On Parameter Estimation for Exponential Dispersion Arma Models

A class of autoregressive moving-average (ARMA) models proposed by Jorgensen and Song [Journal of Applied Probability (1998), Vol. 35, pp. 78-92] with exponential dispersion model margins are useful to deal with non-normal stationary time series with high-order autocorrelation. One property associated with the class of models is that the projection process takes the exact form of the classical Box and Jenkins ARMA representation, leading to considerable ease to establish theories. This paper focuses on the issue of parameter estimation for such models, which has not been thoroughly investigated in Jorgensen and Song's paper. The key of the proposed approach is to treat the residual process associated with the projection essentially as a measurement error, which enables us to formulate directly an ARMA representation for the observed time series. The parameter estimation therefore becomes straightforward using the existing methods for the Box and Jenkins ARMA models such as the quasi-likelihood method. The approach is illustrated by simulation studies and by an analysis of myoclonic seizure counts