Copula-based Regression Models Research Articles

Copula-based regression estimation in the presence of outliers

Ordinary least square estimators for linear regression models are highly influenced by outliers. Copula is a powerful tool for studying the dependence between response and predictor variables and may overcome several limitations associated with the classical linear regression models. In this study, we investigate the efficacy of the copula-based regression models over classical linear regression in the presence of outliers in the x, y, and x-y directions. We also examine the performance of several robust regression estimation techniques when variables are related via elliptical and Archimedean family of copulas and outliers are present in the data. Finally, real and artificial data sets are analyzed to compare the performance of various robust regression techniques.

Analysis of Human Co-exposure to Lead and Cadmium Using Human Biomonitoring (HBM) Data in a Bayesian Copula-Based Regression Framework

The identification of human co-exposure to industrial chemicals or environmental substances is of high interest in human health risk assessment. Due to their ubiquity and persistence in the environment, heavy metals such as cadmium (Cd) and lead (Pb) are of particular concern. Approaches to adequately investigating combinations of these and other often highly correlated variables are lacking. This study proposes a modeling approach to investigate the co-exposure to Cd and Pb, and better understanding the variations of blood Cd and Pb (CdB and PbB, respectively) together with potentially determinant factors. A copula-based regression model was built, using Bayesian inference and Markov Chain Monte Carlo simulation, to relate CdB and PbB of 3- to 14-year-old children participating in the German Environmental Survey for Children (GerES IV) with socio-demographic and ancillary exposure-relevant information. A minor to negligible dependence between CdB and PbB was observed, suggesting that Cd and Pb are subject to differing exposure sources/pathways or kinetics within human body. Despite the resulting low association between CdB and PbB, the developed approach provides methodological bases for enhancing the assessment of the cumulative exposure to multiple substances and for deepening the understanding of the determinants of these exposures.Graphical

The Incorporation of Generalized Linear Models into Bivariate Gaussian Copula and An Application

In non-life insurance mathematics, analyses and premium or reserve calculations are carried out in the presence of dependency between the claim variables in recent years. And, thus over- or underestimation of aggregate loss caused by the assumption of dependency between the claim severity and frequency are prevented. The Gaussian copula function, which is frequently used for dependency modeling, is integrated into the marginal generalized linear models to obtain a mixed copula-based regression model called "copula regression". In this study, a copula regression model is created using a bivariate Gaussian copula, Gamma and Poisson marginal generalized linear models for claim severity and frequency, respectively. An application is performed with a simulated data where there is a dependence between the claim severity and frequency using the R package “CopulaRegression”. The importance of the modeling of dependency between claims is investigated by the comparison of the independent and dependent models and the results of application show that the copula regression model in which dependency is considered has lower relative mean square errors compared the independent marginal generalized linear models.

Conditional expectation formula of copulas for higher dimensions and its application

Due to its simple form, linear regression is the most commonly used model when dealing with a predictive model. However, there are some limitations to the model, such as the constraint of only being able to model variables that have a linear relationship, the assumption of normality on its error, and the multi-collinearity between independent variables which should not occur. One of the alternative models that is free from these limitations is the copula-based regression model defined by the conditional expectation formula of copulas. Leong and Valdez [Claims prediction using copula models, Insurance Math. Econom., 2005] [15] developed a conditional expectation formula of copulas for higher dimensions in the implicit form with bivariate case examples. Crane and Hoek [Conditional expectation formulae for copulas, Aust. N.Z.J. Stat, 2008] [5] provided conditional expectation formula of copulas explicitly for two dimensions with its examples. However, in practice, a predictive model often involves more than two variables, i.e. one dependent variable with more than one independent variable, including a copula-based regression model. With regard to these problems and the limitations of dimension in previous studies, our contribution in this study is extending the copula-based regression model for higher dimensions for class of Farlie-Gumbel-Morgenstern, elliptical, and Archimedean copula. We obtain a closed-form of conditional expectation formula of Farlie-Gumbel-Morgenstern, Gaussian, Student-t, and Clayton copula for n dimensions and provide the formula for Gumbel copula up to four dimensions. We apply our extended formula to estimate KRW/USD currency based on its association with CNY/USD and JPY/USD, and found that the extended function can be used quite accurately.

