The continuous extension of a discrete random variable is amongst the computational methods used for estimation of multivariate normal copula-based models with discrete margins. Its advantage is that the likelihood can be derived conveniently under the theory for copula models with continuous margins, but there has not been a clear analysis of the adequacy of this method. We investigate the asymptotic and small-sample efficiency of two variants of the method for estimating the multivariate normal copula with univariate binary, Poisson, and negative binomial regressions, and show that they lead to biased estimates for the latent correlations, and the univariate marginal parameters that are not regression coefficients. We implement a maximum simulated likelihood method, which is based on evaluating the multidimensional integrals of the likelihood with randomized quasi Monte Carlo methods. Asymptotic and small-sample efficiency calculations show that our method is nearly as efficient as maximum likelihood for fully specified multivariate normal copula-based models. An illustrative example is given to show the use of our simulated likelihood method.
Estimating copulas with discrete marginal distributions is challenging, especially in high dimensions, because computing the likelihood contribution of each observation requires evaluating $2^{J}$ terms, with $J$ the number of discrete variables. Currently, data augmentation methods are used to carry out inference for discrete copula and, in practice, the computation becomes infeasible when $J$ is large. Our article proposes two new fast Bayesian approaches for estimating high dimensional copulas with discrete margins, or a combination of discrete and continuous margins. Both methods are based…
The direct Gaussian copula model with discrete marginal distributions is an appealing data-analytic tool but poses difficult computational challenges due to its intractable likelihood. A number of approximations/surrogates for the likelihood have been proposed, including the continuous extension-based approximation (CE) and the distributional transform-based approximation (DT). The continuous extension approach is exact up to Monte Carlo error but does not scale well computationally. The distributional transform approach permits efficient computation but offers no theoretical guarantee that it…
We propose, for multivariate Gaussian copula models with unknown margins and structured correlation matrices, a rank-based, semiparametrically efficient estimator for the Euclidean copula parameter. This estimator is defined as a one-step update of a rank-based pilot estimator in the direction of the efficient influence function, which is calculated explicitly. Moreover, finite-dimensional algebraic conditions are given that completely characterize efficiency of the pseudo-likelihood estimator and adaptivity of the model with respect to the unknown marginal distributions. For correlation matri…
In this article, we consider the problem of reconstructing networks for continuous, binary, count and discrete ordinal variables by estimating sparse precision matrix in Gaussian copula graphical models. We propose two approaches: $\ell_1$ penalized extended rank likelihood with Monte Carlo Expectation-Maximization algorithm (copula EM glasso) and copula skeptic with pair-wise copula estimation for copula Gaussian graphical models. The proposed approaches help to infer networks arising from nonnormal and mixed variables. We demonstrate the performance of our methods through simulation studies …
This article extends the literature on copulas with discrete or continuous marginals to the case where some of the marginals are a mixture of discrete and continuous components. We do so by carefully defining the likelihood as the density of the observations with respect to a mixed measure. The treatment is quite general, although we focus focus on mixtures of Gaussian and Archimedean copulas. The inference is Bayesian with the estimation carried out by Markov chain Monte Carlo. We illustrate the methodology and algorithms by applying them to estimate a multivariate income dynamics model.
