Estimation of the parameter of a pARMAX model

REVSTAT – Statistical Journal, 8(2) (2010), 139–149
https://infoline.ine.pt/revstat/pdf/rs100204.pdf

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. In fact, they are more convenient for analysis than heavy-tailed ARMA, as their finite-dimensional distributions can easily be written explicitly. The recent power max-autoregressive model (pARMAX) has the interesting feature of describing an
asymptotic independent tail behavior, a property that can be observed in various data series. An estimator of the model parameter c (0 < c < 1) is already available
in the literature, but only in the restrictive case c > 1/2. Here it is presented an estimator for all c 2 (0, 1). Consistency and asymptotic normality are also stated.