Leiden University Medical Centre

Tuesday 4 October 2005

16:00 – 17:00 hours

Lecture room 1or 2, Physiology Building
Wassenaarseweg 62, Leiden

Copulas (Sklar, 1959) enable to specify multivariate distributions with given marginals. Various parametric proposals were made in the literature for these quantities, mainly in the bivariate case (see e.g. Nelsen, 1999). They can be systematically derived from multivariate distributions with known marginals, yielding e.g. the normal and the Student copulas. Alternatively, one can restrict his interest to a sub-family of copulas named Archimedean (Genest & Mackay, 1986). They are characterized by their generator, a strictly decreasing convex function on (0,1) which tends to infinity at zero and which is zero at one.
We propose to approximate a function derived from the generator using B-splines and show how the associated parameters can be estimated using MCMC. The estimation is reasonably quick. The generated chain(s) can be used to build ``credibility envelopes'' for the generator and, thus, to validate simple parametric proposals for the generator. It is a useful complement to the non-parametric procedure proposed by Genest & Rivest (1993).
The fitted generator is smooth and parametric. It can be used together with its posterior distribution to make inference and prediction for derived quantities such as Kendall's tau to cite one. Parameters associated to parametric models for the marginals can be estimated jointly with the copula parameters. This is an interesting alternative to the two-steps procedure (see e.g. Shih & Louis, 1995) which assumes that the regression parameters are fixed known quantities when it comes to copula parameter(s) estimation. Simulated data are used to validate the approach.
The method will be illustrated using a subset of the Framingham Heart study data. We shall analyse the dependence structure underlying the diastolic (DBP) and the systolic (SBP) blood pressures measured on 663 male subjects at their first visit. These responses will be modelled using marginal additive regression models.