Actuarial modelling of extremal events using transformed generalized extreme value distributions and generalized Pareto distributions.

摘要

In 1928, Extreme Value Theory (EVT) originated in work of Fisher and Tippett describing the behavior of maximum of independent and identically distributed random variables. Various applications have been implemented successfully in many fields such as: actuarial science, hydrology, climatology, engineering, and economics and finance.;This paper begins with introducing examples that extreme value theory comes to encounter. Then classical results from EVT are reviewed and the current research approaches are introduced. In particular, statistical methods are emphasized in detail for the modeling of extremal events. A case study of hurricane damages over the last century is presented using the "excess over threshold" (EOT) method.;In most actual cases, the range of the data collected is finite with an upper bound while the fitted Generalized Extreme Value (GEV) and Generalized Pareto (GPD) distributions have infinite tails. Traditionally this is treated as trivial based on the assumption that the upper bound is so large that no significant result is affected when it is replaced by infinity. However, in certain circumstances, the models can be improved by implementing more specific techniques. Different transforms are introduced to rescale the GEV and GPD distributions so that they have finite supports.;All classical methods can be applied directly to transformed models if the upper bound is known. In case the upper bound is unknown, we set up models with one additional parameter based on transformed distributions. Properties of the transform functions are studied and applied to find the cumulative density functions (cdfs) and probability density functions (pdfs) of the transformed distributions. We characterize the transformed distribution from the plots of their cdfs and mean residual life. Then we apply our findings to determine which transformed distribution should be used in the models. At the end some results of parameter estimation are obtained through the maximum likelihood method.