ON GENERALIZED FRACTIONAL PROCESSES

Abstract

Abstract. A class of stationary long‐memory processes is proposed which is an extension of the fractional autoregressive moving‐average (FARMA) model. The FARMA model is limited by the fact that it does not allow data with persistent cyclic (or seasonal) behavior to be considered. Our extension, which includes the FARMA model as a special case, makes use of the properties of the generating function of the Gegenbauer polynomials, and we refer to these models as Gegenbauer autoregressive moving‐average (GARMA) models. While the FARMA model has a peak in the spectrum at f = 0, the GARMA process can model long‐term periodic behavior for any frequency 0 f 0.5. Properties of the GARMA process are examined and techniques for generation of realizations, model identification and parameter estimation are proposed. The use of the GARMA model is illustrated through simulated examples as well as with classical sunspot data.