Copula-based regression models with data missing at random

The existing literature of copula-based regression assumes that complete data are available, but this assumption is violated in many real applications. The present paper allows the regressand and regressors to be missing at random (MAR). We formulate a generalized regression model which unifies many prominent cases such as the conditional mean and quantile regressions. A semiparametric copula and the target regression curve are estimated via the calibration approach. The consistency and asymptotic normality of the estimated regression curve are proved. We show via Monte Carlo simulations that the proposed approach operates well in finite samples, while a benchmark equal-weight approach fails with substantial bias under MAR. An empirical application on revenues and R&D expenses of German manufacturing firms highlights a practical use of our approach.

Gaussian copula-based zero-inflated power series joint models to analyze correlated count data

A Gaussian Copula-based regression model is proposed that accounts for associations between count responses with extra zeros. Our approach entails underlying latent variables to indicate the latent mechanisms which generate the count responses where some of the count responses are inflated in a zero point. The model contains, as special sub-models, several important distributions such as the power series distributions with and without extra zeros, for example, Poisson, negative binomial, zero-inflated Poisson and zero-inflated negative binomial distributions. The full likelihood-based inference method is applied for the estimation of parameters to obtain maximum likelihood estimates of the parameters. Modified Pearson residuals, where the correlation between responses is taken into account, are used for finding abnormal observations. To illustrate the utility of the models, some simulations are illustrated. Finally, the proposed models are applied to an insurance data set for insurers, obtained from an observational study, where the number of automobile claims and the number of third party claims are the correlated count responses. The effects of car age and the type of car, driving place on both responses are investigated simultaneously.

Copula-Based Regression Models with Responses Missing at Random: A Unified Approach

The existing literature of copula-based regression assumes that complete data are available, but this assumption is violated in many real applications. The present paper allows the regressand and regressors to be missing at random (MAR). We formulate a generalized regression model which unifies many prominent cases such as the conditional mean and quantile regressions. A semiparametric copula and the target regression curve are estimated via the calibration approach. The consistency and asymptotic normality of the estimated regression curve are proved. We show via Monte Carlo simulations that the proposed approach operates well in finite samples, while a benchmark equal-weight approach fails with substantial bias under MAR. An empirical application on revenues and R\&D expenses of German manufacturing firms highlights a practical use of our approach.

3128 Copula-Based Regression Models for Correlated Bivariate Binary Outcomes: Application to Ophthalmologic Data Structures

OBJECTIVES/SPECIFIC AIMS: To account for association between the pair of binary outcomes, we adopt the Clayton and Frank copulas to indirectly specify their joint distributions. METHODS/STUDY POPULATION: We propose a regression model for the joint modelling of correlated bivariate outcomes using copulas. RESULTS/ANTICIPATED RESULTS: develop full maximum likelihood inference.

Bivariate Severity Analysis of Train Derailments using Copula-Based Regression Models

The far-reaching consequences of train derailments have been a major concern to industry and government despite their relatively low occurrence. These consequences include injury, loss of life and property, interruption of services, and destruction of the environment. Thus, it is imperative to carefully examine train derailment severity. The majority of extant literature has failed to consider the multivariate nature of derailment severity and has instead focused mainly on only one severity outcome, namely, the number of derailed cars. However, it is also important to analyze the monetary damage incurred by railroads during derailments. In this paper, a joint mixed copula-based model for derailed cars and monetary damage is presented for the combined analysis of their relationship with a set of covariates that might affect both outcomes. Marginal generalized regression linear models are combined with a bivariate copula, which characterizes the dependence between the two variables. Copulas also address endogeneity due to similar unobserved or omitted variables that may affect both response variables. The copula-based regression model was found to be more appropriate than the independent multivariate regression model. The incorporation of the copula to characterize the dependence resulted in a greater effect on the dispersion estimate than the point estimates. Derailment speed was found to have the most pronounced effect on both response variables. However, it was found to have a greater impact on monetary damage than the number of derailed cars.

