Errors-in-Variables-Based Methodology of Estimation and Testing for Infinite-Variance Periodic Autoregressive Models with Additive Noise

Cyclostationary models, characterized by periodic statistical functions, are prevalent in various real phenomena. A common example is the periodic autoregressive (PAR) time series, an extension of the well-known autoregressive model. However, practical scenarios often involve signals disturbed by additive noise that masks the cyclostationary behavior. Furthermore, the cyclostationary model as well as the additive noise may exhibit impulsiveness with a heavy-tailed infinite-variance distribution. This paper introduces a novel approach for estimating parameters in PAR models affected by additive infinite-variance noise. In our approach we assume that PAR time series may also have heavy-tailed non-Gaussian distribution. Building upon the errors-in-variables (EIV) method designed for finite-variance cases, we adapt it for the infinite-variance scenario. To overcome challenges in defining autocovariance in this context, we propose using the fractional lower-order covariance (FLOC), which is applicable to models with finite fractional moments of appropriate order. The proposed estimation method is used in the methodology for statistical testing whether a signal originates from a “pure” or noise-corrupted PAR model. The efficiency of our approach is demonstrated on an exemplary infinite-variance distribution of PAR model and additive noise, namely the alpha-stable distribution. Comparative analysis is performed with the FLOC-based high-order Yule-Walker method.