Joanna Landmesser, Ph.D., Econometrics and Statistics, Warsaw University of Life Sciences, ul. Nowoursynowska 166, Warszawa, 02-787, Poland
Copulas, introduced by Sklar in 1959, are a useful method for deriving joint distributions given the marginals especially when we work with nonnormal distributions. The aim of our paper is to use this idea in the context of joint duration distributions for the modeling of multivariate survival functions.
We consider the case of parallel duration spells that are dependent. Survival times are assumed to be continuous, the spells are not censored and have parametric distributions.
A q-dimensional multivariate survival function S(t)=S(t1, …, tq)=Pr[T1>t1,…,Tq>tq] has a corresponding copula representation C(S1(t1),…,Sq(tq)).
For the case q=2 we obtain S(t1,t2)=1-F(t1)-F(t2)+F(t1,t2)=S1(t1)+S2(t2)-1+C(1-S1(t1),1-S2(t2)), where the function C(.) is the survival copula and S(t1,t2) is now a function of the marginal survival functions only. There are many examples of bivariate copula functions in the published literature (e.g. Normal, Clayton or Frank copulas).
In our paper a numerical example is considered, which is based on the data from the German Socio-Economic Panel Study from 1984 to 2006. We model the dependence structure among durations using copula approach. Marginal models are fitted and tested using standard univariate survival models, and the dependence parameter is estimated in a sequential second-stage procedure.