9:30 An introduction to copulas for modelling continuous data: Parametric families–methods of inference.
Studying associations among multivariate outcomes 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 thus, limiting their use. Unfortunately, these limitations occur very often when working with multivariate distributions. To solve this problem, copulas seem to be a promising solution. Copulas, see e.g. the books by Joe (1997, 2014) or Nelsen (2006), are a currently fashionable way to model multivariate data as they account for the dependence structure and provide a flexible representation of the multivariate distribution. We will provide a survey of parametric families that are appropriate as models for multivariate continuous data with different dependence structures. Inferential procedures will be also discussed.
11:00 Cofee/Tea and Refreshments
11:30 Insurance company loss and expense application
The methods in the previous lecture are illustrated in the loss-ALAE data set in Frees and Valdez (1998). The data comprise general liability claims randomly chosen from late settlement lags and were provided by InsuranceServices Office, Inc. Each claim consists of an indemnity payment (the loss) and an allocated loss adjustment expense (ALAE). The objective is to describe the joint distribution of losses and expenses. Before that, we calculate some simple descriptive statistics and diagnostics to choose potential copula models.
14:00 CopulaModel package by Joe and Krupskii
In this lecture we will illustrate the software CopulaModel for the analysis of the loss-ALAE data set in Frees and Valdez (1998). We will focus on (a) Implementation of the diagnostics for dependence and tail asymmetry, and, (b) estimation of copula-based models. Before that, we will show that R is well-suited for programming your own maximum likelihood routines. Our focus will be the nlm command, which implements a quasi Newton algorithm for non-linear optimisation. Optimisation through nlm is relatively straightforward, since it is usually not necessary to provide analytic first and second derivatives.
- Frees, E. W. and Valdez, E. A. (1996). Understanding relationships using copulas, North American Actuarial Journal, 1–25
- Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall, London.
- Joe, H. (2014). Dependence Modeling with copulas. Chapman & Hall, London.
- Joe, H. and Krupskii, P. (2014). CopulaModel: Dependence Modeling with Copulas. R package version 0.6.
- Nelsen, R. B. (2006). An Introduction to Copulas. Springer-Verlag, New York.