Asymptotic Theory of the Least Squares Estimators of Sinusoidal Signal

Abstract

The consistency and the asymptotic normality of the least squares estimators are derived of the sinusoidal model under the assumption of stationary random error. It is observed that the model does not satisfy the standard sufficient conditions of Jennrich (1969), Wu (1981) or Kundu (1991). Recently the consistency and the asymptotic normality are derived for the sinusoidal signal under the assumption of normal error (Kundu; 1993) and under the assumptions of independent and identically distributed random variables in Kundu and Mitra (1996). This paper will generalize them. Hannan (1971) also considered the similar kind of model and establish the result after making the Fourier transform of the data for one parameter model. We establish the result without making the Fourier transform of the data. We give an explicit expression of the asymptotic distribution of the multiparameter case, which is not available in the literature. Our approach is different from Hannan's approach. We do some simulations study to see the small sample properties of the two types of estimators.

AMS Subject Classifications (1985)::

• 62J02
• 62C05

Keywords:

• Asymptotic distribution
• strong consistency
• least squares estimators and stationary distribution