On Weighted Least Squares Estimators of Parameters of a Chirp Model

The least squares method seems to be a natural choice in estimating the parameters of a chirp model. But the least squares estimators are very sensitive to the outliers. Even in presence of very few outliers, the performance of the least squares estimators becomes quite unsatisfactory. Due to this reason, the least absolute deviation method has been proposed in the literature. But implementing the least absolute deviation method is quite challenging particularly for the multicomponent chirp model. In this paper, we propose to use the weighted least squares estimators, which seem to be more robust in presence of a few outliers. First, we consider the weighted least squares estimators of the unknown parameters of a single component chirp signal model. It is assumed that the weight function is a finite degree polynomial and the errors are independent and identically distributed random variables with mean zero and finite variance. It is observed that the weighted least squares estimators are strongly consistent and they have the same convergence rate as the least squares estimators. The weighted least squares estimators can be obtained by solving a two dimensional optimization problem. In case of the multicomponent chirp signal, we provide a sequential weighted least squares estimators and provide the consistency and asymptotic normality properties of these sequential weighted least squares estimators. To compute the sequential weighted least squares estimators one needs to solve only one two dimensional optimization problem at each stage. Extensive simulations have been performed to see the performances of the proposed estimators. Two data sets have been analyzed for illustrative purposes.