Asymptotic Theory of A Frequency Estimator

Traditional methods of estimating the frequency of sinusoids from noisy data include the periodogram maximization and the nonlinear least squares. It is well-known that these methods lead to efficient frequency estimates whose asymptotic standard error is of order O(n−3/2). To actually compute the estimates, some sort of iterative search procedures must be employed in view of the high nonlinearity in the parameters. The presence of many local extrema in this problem requires iterative search algorithms to be started with an accurate initial guess—the required precision of the starting values is typically O(n−1), which is not readily available. In this paper we investigate a recent promising approach, contraction-mapping (CM) method for frequency estimation. The CM method is based on an iterative filtering idea in which the estimated first-order autocorrelation of the filtered process contracts to a fixed point. The CM frequency estimator is derived from this fixed point. The critical issue of how the accuracy of the initial guess for the fixed-point iteration controls the precision of the CM estimator is studied, and the asymptotic relationship between the initial estimator and the CM estimator is quantified together with the limiting distributions and almost sure convergence of the fixed points. It is shown that we can start this CM algorithm with poor initial values of precision O(1) and end with a final estimator whose precision can be made arbitrarily close to O(n−3/2). Moreover, this procedure is guaranteed to converge. AMS 1991 subject classifications. Primary 62M10; Secondary 60G35, 93E12.