margins are constant. The goal of this article is to propose a test for detection of a break in the copula when a change in marginal distribution occurs at a kn…
This paper proposes a maximum-likelihood approach to jointly estimate marginal conditional quantiles of multivariate response variables in a linear regression framework. We consider a slight reparameterization of the Multivariate Asymmetric Laplace distribution proposed by Kotz et al (2001) and exploit its location-scale mixture representation to implement a new EM algorithm for estimating model parameters. The idea is to extend the link between the Asymmetric Laplace distribution and the well-known univariate quantile regression model to a multivariate context, i.e. when a multivariate depend…
Based on the concept of a Lévy copula to describe the dependence structure of a multivariate Lévy process we present a new estimation procedure. We consider a parametric model for the marginal Lévy processes as well as for the Lévy copula and estimate the parameters by a two-step procedure. We first estimate the parameters of the marginal processes, and then estimate in a second step only the dependence structure parameter. For infinite Lévy measures we truncate the small jumps and base our statistical analysis on the large jumps of the model. Prominent example will be a bivariate stable \lp, …
The prevalence of spatially referenced multivariate data has impelled researchers to develop a procedure for the joint modeling of multiple spatial processes. This ordinarily involves modeling marginal and cross-process dependence for any arbitrary pair of locations using a multivariate spatial covariance function. However, building a flexible multivariate spatial covariance function that is nonnegative definite is challenging. Here, we propose a semiparametric approach for multivariate spatial covariance function estimation with approximate Matérn marginals and highly flexible cross-covarianc…
We introduce a flexible parametric mixed effects model for correlated binary data, with parameters that can be directly interpreted as marginal odds ratios. This leads to a robust estimation equation with an optimal weighting matrix being the inverse of a genuine model-based covariance matrix. Flexible correlation structures can be imposed by correlated random effects, and correlation parameters can be estimated by solving a composite likelihood score function. Marginal parameters are consistently estimated even when the conditional parametric model is misspecified, and the robust estimation p…
The skew-normal and related families are flexible and asymmetric parametric models suitable for modelling a diverse range of systems. We show that the multivariate maximum of a high-dimensional extended skew-normal random sample has asymptotically independent components and derive the speed of convergence of the joint tail. To describe the possible dependence among the components of the multivariate maximum, we show that under appropriate conditions an approximate multivariate extreme-value distribution that leads to a rich dependence structure can be derived.
Multivariate density estimation is a popular technique in statistics with wide applications including regression models allowing for heteroskedasticity in conditional variances. The estimation problems become more challenging when observations are missing in one or more variables of the multivariate vector. A flexible class of mixture of tensor products of kernel densities is proposed which allows for easy implementation of imputation methods using Gibbs sampling and shown to have superior performance compared to some of the exisiting imputation methods currently available in literature. Numer…
In environmental sciences, it is often of interest to assess whether the dependence between extreme measurements has changed during the observation period. The aim of this work is to propose a statistical test that is particularly sensitive to such changes. The resulting procedure is also extended to allow the detection of changes in the extreme-value dependence under the presence of known breaks in the marginal distributions. Simulations are carried out to study the finite-sample behavior of both versions of the proposed test. Illustrations on hydrological data sets conclude the work.