Copula-based regression modeling of bivariate severity of temporary disability and permanent motor injuries

The analysis of factors influencing the severity of the personal injuries suffered by victims of motor accidents is an issue of major interest. Yet, most of the extant literature has tended to address this question by focusing on either the severity of temporary disability or the severity of permanent injury. In this paper, a bivariate copula-based regression model for temporary disability and permanent injury severities is introduced for the joint analysis of the relationship with the set of factors that might influence both categories of injury. Using a motor insurance database with 21,361 observations, the copula-based regression model is shown to give a better performance than that of a model based on the assumption of independence. The inclusion of the dependence structure in the analysis has a higher impact on the variance estimates of the injury severities than it does on the point estimates. By taking into account the dependence between temporary and permanent severities a more extensive factor analysis can be conducted. We illustrate that the conditional distribution functions of injury severities may be estimated, thus, providing decision makers with valuable information.

Simultaneous inference in structured additive conditional copula regression models: a unifying Bayesian approach

While most regression models focus on explaining distributional aspects of one single response variable alone, interest in modern statistical applications has recently shifted towards simultaneously studying multiple response variables as well as their dependence structure. A particularly useful tool for pursuing such an analysis are copula-based regression models since they enable the separation of the marginal response distributions and the dependence structure summarised in a specific copula model. However, so far copula-based regression models have mostly been relying on two-step approaches where the marginal distributions are determined first whereas the copula structure is studied in a second step after plugging in the estimated marginal distributions. Moreover, the parameters of the copula are mostly treated as a constant not related to covariates and most regression specifications for the marginals are restricted to purely linear predictors. We therefore propose simultaneous Bayesian inference for both the marginal distributions and the copula using computationally efficient Markov chain Monte Carlo simulation techniques. In addition, we replace the commonly used linear predictor by a generic structured additive predictor comprising for example nonlinear effects of continuous covariates, spatial effects or random effects and furthermore allow to make the copula parameters covariate-dependent. To facilitate Bayesian inference, we construct proposal densities for a Metropolis---Hastings algorithm relying on quadratic approximations to the full conditionals of regression coefficients avoiding manual tuning. The performance of the resulting Bayesian estimates is evaluated in simulations comparing our approach with penalised likelihood inference, studying the choice of a specific copula model based on the deviance information criterion, and comparing a simultaneous approach with a two-step procedure. Furthermore, the flexibility of Bayesian conditional copula regression models is illustrated in two applications on childhood undernutrition and macroecology.

Total loss estimation using copula-based regression models

We present a joint copula-based model for insurance claims and sizes. It uses bivariate copulae to accommodate for the dependence between these quantities. We derive the general distribution of the policy loss without the restrictive assumption of independence. We illustrate that this distribution tends to be skewed and multi-modal, and that an independence assumption can lead to substantial bias in the estimation of the policy loss. Further, we extend our framework to regression models by combining marginal generalized linear models with a copula. We show that this approach leads to a flexible class of models, and that the parameters can be estimated efficiently using maximum-likelihood. We propose a test procedure for the selection of the optimal copula family. The usefulness of our approach is illustrated in a simulation study and in an analysis of car insurance policies.

Analysis of directional dependence using asymmetric copula-based regression models

The directional dependence between variables using asymmetric copula regression has drawn much attention in recent years. There are, however, some critical issues which have not been properly addressed in regards to the statistical inference of the directional dependence. For example, the previous use of asymmetric copulas failed to fully capture the dependence patterns between variables, and the method used for the parameter estimation was not optimal. In addition, no method was considered for selecting a suitable asymmetric copula or for computing the general measurements of the directional dependence when there are no closed-form expressions. In this paper, we propose a generalized multiple-step procedure for the full inference of the directional dependence in joint behaviour based on the asymmetric copula regression. The proposed procedure utilizes several novel methodologies that have not been considered in the literature of the analysis of directional dependence. The performance and advantages of the proposed procedure are illustrated using two real data examples, one from biological research on histone genes, and the other from developmental research on attention deficit hyperactivity disorder.