Non-Gaussian ARMA Models

Abstract

We discuss stationary AR and ARMA time series models for sequences of integer-valued random variables and continuous random variables. Stationary distribution of these models is non-Gaussian. Such models can be broadly described as extensions of Gaussian ARMA models, which have been very widely discussed in the time series literature. These non-Gaussian AR models share two important properties with a linear AR(1) model: (i) the conditional expectation of \(X_t\) is a linear function of the past observation and (ii) the auto-correlation function (ACF) has an exponential decay. However, the conditional variance of an observation is frequently a function of the past observations. These models are formed so as to have a specific form of the stationary distribution. Stationary distributions include standard discrete distributions such as binomial, geometric, Poisson, and continuous distributions such as exponential, Weibull, gamma, inverse Gaussian, and Cauchy. In some cases, maximum likelihood estimation is tractable. In other cases, regularity conditions are not met. Estimation is then carried out based on properties of the marginal distribution of the process and mixing properties such as strong or \(\phi \)-mixing are useful to derive properties of the estimators.