In this paper, a nonparametric estimator is proposed for estimating the L1-median for multivariate conditional distribution when the covariates take values in an infinite dimensional space. The multivariate case is more appropriate to predict the components of a vector of random variables simultaneously rather than predicting each of them separately. While estimating the conditional L1-median function using the well-known Nadarya-Waston estimator, we establish the strong consistency of this estimator as well as the asymptotic normality. We also present some simulations and provide how to built…
Several researchers have described two-part models with patient-specific stochastic processes for analysing longitudinal semicontinuous data. In theory, such models can offer greater flexibility than the standard two-part model with patient-specific random effects. However, in practice the high dimensional integrations involved in the marginal likelihood (i.e. integrated over the stochastic processes) significantly complicates model fitting. Thus non-standard computationally intensive procedures based on simulating the marginal likelihood have so far only been proposed. In this paper, we descr…
We propose a novel and flexible rank-breaking-then-composite-marginal-likelihood (RBCML) framework for learning random utility models (RUMs), which include the Plackett-Luce model. We characterize conditions for the objective function of RBCML to be strictly log-concave by proving that strict log-concavity is preserved under convolution and marginalization. We characterize necessary and sufficient conditions for RBCML to satisfy consistency and asymptotic normality. Experiments on synthetic data show that RBCML for Gaussian RUMs achieves better statistical efficiency and computational efficien…
Multivariate distributions are explored using the joint distributions of marginal sample quantiles. Limit theory for the mean of a function of order statistics is presented. The results include a multivariate central limit theorem and a strong law of large numbers. A result similar to Bahadur's representation of quantiles is established for the mean of a function of the marginal quantiles. In particular, it is shown that \[\sqrt{n}\Biggl(\frac{1}{n}\sum_{i=1}^nϕ\bigl(X_{n:i}^{(1)},...,X_{n:i}^{(d)}\bigr)-\barγ\Biggr)=\frac{1}{\sqrt{n}}\sum_{i=1}^nZ_{n,i}+\mathrm{o}_P(1)\] as $n\rightarrow\inft…
The univariate distorted distribution were introduced in risk theory to represent changes (distortions) in the expected distributions of some risks. Later they were also applied to represent distributions of order statistics, coherent systems, proportional hazard rate (PHR) and proportional reversed hazard rate (PRHR) models, etc. In this paper we extend this concept to the multivariate setup. We show that, in some cases, they are a valid alternative to the copula representations especially when the marginal distributions may not be easily handled. Several relevant examples illustrate the appl…
Most of the available multivariate statistical models dictate on fitting different parameters for the covariate effects on each multiple responses. This might be unnecessary and inefficient for some cases. In this article, we propose a modeling framework for multivariate marginal models to analyze multivariate longitudinal data which provides flexible model building strategies. We show that the model handles several response families such as binomial, count and continuous. We illustrate the model on the Mother's Stress and Children's Morbidity data set. A simulation study is conducted to exami…
We propose to model multivariate volatility processes based on the newly defined conditionally uncorrelated components (CUCs). This model represents a parsimonious representation for matrix-valued processes. It is flexible in the sense that we may fit each CUC with any appropriate univariate volatility model. Computationally it splits one high-dimensional optimization problem into several lower-dimensional subproblems. Consistency for the estimated CUCs has been established. A bootstrap test is proposed for testing the existence of CUCs. The proposed methodology is illustrated with both simula…
This article is devoted to the study of tail index estimation based on i.i.d. multivariate observations, drawn from a standard heavy-tailed distribution, i.e. of which 1-d Pareto-like marginals share the same tail index. A multivariate Central Limit Theorem for a random vector, whose components correspond to (possibly dependent) Hill estimators of the common shape index alpha, is established under mild conditions. Motivated by the statistical analysis of extremal spatial data in particular, we introduce the concept of (standard) heavy-tailed random field of tail index alpha and show how this l…
The contribution of this study is twofold: First, we propose an efficient algorithm for the computation of the (weighted) maximum likelihood estimators for the parameters of the multivariate Student-$t$ distribution, which we call generalized multivariate myriad filter. Second, we use the generalized multivariate myriad filter in a nonlocal framework for the denoising of images corrupted by different kinds of noise. The resulting method is very flexible and can handle heavy-tailed noise such as Cauchy noise, as well as the other extreme, namely Gaussian noise. Furthermore, we detail how the li…
We derive convenient uniform concentration bounds and finite sample multivariate normal approximation results for quadratic forms, then describe some applications involving variance components estimation in linear random-effects models. Random-effects models and variance components estimation are classical topics in statistics, with a corresponding well-established asymptotic theory. However, our finite sample results for quadratic forms provide additional flexibility for easily analyzing random-effects models in non-standard settings, which are becoming more important in modern applications (…
