Fractional lower-order covariance (FLOC)-based estimation for multidimensional PAR(1) model with alpha-stable noise

Many real data exhibit periodic behavior. The periodic autoregressive moving average (PARMA) is one of the most common and useful model to describe these data. In the classical case, the PARMA model is considered by the assumption of Gaussian (or finite-variance) distribution of the noise. However, the Gaussian distribution seems to be unsuitable in many real applications, especially when the corresponding data exhibit impulsive-like behavior. Therefore, the extensions of the classical PARMA models are considered and the Gaussian distribution of the noise is replaced by the so-called heavy-tailed distribution. One of the most known distribution that can be used here is the alpha-stable one. In this paper, we introduce a new estimation technique for the parameters of the multi-dimensional periodic autoregressive time series of order 1 (i.e., PAR(1)), which is based on fractional lower-order covariance, the alternative dependence measure adequate for alpha-stable distributed models. From theoretical point of view, the use of this technique is justified as in this case the classical measure (i.e., covariance) is not defined. The practical aspect of this technique is discussed. The efficiency of the technique on simulated data is demonstrated using the Monte Carlo approach in different contexts, including the sample size and index of stability a of the noise's distribution. Lastly, we present the real data analysis.