Multivariate response data abound in many application areas including insurance, risk management, finance, biology, psychometrics, health and environmental sciences. Studying associations among multivariate response data is an interesting problem in statistical science. The dependence between random variables is completely described by their multivariate distribution. When the multivariate distribution has a simple form, standard methods can be used to make inference. On the other hand one may create multivariate distributions based on particular assumptions, limiting thus their use. For example, most existing models assume rigid margins of the same form (e.g., Gaussian, Student, exponential, Gamma, Poisson, etc.) and/or limited dependence structure.

To solve this problem copulas seem to be a promising solution. Copulas are a currently popular and useful way to model multivariate response data, as they account for the dependence structure and provide a flexible representation of the multivariate distribution. They allow for flexible dependence modeling, different from assuming simple linear correlation structures and normality. Copulas, essentially,  enable you to break the model building process into two separate steps:

(a) Choice of arbitrary marginal distributions:
(i) They could take different forms;
(ii) They could involve covariates.

(b) Choice of an arbitrary copula function (dependence structure).

That makes them particularly well suited to many applications in finance, insurance and medicine, among others.

This project will focus on dependence modeling with copulas for non-normal multivariate/longitudinal response data. Such data have different dependence structures including features such as tail dependence and/or negative dependence. To this end, the desiderata properties of multivariate copula families for modeling multivariate/longitudinal response data are given below (see e.g., Nikoloulopoulos and Karlis (2009) and Nikoloulopoulos et al. (2012)):

(a) Wide range of dependence, allowing both positive and negative dependence.
(b) Flexible dependence (including tail dependence for continuous data), meaning that the number of bivariate margins is (approximately) equal to the number of dependence parameters.
(c) Computationally feasible joint cumulative distribution function (discrete data) or density (continuous data) for likelihood estimation.
(d) Closure property under marginalization, meaning that lower-order margins belong to the same parametric family.
(e)  No joint constraints for the dependence parameters, meaning that the use of covariate functions for the dependence parameters is straightforward.

In the existing literature, none of the existing parametric families of multivariate copulas satisfy all these conditions; hence there are many challenges for copula-based models for multivariate/longitudinal response data.

The primary goal of this project is to develop parametric families of copulas, see for example Nikoloulopoulos and Karlis (2009); Joe et al. (2010); Nikoloulopoulos et al. (2009, 2012), that are appropriate as models for multivariate data and inferential methods, see for example Nikoloulopoulos et al. (2011), to overcome computational complexities imposed by some existing families. A secondary goal is to apply the developed models and methods to the aforementioned application areas.

(i) Joe, H., Li, H., and Nikoloulopoulos, A. K. (2010). Tail dependence functions and vine copulas. Journal of Multivariate Analysis, 101:252-270.
(ii) Nikoloulopoulos, A. K., Joe, H., and Chaganty, N. R. (2011). Weighted scores method for regression models with dependent data. Biostatistics, 12:653-665.
(iii) Nikoloulopoulos, A. K., Joe, H., and Li, H. (2009). Extreme value properties of multivariate t copulas. Extremes, 12:129-148.
(iv) Nikoloulopoulos, A. K., Joe, H., and Li, H. (2012). Vine copulas with asymmetric tail dependence and applications to financial return data. Computational Statistics & Data Analysis, 56:3659-3673.
(v) Nikoloulopoulos, A. K. and Karlis, D. (2009). Finite normal mixture copulas for multivariate discrete data modeling. Journal of Statistical Planning and Inference, 139:3878-3890.