The problem of reducing the bias of maximum likelihood estimator in a general multivariate elliptical regression model is considered. The model is very flexible and allows the mean vector and the dispersion matrix to have parameters in common. Many frequently used models are special cases of this general formulation, namely: errors-in-variables models, nonlinear mixed-effects models, heteroscedastic nonlinear models, among others. In any of these models, the vector of the errors may have any multivariate elliptical distribution. We obtain the second-order bias of the maximum likelihood estimat…
This paper investigates improved testing inferences under a general multivariate elliptical regression model. The model is very flexible in terms of the specification of the mean vector and the dispersion matrix, and of the choice of the error distribution. The error terms are allowed to follow a multivariate distribution in the class of the elliptical distributions, which has the multivariate normal and Student-t distributions as special cases. We obtain Skovgaard's adjusted likelihood ratio statistics and Barndorff-Nielsen's adjusted signed likelihood ratio statistics and we conduct a simula…
The integration of physical relationships into stochastic models is of major interest e.g. in data assimilation. Here, a multivariate Gaussian random field formulation is introduced, which represents the differential relations of the two-dimensional wind field and related variables such as streamfunction, velocity potential, vorticity and divergence. The covariance model is based on a flexible bivariate Matérn covariance function for streamfunction and velocity potential. It allows for different variances in the potentials, non-zero correlations between them, anisotropy and a flexible smoothne…
A semi-analytic method is proposed for the generation of realizations of a multivariate process of a given linear correlation structure and marginal distribution. This is an extension of a similar method for univariate processes, transforming the autocorrelation of the non-Gaussian process to that of a Gaussian process based on a piece-wise linear marginal transform from non-Gaussian to Gaussian marginal. The extension to multivariate processes involves the derivation of the autocorrelation matrix from the marginal transforms, which determines the generating vector autoregressive process. The …
We give yet another example of the usefulness of working with marginals of multivariate Gibbs distributions (Cerquetti, 2013) in deriving Bayesian nonparametric estimators under Gibbs priors in species sampling problems. Here in particular we substantially reduce length and complexity of the proofs in Bacallado et al. (2013, Th. 1, and Th. 2) for looking backward probabilities under incomplete information.
Marginally specified models have recently become a popular tool for discrete longitudinal data analysis. Nonetheless, they introduce complex constraint equations and model fitting algorithms. Moreover, there is a lack of available software to fit these models. In this paper, we propose a three-level marginally specified model for analysis of multivariate longitudinal binary response data. The implicit function theorem is introduced to approximately solve the marginal constraint equations explicitly. Furthermore, the use of \textit{probit} link enables direct solutions to the convolution equati…
Efficiently Computed Marginal Likelihoods for Multivariate Mixture Models
Laboratoire de Mathématiques Blaise Pascal, CNRS, Université Clermont Auvergne（布莱斯·帕斯卡数学实验室，法国国家科学研究中心，克莱蒙奥弗涅大学）；CNRS, MAP5, Université Paris Cité（法国国家科学研究中心，MAP5，巴黎西岱大学）；Departments of Statistics and Sociology, University of Washington（华盛顿大学统计与社会学系）
Models based on multivariate t distributions are widely applied to analyze data with heavy tails. However, all the marginal distributions of the multivariate t distributions are restricted to have the same degrees of freedom, making these models unable to describe different marginal heavy-tailedness. We generalize the traditional multivariate t distributions to non-elliptically contoured multivariate t distributions, allowing for different marginal degrees of freedom. We apply the non-elliptically contoured multivariate t distributions to three widely-used models: the Heckman selection model w…
When the number of features exponentially outnumbers the number of samples, feature screening plays a pivotal role in reducing the dimension of the feature space and developing models based on such data. While most extant feature screening approaches are only applicable to data having univariate response, we propose a new method (GenCorr) that admits a multivariate response. Such an approach allows us to more appropriately model multiple responses as a single unit, rather than as unrelated entities, which avails more robust analyses in relation to complex traits embedded in the covariance stru…
An MCMC simulation method based on a two stage delayed rejection Metropolis-Hastings algorithm is proposed to estimate a factor multivariate stochastic volatility model. The first stage uses kstep iteration towards the mode, with k small, and the second stage uses an adaptive random walk proposal density. The marginal likelihood approach of Chib (1995) is used to choose the number of factors, with the posterior density ordinates approximated by Gaussian copula. Simulation and real data applications suggest that the proposed simulation method is computationally much more efficient than the appr…
