Multivariate generalised Pareto distributions
Nader Tajvidi · 1995
this paper we give a multivariate analogue of the GPD and consider estimation of parameters in some speciøc bivariate generalised Pareto distributions (BGPD 's). In the latter case a model is øtted to the joint distribution of a bivariate observation subject to condition that at least one component exceeds a high threshold. This allows us to study dependent extremes and for example to estimate a bivariate upper quantile curve or calculate the probability that the thresholds are simultaneously exceeded by two variables. This approach doesn't require multivariate ordering. It permits simultaneous estimation of marginal and dependence parameters. We begin, in Sections 2 and 3, with a brief summary of univariate extreme value distributions and the corresponding generalised Pareto distributions. In Section 4 we present the main theorem which motivates our deønition of multivariate generalised Pareto distribution (MGPD). The family of multivariate extreme value distributions is inønite dimensional but there exists several characterisation of these distributions. In Section 5 we present a summary of dioeerent characterisations. We develop multivariate Pareto distributions which correspond to a characterisation by Resnick and De Haan ([19] and [11]). Usually we call these distributions ibivariate Pareto distributionsj (BPD's) or imultivariate Pareto distributionsj (MPD's). As we will see later these correspond to multivariate generalised Pareto distributions with so called standard Pareto marginals. We use ibivariate generalised Pareto distributionj or imultivariate generalised Pareto distributionj to refer to distributions with the generalised Pareto marginals. In Section 6 we consider the general form of the BPD and give some parametric subfamilies of this distribution. Sec...