Kristian Åkesson

Title: Modelling Dependence with Archimedean Copulas

Abstract: The study of multivariate random variables is a basic
problem in statistics and has for long been dominated by the
multivariate normal distribution. Not only can the distribution be a
good approximation in most situations, due to the Central Limit
Theorem (CLT), but it has also many attractive mathematical properties
that make analysis simple. Although the Gaussian framework has its
practical advantages many have recognized the need for examining
alternative approaches. This thesis introduces the concept of copulas,
a statistical tool for understanding dependence among random
variables, not necessarily normally distributed. A copula is a
function that links univariate marginal distributions to the full
multivariate distribution. They provide a straightforward way to
extend modelling from the usual joint normality assumption to more
general joint distributions. The objective of this thesis is to
analyze dependence between multivariate random variables using a
particular class of copulas called Archimedean copulas. Our focus is
the modelling of bivariate distributions. We describe the main methods
of inference for copulas and also conduct a small Monte Carlo
simulation to study the performance of the estimators. One of the main
difficulties using copulas is evaluating the fit of estimated
copulas. In this thesis we have provided some methods to solve this
problem. The procedures described are applied to financial data, for
which we perform portfolio optimization between two stock market
indices, and to temperature data, for which we fit a bivariate
distribution to annual maximum temperature observations.