Multivariate space-time data are increasingly available in various scientific disciplines. When analyzing these data, one of the key issues is to describe the multivariate space-time dependencies. Under the Gaussian framework, one needs to propose relevant models for multivariate space-time covariance functions, i.e. matrix-valued mappings with the additional requirement of non-negative definiteness. We propose a flexible parametric class of cross-covariance functions for multivariate space-time Gaussian random fields. Space-time components belong to the (univariate) Gneiting class of space-ti…
We introduce a flexible individual frailty model for clustered right-censored data, in which covariate effects can be marginally interpreted as log failure odds ratios. Flexible correlation structures can be imposed by introducing multivariate exponential distributed frailties, constructed from a set of multivariate Gaussian random variables. Finite and infinite dimensional parameters are consistently estimated by maximizing a composite contributing marginal likelihood and a consistent estimate for their asymptotic covariance is proposed. Parameter estimation is implemented through a hybrid ex…
Copulas allow a flexible and simultaneous modeling of complicated dependence structures together with various marginal distributions. Especially if the density function can be represented as the product of the marginal density functions and the copula density function, this leads to both an intuitive interpretation of the conditional distribution and convenient estimation procedures. However, this is no longer the case for copula models with mixed discrete and continuous marginal distributions, because the corresponding density function cannot be decomposed so nicely. In this paper, we introdu…
This paper addresses the problem of finding parametric constraints that ensure the validity of the multivariate Mat{é}rn covariance for modeling the spatial correlation structure of coregionalized variables defined in an Euclidean space. To date, much attention has been given to the bivariate setting, while the multivariate setting has been explored to a limited extent only. The existing conditions often imply severe restrictions on the upper bounds for the collocated correlation coefficients, which makes the multivariate Mat{é}rn model appealing for the case of weak spatial cross-dependence o…
Hydroclimatic processes are characterized by heterogeneous spatiotemporal correlation structures and marginal distributions that can be continuous, mixed-type, discrete or even binary. Simulating exactly such processes can greatly improve hydrological analysis and design. Yet this challenging task is accomplished often by ad hoc and approximate methodologies that are devised for specific variables and purposes. In this study, a single framework is proposed allowing the exact simulation of processes with any marginal and any correlation structure. We unify, extent, and improve of a general-purp…
We study a class of missingness mechanisms, called sequentially additive nonignorable, for modeling multivariate data with item nonresponse. These mechanisms explicitly allow the probability of nonresponse for each variable to depend on the value of that variable, thereby representing nonignorable missingness mechanisms. These missing data models are identified by making use of auxiliary information on marginal distributions, such as marginal probabilities for multivariate categorical variables or moments for numeric variables. We present theory proving identification results, and illustrate t…
Hybrid Bernstein Normalizing Flows for Flexible Multivariate Density Regression with Interpretable Marginals
Institute for Applied Research HTWG - University of Applied Sciences（应用研究研究所 HTWG - 高等专业学院）；Chair of Statistics and Campus Institute Data Science (CIDAS) University of Göttingen（统计学教授及校园数据科学研究所 (CIDAS) 雷姆森大学）；Department of Statistics LMU Munich（统计学系 慕尼黑大学）；Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）
We develop a new class of dynamic multivariate Poisson count models that allow for fast online updating and we refer to these models as multivariate Poisson-scaled beta (MPSB). The MPSB model allows for serial dependence in the counts as well as dependence across multiple series with a random common environment. Other notable features include analytic forms for state propagation and predictive likelihood densities. Sequential updating occurs through the updating of the sufficient statistics for static model parameters, leading to a fully adapted particle learning algorithm and a new class of p…
The problem we consider considers estimating a multivariate longitudinal panel data model whose outcomes can be a combination of discrete and continuous variables. This problem is challenging because the likelihood is usually analytically intractable. Our article makes both a methodological contribution and also a substantive contribution to the application. The methodological contribution is to introduce into the panel data literature a particle Metropolis within Gibbs method to carry out Bayesian inference, using a Hamiltonian Monte Carlo (Neal 2011} proposal for sampling the vector of unkno…
We develop a novel multi-factor copula model for multivariate spatial extremes, which is designed to capture the different combinations of marginal and cross-extremal dependence structures within and across different spatial random fields. Our proposed model, which can be seen as a multi-factor copula model, can capture all possible distinct combinations of extremal dependence structures within each individual spatial process while allowing flexible cross-process extremal dependence structures for both upper and lower tails. We show how to perform Bayesian inference for the proposed model usin…
