One main theme of my research is dependence modelling, in which the multivariate responses can be binary, categorical, extreme value, heavy-tailed, etc. Copulas (multivariate distribution with uniform univariate margins) have an important role in these models. Non-normal time series can be considered as a special case, in which dependence is decreasing with lag. The theory is quite different from classical multivariate statistics.

The research problems here are challenging, as one must deal with existence of models with some desirable properties. This area of research requires the study of methods for constructing parametric families and the search for new stochastic constructions and representations for models. In extreme value inference, the concept of tail dependence of multivariate models must be considered. More recently, the use of the tail order has been useful for tail behaviour of copulas; copula families can then be compared in an analogous use of tailweights for univariate densities.

A general approach is dependence modelling with copulas. Since 2008, one of the biggest advances for high-dimensional copula models and applications has been the development of vine pair-copula constructions that cover continuous and discrete variables. The vine is a graphical model with trees that shows dependence among variables; bivariate copulas are assigned to edges of the vine trees and they are algebraically independent. The vine extends the Markov tree (a 1-truncated vine) to allow for possible layers of conditional dependences. For the special case of multivariate Gaussian, the edges of the vine are correlations for tree 1 and partial correlations for the remaining trees of the vine.

For high-dimensional data, truncated vine models can be used to summarize parsimonious dependence structures. When latent variables are included in the vine, truncated vine models include the copula versions of Gaussian-based common and structured factor models and copula versions of structural equation models.

Inferential techniques have been developed so that the models are computationally feasible to work with. Inference methods based on low-dimensional margins have been studied; examples are two-stage estimation (or inference functions for margins) and composite likelihood.

Specific situations where I have made use of modern multivariate concepts include:

• extreme value inference for multivariate maxima,
• models for repeated measures or multivariate discrete (binary, ordinal, count) response in biostatistical applications
• time series models for non-normal response with covariates,
• models for familial data, in which families have different sizes and different pedigree structure,
• psychometrics models for item response data,
• copula-GARCH models for financial returns.

Current and future research includes development of models and inference procedures for multivariate applications in biostatistics, econometrics and finance, genetics, psychometrics and machine learning.