The distributional transform (DT) is amongst the computational methods used for estimation of high-dimensional multivariate normal copula models with discrete responses. Its advantage is that the likelihood can be derived conveniently under the theory for copula models with continuous margins, but there has not been a clear analysis of the adequacy of this method. We investigate the small-sample and asymptotic efficiency of the method for estimating high-dimensional multivariate normal copula models with univariate Bernoulli, Poisson, and negative binomial margins, and show that the DT approxi…
Copulas provide a modular parameterization of multivariate distributions that decouples the modeling of marginals from the dependencies between them. Gaussian Mixture Copula Model (GMCM) is a highly flexible copula that can model many kinds of multi-modal dependencies, as well as asymmetric and tail dependencies. They have been effectively used in clustering non-Gaussian data and in Reproducibility Analysis, a meta-analysis method designed to verify the reliability and consistency of multiple high-throughput experiments. Parameter estimation for GMCM is challenging due to its intractable likel…
Quantitative studies in many fields involve the analysis of multivariate data of diverse types, including measurements that we may consider binary, ordinal and continuous. One approach to the analysis of such mixed data is to use a copula model, in which the associations among the variables are parameterized separately from their univariate marginal distributions. The purpose of this article is to provide a method of semiparametric inference for copula models via the construction of what we call a marginal set likelihood function for the association parameters. The proposed method of inference…
We propose a new class of multivariate survival models based on archimedean copulas with margins modeled by the Yang and Prentice (YP) model. The Ali-Mikhail-Haq (AMH), Clayton, Frank, Gumbel-Hougaard (GH), and Joe copulas are employed to accommodate the dependency among marginal distributions. Baseline distributions are modeled semiparametrically by the piecewise exponential (PE) distribution and the Bernstein polynomials. The new class of models possesses some attractive features: i) the ability to take into account survival data with crossing survival curves; ii) the inclusion of the well-k…
We define a copula process which describes the dependencies between arbitrarily many random variables independently of their marginal distributions. As an example, we develop a stochastic volatility model, Gaussian Copula Process Volatility (GCPV), to predict the latent standard deviations of a sequence of random variables. To make predictions we use Bayesian inference, with the Laplace approximation, and with Markov chain Monte Carlo as an alternative. We find both methods comparable. We also find our model can outperform GARCH on simulated and financial data. And unlike GARCH, GCPV can easil…
A method that uses order statistics to construct multivariate distributions with fixed marginals and which utilizes a representation of the Bernstein copula in terms of a finite mixture distribution is proposed. Expectation-maximization (EM) algorithms to estimate the Bernstein copula are proposed, and a local convergence property is proved. Moreover, asymptotic properties of the proposed semiparametric estimators are provided. Illustrative examples are presented using three real data sets and a 3-dimensional simulated data set. These studies show that the Bernstein copula is able to represent…
The composite likelihood (CL) is amongst the computational methods used for the estimation of high-dimensional multivariate normal (MVN) copula models with discrete responses. Its computational advantage, as a surrogate likelihood method, is that is based on the independence likelihood for the univariate regression and non-regression parameters and pairwise likelihood for the correlation parameters, but the efficiency of estimating the univariate regression and non-regression parameters can be low. For a high-dimensional discrete response, we propose weighted versions of the composite likeliho…
Maximum pseudo-likelihood (MPL) is a semiparametric estimation method often used to obtain the dependence parameters in copula models from data. It has been shown that despite being consistent, and in some cases efficient, MPL estimation can overestimate the level of dependence especially for small weakly dependent samples. We show that the MPL method uses the expected value of order statistics and we propose to use instead the median or the mode of the same order statistics. In a simulation study we compare the finite-sample performance of the proposed estimators with that of the original MPL…
We present an algorithm to obtain the maximum likelihood estimates of the correlation parameters of elliptical copulas. Previously existing methods for this task were either fast but only approximate or exact but very time-consuming, especially for high-dimensional problems. Our proposal combines the advantages of both, since it obtains the exact estimates and its performance makes it suitable for most practical applications. The algorithm is given with explicit expressions for the Gaussian and Student's t copulas.
