MARKOV-CHAIN MODELS FOR THRESHOLD EXCEEDANCES

In recent research on extreme value statistics, there has been an exte nsive development of threshold methods, first in the univariate case a nd subsequently in the multivariate case as well. In this paper, an al ternative methodology for extreme values of univariate time series is developed, by assuming that the time series is Markovian and using biv ariate extreme value theory to suggest appropriate models for the tran sition distributions, A new likelihood representation for threshold me thods is presented which we apply to a Markovian time series. An impor tant motivation for developing this kind of theory is the possibility of calculating probability distributions for functionals of extreme ev ents. We address this issue by showing how a theory of compound Poisso n limits for additive functionals can be combined with simulation to o btain numerical solutions for problems of practical interest. The meth ods are illustrated by application to temperature data.