Estimation of frequencies in presence of heavy tail errors

In this paper, we consider the problem of estimating the sinusoidal frequencies in presence of additive white noise. The additive white noise has mean zero but it may not have finite variance. We propose to use the least-squares estimators or the approximate least-squares estimators to estimate the unknown parameters. It is observed that the least-squares estimators and the approximate least-squares estimators are asymptotically equivalent and both of them provide consistent estimators of the unknown parameters. We obtain the asymptotic distribution of the least-squares estimators under the assumption that the errors are from a symmetric stable distribution. We propose different methods of constructing confidence intervals and compare their performances through Monte Carlo simulations. We also discuss the properties of the estimators if the errors are correlated and finally we discuss some open problems.