A copula model for marked point processes

Many chronic diseases feature recurring clinically important events. In addition, however, there often exists a random variable which is realized upon the occurrence of each event reflecting the severity of the event, a cost associated with it, or possibly a short term response indicating the effect of a therapeutic intervention. We describe a novel model for a marked point process which incorporates a dependence between continuous marks and the event process through the use of a copula function. The copula formulation ensures that event times can be modeled by any intensity function for point processes, and any multivariate model can be specified for the continuous marks. The relative efficiency of joint versus separate analyses of the event times and the marks is examined through simulation under random censoring. An application to data from a recent trial in transfusion medicine is given for illustration.     

Nonparametric estimation of copula functions for dependence modelling

Copulas characterize the dependence among components of random vectors. Unlike marginal and joint distributions, which are directly observable, the copula of a random vector is a hidden dependence structure that links the joint distribution with its margins. Choosing a parametric copula model is thus a nontrivial task but it can be facilitated by relying on a nonparametric estimator. Here the authors propose a kernel estimator of the copula that is mean square consistent everywhere on the support. They determine the bias and variance of this estimator. They also study the effects of kernel smoothing on copula estimation. They then propose a smoothing bandwidth selection rule based on the derived bias and variance. After confirming their theoretical findings through simulations, they use their kernel estimator to formulate a goodness-of-fit test for parametric copula models.     

Directional dependence via Gaussian copula beta regression model with asymmetric GARCH marginals

This article proposes a new directional dependence by using the Gaussian copula beta regression model. In particular, we consider an asymmetric Generalized AutoRegressive Conditional Heteroscedasticity (GARCH) model for the marginal distribution of standardized residuals to make data exhibiting conditionally heteroscedasticity to white noise process. With the simulated data generated by an asymmetric bivariate copula, we verify our proposed directional dependence method. For the multivariate direction dependence by using the Gaussian copula beta regression model, we employ a three-dimensional archemedian copula to generate trivariate data and then show the directional dependence for one random variable given two other random variables. With West Texas Intermediate Daily Price (WTI) and the Standard & Poor’s 500 (S&P 500), our proposed directional dependence by the Gaussian copula beta regression model reveals that the directional dependence from WTI to S&P 500 is greater than that from S&P 500 to WTI. To validate our empirical result, the Granger causality test is conducted, confirming the same result produced by our method.     

Max‐infinitely divisible models and inference for spatial extremes

For many environmental processes, recent studies have shown that the dependence strength is decreasing when quantile levels increase. This implies that the popular max‐stable models are inadequate to capture the rate of joint tail decay, and to estimate joint extremal probabilities beyond observed levels. We here develop a more flexible modeling framework based on the class of max‐infinitely divisible processes, which extend max‐stable processes while retaining dependence properties that are natural for maxima. We propose two parametric constructions for max‐infinitely divisible models, which relax the max‐stability property but remain close to some popular max‐stable models obtained as special cases. The first model considers maxima over a finite, random number of independent observations, while the second model generalizes the spectral representation of max‐stable processes. Inference is performed using a pairwise likelihood. We illustrate the benefits of our new modeling framework on Dutch wind gust maxima calculated over different time units. Results strongly suggest that our proposed models outperform other natural models, such as the Student‐t copula process and its max‐stable limit, even for large block sizes.     

A dynamic double asymmetric copula generalized autoregressive conditional heteroskedasticity model: application to China's and US stock market

Modeling the relationship between multiple financial markets has had a great deal of attention in both literature and real-life applications. One state-of-the-art technique is that the individual financial market is modeled by generalized autoregressive conditional heteroskedasticity (GARCH) process, while market dependence is modeled by copula, e.g. dynamic asymmetric copula-GARCH. As an extension, we propose a dynamic double asymmetric copula (DDAC)-GARCH model to allow for the joint asymmetry caused by the negative shocks as well as by the copula model. Furthermore, our model adopts a more intuitive way of constructing the sample correlation matrix. Our new model yet satisfies the positive-definite condition as found in dynamic conditional correlation-GARCH and constant conditional correlation-GARCH models. The simulation study shows the performance of the maximum likelihood estimate for DDAC-GARCH model. As a case study, we apply this model to examine the dependence between China and US stock markets since 1990s. We conduct a series of likelihood ratio test tests that demonstrate our extension (dynamic double joint asymmetry) is adequate in dynamic dependence modeling. Also, we propose a simulation method involving the DDAC-GARCH model to estimate value at risk (VaR) of a portfolio. Our study shows that the proposed method depicts VaR much better than well-established variance–covariance method.     

