On some parameter estimation problems in alpha-stable processes

Current algorithms for estimating the parameters of a symmetric alpha-stable ARMA process are either highly non-linear, or assume small MA orders (q less than or equal to 3), or invoke the minimum-phase assumption. We use results from the statistics literature to show that the normalized correlation is well-defined; we show that the normalized cumulants are also well-behaved. We propose to use the correlation to estimate the spectrally-equivalent minimum-phase (SEMP) parameters, and then to use the cumulants to resolve the phase of the model. We also show that correlation-based techniques (such as ESPRIT) work well for estimating the parameters of harmonics observed in alpha-stable noise. Correlation-based algorithms are shown to work well despite the infinite variance of the alpha-stable process.