Multivariate models and inference

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In research studies in many areas, such as finance, measured variables do not follow the classical bell-shaped distribution individually and do not have dependence that can be multivariate Gaussian after transforms. In the past 10 years, the vine pair-copula construction has been the most flexible multivariate non-Gaussian distributions for applications. The vine pair-copula construction can handle measured variables that are continuous, discrete, binary or ordinal. In studies that required prediction of one (response) variable from explanatory variables and where the variables are simultaneously observed, the vine copula approach fits a joint distribution followed by inference from the conditional distribution of the response given the explanatory variables. Compare with classical methods of multiple regression, generalized linear models and survival regression, this approach allows prediction with models for flexible shapes for conditional quantiles and conditional variance as a function of the explanatory variables. The approach is especially useful for unbounded predictor spaces, as classical methods do not handled this situation well. With use of copulas in areas of applications involving risk, tail inference is important. To evaluate suitability of different models, we have developed tail-weighted measures of dependence as diagnostics in addition to the usual measures of central dependence. Our derivations are based on a tail form of copulas that allow for non-parametric estimation of joint tail (risk) probabilities and assess fits of parametric models. In some applications, we use latent variables similar to classical models, but with the use of copulas linking observed variables to latent variables to allow for more flexibility in tail dependence behaviour and better fits to multivariate data. The main objectives of some recent and proposed future research is to further develop multivariate models and inference/computing procedures for multivariate non-Gaussian response. With a larger number of variables, it is desirable to have multivariate models with parsimonious dependence structures and flexible tail behaviour. For predictions from conditional distributions after fitting a flexible copula, research is underway to compare models based on different vine copulas with models using machine learning methods; criteria include interpretability and out-of-sample performance. Variation selection is part of the model comparison.