In many studies multivariate event time data are generated from clusters having a possibly complex association pattern. Flexible models are needed to capture this dependence. Vine copulas serve this purpose. Inference methods for vine copulas are available for complete data. Event time data, however, are often subject to right-censoring. As a consequence, the existing inferential tools, e.g. likelihood estimation, need to be adapted. A two-stage estimation approach is proposed. First, the marginal distributions are modeled. Second, the dependence structure modeled by a vine copula is estimated…
Model selection is an important activity in modern data analysis and the conventional Bayesian approach to this problem involves calculation of marginal likelihoods for different models, together with diagnostics which examine specific aspects of model fit. Calculating the marginal likelihood is a difficult computational problem. Our article proposes some extensions of the Laplace approximation for this task that are related to copula models and which are easy to apply. Variations which can be used both with and without simulation from the posterior distribution are considered, as well as use …
We present a vine copula based composite likelihood approach to model spatial dependencies, which allows to perform prediction at arbitrary locations. This approach combines established methods to model (spatial) dependencies. On the one hand the geostatistical concept utilizing spatial differences between the variable locations to model the extend of spatial dependencies is applied. On the other hand the flexible class of C-vine copulas is utilized to model the spatial dependency structure locally. These local C-vine copulas are parametrized jointly, exploiting an existing relationship betwee…
Quantile regression, that is the prediction of conditional quantiles, has steadily gained importance in statistical modeling and financial applications. The authors introduce a new semiparametric quantile regression method based on sequentially fitting a likelihood optimal D-vine copula to given data resulting in highly flexible models with easily extractable conditional quantiles. As a subclass of regular vine copulas, D-vines enable the modeling of multivariate copulas in terms of bivariate building blocks, a so-called pair-copula construction (PCC). The proposed algorithm works fast and acc…
A new class of copulas based on order statistics was introduced by Baker (2008). Here, further properties of the bivariate and multivariate copulas are described, such as that of likelihood ratio dominance (LRD), and further bivariate copulas are introduced that generalize the earlier work. One of the new copulas is an integral of a product of Bessel functions of imaginary argument, and can attain the Fréchet bound. The use of these copulas for fitting data is described, and illustrated with examples. It was found empirically that the multivariate copulas previously proposed are not flexible e…
Quantification of microbial interactions from 16S rRNA and meta-genomic sequencing data is difficult due to their sparse nature, as well as the fact that the data only provides measures of relative abundance. In this paper, we propose using copula models with mixed zero-beta margins for estimation of taxon-taxon interactions using the normalized microbial relative abundances. Copulas allow for separate modeling of the dependence structure from the margins, marginal covariate adjustment, and uncertainty measurement. Our method shows that a two-stage maximum likelihood approach provides accurate…
Using the classical estimation method of moments, we propose a new semiparametric estimation procedure for multi-parameter copula models. Consistency and asymptotic normality of the obtained estimators are established. By considering an Archimedean copula model, an extensive simulation study, comparing these estimators with the pseudo maximum likelihood, rho-inversion and tau-inversion ones, is carried out. We show that, with regards to the other methods, the moment based estimation is quick and simple to use with reasonable bias and root mean squared error.
This paper introduces a new class of observation driven dynamic models. The time evolving parameters are driven by innovations of copula form. The resulting models can be made strictly stationary and the innovation term is typically chosen to be Gaussian. The innovations are formed by applying a copula approach for the conditional score function which has close connections the existing literature on GAS models. This new method provides a unified framework for observation-driven models allowing the likelihood to be explicitly computed using the prediction decomposition. The approach may be used…
Vine copulas are a flexible tool for multivariate non-Gaussian distributions. For data from an observational study where the explanatory variables and response variables are measured together, a proposed vine copula regression method uses regular vines and handles mixed continuous and discrete variables. This method can efficiently compute the conditional distribution of the response variable given the explanatory variables. The performance of the proposed method is evaluated on simulated data sets and a real data set. The experiments demonstrate that the vine copula regression method is super…
We describe a simple method for making inference on a functional of a multivariate distribution. The method is based on a copula representation of the multivariate distribution and it is based on the properties of an Approximate Bayesian Monte Carlo algorithm, where the proposed values of the functional of interest are weighed in terms of their empirical likelihood. This method is particularly useful when the "true" likelihood function associated with the working model is too costly to evaluate or when the working model is only partially specified.
