Modeling rare events through a pRARMAX process

Ferreira and Canto e Castro (2007, 2008) introduce a power max-autoregressive process, in short pARMAX, as an alternative to heavy tailed ARMA when modeling rare events. In this paper, an extension of pARMAX is considered, by including a random component which makes the model more applicable to real data. We will see conditions under which this new model, here denoted as pRARMAX, has unique stationary distribution and we analyze its extremal behavior. Based on Bortot and Tawn (1998), we derive a threshold-dependent extremal index which is a functional of the coefficient of tail dependence of Ledford and Tawn (1996, 1997) which in turn relates with the pRARMAX parameter. In order to fit a pRARMAX model to an observed data series, we present a methodology based on minimizing the Bayes risk in classification theory and analyze this procedure through a simulation study. We illustrate with an application to financial data.