Parameter estimation for Gegenbaeur Arfisma processes with infinite variance innovations

Abstract

. In this study, we consider the Gegenbauer ARFISMA process with α-stable innovations which belong to the class of infinite variance time series. This is a finite parameter model that exhibits long-range dependence, high variability, and Seasonal and /or Cyclical Long Memory (SCLM) variations. The Gegenbauer ARFISMA time series with α-stable innovations can be rewritten as linear processes (infinite order moving averages) with coefficients that decay slowly to zero and with innovations that are in the domain of attraction of a α-stable distribution with stability parameter ( 1 < α ≤ 2 ) . We assume that the coefficients of the linear process depend on an unknown parameter vector ξ which should be estimated from samples of length n. We prove that under some assumptions, a Whittle-type estimator ξ n for ξ is consistent (ξ n converges to the true value ξ 0 in probability as n ⟶ + ∞ ), and, we derive the limiting distribution of the resealed differences ( n / log ⁡ ( n ) ) 1 / α ( ξ n − ξ 0 ) . We perform some simulations to illustrate the consistency of the Whittle-type estimator ξ n for ξ.