Bivariate degradation modelling with marginal heterogeneous stochastic processes

In this paper, we consider that the degradation of two performance characteristics of a product can be modelled by stochastic processes and jointly by copula functions, but different stochastic processes govern the behaviour of each performance characteristic (PC) degradation. Different heterogeneous and homogeneous models are presented considering copula functions and different combinations of the most used stochastic processes in degradation analysis as marginal distributions. This is an important aspect to consider because the behaviour of the degradation of each PC may be different in its nature. As the joint distributions of the proposed models result in complex distributions, the estimation of the parameters of interest is performed via MCMC. A simulation study is performed to compare heterogeneous and homogeneous models. In addition, the proposed models are implemented to crack propagation data of two terminals of an electronic device, and some insights are provided about the product reliability under heterogeneous models.     

On composite likelihood in bivariate meta-analysis of diagnostic test accuracy studies

The composite likelihood is amongst the computational methods used for estimation of the generalized linear mixed model (GLMM) in the context of bivariate meta-analysis of diagnostic test accuracy studies. Its advantage is that the likelihood can be derived conveniently under the assumption of independence between the random effects, but there has not been a clear analysis of the merit or necessity of this method. For synthesis of diagnostic test accuracy studies, a copula mixed model has been proposed in the biostatistics literature. This general model includes the GLMM as a special case and can also allow for flexible dependence modelling, different from assuming simple linear correlation structures, normality and tail independence in the joint tails. A maximum likelihood (ML) method, which is based on evaluating the bi-dimensional integrals of the likelihood with quadrature methods, has been proposed, and in fact it eases any computational difficulty that might be caused by the double integral in the likelihood function. Both methods are thoroughly examined with extensive simulations and illustrated with data of a published meta-analysis. It is shown that the ML method has no non-convergence issues or computational difficulties and at the same time allows estimation of the dependence between study-specific sensitivity and specificity and thus prediction via summary receiver operating curves.     

Non‐parametric Copula Estimation Under Bivariate Censoring

In this paper, we consider non‐parametric copula inference under bivariate censoring. Based on an estimator of the joint cumulative distribution function, we define a discrete and two smooth estimators of the copula. The construction that we propose is valid for a large range of estimators of the distribution function and therefore for a large range of bivariate censoring frameworks. Under some conditions on the tails of the distributions, the weak convergence of the corresponding copula processes is obtained in l^∞([0,1]²). We derive the uniform convergence rates of the copula density estimators deduced from our smooth copula estimators. Investigation of the practical behaviour of these estimators is performed through a simulation study and two real data applications, corresponding to different censoring settings. We use our non‐parametric estimators to define a goodness‐of‐fit procedure for parametric copula models. A new bootstrap scheme is proposed to compute the critical values.     

Semiparametric models of longitudinal and time-to-event data with applications to HIV viral dynamics and CD4 counts

We propose a semiparametric approach based on proportional hazards and copula method to jointly model longitudinal outcomes and the time-to-event. The dependence between the longitudinal outcomes on the covariates is modeled by a copula-based times series, which allows non-Gaussian random effects and overcomes the limitation of the parametric assumptions in existing linear and nonlinear random effects models. A modified partial likelihood method using estimated covariates at failure times is employed to draw statistical inference. The proposed model and method are applied to analyze a set of progression to AIDS data in a study of the association between the human immunodeficiency virus viral dynamics and the time trend in the CD4/CD8 ratio with measurement errors. Simulations are also reported to evaluate the proposed model and method.     