The purpose of this paper is to introduce two semiparametric methods for the estimation of copula parameter. These methods are based on minimum Alpha-Divergence between a non-parametric estimation of copula density using local likelihood probit transformation method and a true copula density function. A Monte Carlo study is performed to measure the performance of these methods based on Hellinger distance and Neyman divergence as special cases of Alpha-Divergence. Simulation results are compared to the Maximum Pseudo-Likelihood (MPL) estimation as a conventional estimation method in well-known …
A novel copula-based multivariate panel ordinal model is developed to estimate structural relations among components of well-being. Each ordinal time-series is modelled using a copula-based Markov model to relate the marginal distributions of the response at each time of observation and then, at each observation time, the conditional distributions of each ordinal time-series are joined using a multivariate t copula. Maximum simulated likelihood based on evaluating the multidimensional integrals of the likelihood with randomized quasi Monte Carlo methods is used for the estimation. Asymptotic c…
This research is motivated by discovering and underpinning genetic causes for the progression of a bilateral eye disease, Age-related Macular Degeneration (AMD), of which the primary outcomes, progression times to late-AMD, are bivariate and interval-censored due to intermittent assessment times. We propose a novel class of copula-based semiparametric transformation models for bivariate data under general interval censoring, which includes the case 1 interval censoring (current status data) and case 2 interval censoring. Specifically, the joint likelihood is modeled through a two-parameter Arc…
Copula models have become one of the most widely used tools in the applied modelling of multivariate data. Similarly, Bayesian methods are increasingly used to obtain efficient likelihood-based inference. However, to date, there has been only limited use of Bayesian approaches in the formulation and estimation of copula models. This article aims to address this shortcoming in two ways. First, to introduce copula models and aspects of copula theory that are especially relevant for a Bayesian analysis. Second, to outline Bayesian approaches to formulating and estimating copula models, and their …
Implicit copulas are the most common copula choice for modeling dependence in high dimensions. This broad class of copulas is introduced and surveyed, including elliptical copulas, skew $t$ copulas, factor copulas, time series copulas and regression copulas. The common auxiliary representation of implicit copulas is outlined, and how this makes them both scalable and tractable for statistical modeling. Issues such as parameter identification, extended likelihoods for discrete or mixed data, parsimony in high dimensions, and simulation from the copula model are considered. Bayesian approaches t…
This paper deals with robust inference for parametric copula models. Estimation using Canonical Maximum Likelihood might be unstable, especially in the presence of outliers. We propose to use a procedure based on the Maximum Mean Discrepancy (MMD) principle. We derive non-asymptotic oracle inequalities, consistency and asymptotic normality of this new estimator. In particular, the oracle inequality holds without any assumption on the copula family, and can be applied in the presence of outliers or under misspecification. Moreover, in our MMD framework, the statistical inference of copula model…
We show how to extract the implicit copula of a response vector from a Bayesian regularized regression smoother with Gaussian disturbances. The copula can be used to compare smoothers that employ different shrinkage priors and function bases. We illustrate with three popular choices of shrinkage priors --- a pairwise prior, the horseshoe prior and a g prior augmented with a point mass as employed for Bayesian variable selection --- and both univariate and multivariate function bases. The implicit copulas are high-dimensional, have flexible dependence structures that are far from that of a Gaus…
In this article, we discuss two specific classes of models - Gaussian Mixture Copula models and Mixture of Factor Analyzers - and the advantages of doing inference with gradient descent using automatic differentiation. Gaussian mixture models are a popular class of clustering methods, that offers a principled statistical approach to clustering. However, the underlying assumption, that every mixing component is normally distributed, can often be too rigid for several real life datasets. In order to to relax the assumption about the normality of mixing components, a new class of parametric mixtu…
Joint modelling of longitudinal and time-to-event data is usually described by a joint model which uses shared or correlated latent effects to capture associations between the two processes. Under this framework, the joint distribution of the two processes can be derived straightforwardly by assuming conditional independence given the random effects. Alternative approaches to induce interdependency into sub-models have also been considered in the literature and one such approach is using copulas to introduce non-linear correlation between the marginal distributions of the longitudinal and time…
A Copula density estimation method that is based on a finite mixture of heterogeneous parametric copula densities is proposed here. More specifically, the mixture components are Clayton, Frank, Gumbel, T, and normal copula densities, which are capable of capturing lower tail,strong central, upper tail, heavy tail, and symmetrical elliptical dependence, respectively. The model parameters are estimated by an interior-point algorithm for the constrained maximum likelihood problem. The interior-point algorithm is compared with the commonly used EM algorithm. Simulation and real data application sh…
Capturing complex dependence structures between outcome variables (e.g., study endpoints) is of high relevance in contemporary biomedical data problems and medical research. Distributional copula regression provides a flexible tool to model the joint distribution of multiple outcome variables by disentangling the marginal response distributions and their dependence structure. In a regression setup each parameter of the copula model, i.e. the marginal distribution parameters and the copula dependence parameters, can be related to covariates via structured additive predictors. We propose a frame…
We propose a new copula model for replicated multivariate spatial data. Unlike classical models that assume multivariate normality of the data, the proposed copula is based on the assumption that some factors exist that affect the joint spatial dependence of all measurements of each variable as well as the joint dependence among these variables. The model is parameterized in terms of a cross-covariance function that may be chosen from the many models proposed in the literature. In addition, there are additive factors in the model that allow tail dependence and reflection asymmetry of each vari…
