Asymptotic and pre-asymptotic tail behavior of a power max-autoregressive model

Abstract

Max-autoregressive models for time series data are useful when we want to make inference about rare events, mainly in areas like hydrology, geophysics and finance. Here we present a power max-autoregressive ($p$ARMAX) process, $\{X_i\}$, defined in such a way that the asymptotic tail dependence coefficient of Ledford and Tawn, computed for observations lag $m$ apart ($\eta_m$), exhibits a power decay with $m$ for larger values of $c$, the main parameter of the process, namely, $\eta_m=c^m$, $c\in(1/2,1)$. We also look at the threshold-dependent form of the extremal index, which is an important functional when extending discussions of extreme values from independent and identically distributed (i.i.d.) sequences to stationary ones. We state an approach for this functional as well as its connection with the coefficient $\eta$ for the $p$ARMAX process.