Invariance Theorems for Fisher Information

In multivariate and multi-parameter contexts, new expressions for Fisher Information are derived using the copula representation of the joint distribution of random variables. Invariance of Fisher Information to margins of the joint distribution is then demonstrated.     

Mutual information as a measure of multivariate association: analytical properties and statistical estimation

Mutual information (also known as Kullback–Leibler divergence) can be viewed as a measure of multivariate association in a random vector. The definition incorporates the joint density as well as the marginal densities. We will focus on a representation of mutual information in terms of copula densities that is thus independent of the marginal distributions. This representation yields a different approach to estimating mutual information than the original definition does, as only the copula density has to be estimated. We review analytical properties and examples for selected distributions and discuss methods of nonparametric estimation of copula densities and hence of the mutual information from a sample. Based on a simulation study, we compare the performance of these estimators with respect to bias, standard deviation, and the root mean squared error. The Gauss and the Frank copula are considered as examples.     

Measures of biomarker dependence using a copula-based multivariate epsilon–skew–normal family of distributions

In this note we develop a new multivariate copula model based on epsilon–skew–normal marginal densities for the purpose of examining biomarker dependency structures. We illustrate the flexibility and utility of this model via a variety of graphical tools and a data analysis example pertaining to salivary biomarker. The multivariate normal model is a sub-model of the multivariate epsilon–skew–normal distribution.     

Gaussian copula distributions for mixed data,with application in discrimination

The construction of a joint model for mixed discrete and continuous random variables that accounts for their associations is an important statistical problem in many practical applications. In this paper, we use copulas to construct a class of joint distributions of mixed discrete and continuous random variables. In particular, we employ the Gaussian copula to generate joint distributions for mixed variables. Examples include the robit-normal and probit-normal-exponential distributions, the first for modelling the distribution of mixed binary-continuous data and the second for a mixture of continuous, binary and trichotomous variables. The new class of joint distributions is general enough to include many mixed-data models currently available. We study properties of the distributions and outline likelihood estimation; a small simulation study is used to investigate the finite-sample properties of estimates obtained by full and pairwise likelihood methods. Finally, we present an application to discriminant analysis of multiple correlated binary and continuous data from a study involving advanced breast cancer patients.     

Maximum likelihood analysis of semicompeting risks data with semiparametric regression models

The “semicompeting risks” include a terminal event and a non-terminal event. The terminal event may censor the non-terminal event but not vice versa. Because times to the two events are usually correlated, the non-terminal event is subject to dependent/informative censoring by the terminal event. We seek to conduct marginal regressions and joint association analyses for the two event times under semicompeting risks. The proposed method is based on the modeling setup where the semiparametric transformation models are assumed for marginal regressions, and a copula model is assumed for the joint distribution. We propose a nonparametric maximum likelihood approach for inferences, which provides a martingale representation for the score function and an analytical expression for the information matrix. Direct theoretical developments and computational implementation are allowed for the proposed approach. Simulations and a real data application demonstrate the utility of the proposed methodology.     

The copula directional dependence by stochastic volatility models

This paper proposes a copula directional dependence by using a bivariate Gaussian copula beta regression with Stochastic Volatility (SV) models for marginal distributions. With the asymmetric copula generated by the composition of two Plackett copulas, we show that our SV copula directional dependence by the Gaussian copula beta regression model is superior to the Kim and Hwang (2016) copula directional dependence by an asymmetric GARCH model in terms of the percent relative efficiency of bias and mean squared error. To validate our proposed method with the real data, we use Brent Crude Daily Price (BRENT), West Texas Intermediate Daily Price (WTI), the Standard & Poor’s 500 (SP) and US 10-Year Treasury Constant Maturity Rate (TCM) so that our copula SV directional dependence is overall superior to the Kim and Hwang (2016) copula directional dependence by an asymmetric GARCH model in terms of precision by the percent relative efficiency of mean squared error. In terms of forecasting using the real financial data, we also show that the Bayesian SV model of the uniform transformed data by a copula conditional distribution yields an improvement on the volatility models such as GARCH and SV.     

Copula directed acyclic graphs

A new methodology for selecting a Bayesian network for continuous data outside the widely used class of multivariate normal distributions is developed. The ‘copula DAGs’ combine directed acyclic graphs and their associated probability models with copula C/D-vines. Bivariate copula densities introduce flexibility in the joint distributions of pairs of nodes in the network. An information criterion is studied for graph selection tailored to the joint modeling of data based on graphs and copulas. Examples and simulation studies show the flexibility and properties of the method.     

Regression analysis of longitudinal data with outcome‐dependent sampling and informative censoring

We consider a regression analysis of longitudinal data in the presence of outcome‐dependent observation times and informative censoring. Existing approaches commonly require a correct specification of the joint distribution of longitudinal measurements, the observation time process, and informative censoring time under the joint modeling framework and can be computationally cumbersome due to the complex form of the likelihood function. In view of these issues, we propose a semiparametric joint regression model and construct a composite likelihood function based on a conditional order statistics argument. As a major feature of our proposed methods, the aforementioned joint distribution is not required to be specified, and the random effect in the proposed joint model is treated as a nuisance parameter. Consequently, the derived composite likelihood bypasses the need to integrate over the random effect and offers the advantage of easy computation. We show that the resulting estimators are consistent and asymptotically normal. We use simulation studies to evaluate the finite‐sample performance of the proposed method and apply it to a study of weight loss data that motivated our investigation.     

Positive quadrant dependence tests for copulas

In this paper the interest is in testing the null hypothesis of positive quadrant dependence (PQD) between two random variables. Such a testing problem is important since prior knowledge of PQD is a qualitative restriction that should be taken into account in further statistical analysis, for example, when choosing an appropriate copula function to model the dependence structure. The key methodology of the proposed testing procedures consists of evaluating a “distance” between a nonparametric estimator of a copula and the independence copula, which serves as a reference case in the whole set of copulas having the PQD property. Choices of appropriate distances and nonparametric estimators of copula are discussed, and the proposed methods are compared with testing procedures based on bootstrap and multiplier techniques. The consistency of the testing procedures is established. In a simulation study the authors investigate the finite sample size and power performances of three types of test statistics, Kolmogorov–Smirnov, Cramér–von‐Mises, and Anderson–Darling statistics, together with several nonparametric estimators of a copula, including recently developed kernel type estimators. Finally, they apply the testing procedures on some real data. The Canadian Journal of Statistics 38: 555–581; 2010 © 2010 Statistical Society of Canada     

Multi-type insurance claim processes with high-dimensional covariates

Multi-type insurance claim processes have attracted considerable research interest in the literature. The existing statistical inference for such processes, however, may encounter “curse of dimensionality” due to high-dimensional covariates. In this article, a technique of sufficient dimension reduction is applied to multiple-type insurance claim data, which uses a copula to model the dependence between different types of claim processes, and incorporates a one-dimensional frailty to fit the dependence of claims “within” the same claim process. A two-step procedure is proposed to estimate model parameters. The first step develops nonparametric estimators of the baseline, the basis of the central subspace and its dimension, and the regression function. Then the second step estimates the copula parameter. Simulations are performed to evaluate and confirm the theoretical results.     

Bootstrapping the conditional copula

This paper is concerned with inference about the dependence or association between two random variables conditionally upon the given value of a covariate. A way to describe such a conditional dependence is via a conditional copula function. Nonparametric estimators for a conditional copula then lead to nonparametric estimates of conditional association measures such as a conditional Kendall's tau. The limiting distributions of nonparametric conditional copula estimators are rather involved. In this paper we propose a bootstrap procedure for approximating these distributions and their characteristics, and establish its consistency. We apply the proposed bootstrap procedure for constructing confidence intervals for conditional association measures, such as a conditional Blomqvist beta and a conditional Kendall's tau. The performances of the proposed methods are investigated via a simulation study involving a variety of models, ranging from models in which the dependence (weak or strong) on the covariate is only through the copula and not through the marginals, to models in which this dependence appears in both the copula and the marginal distributions. As a conclusion we provide practical recommendations for constructing bootstrap-based confidence intervals for the discussed conditional association measures.