Sampling issues in least squares, Fourier analytic, and number theoretic methods in parameter estimation

Abstract

Given, noisy data from a periodic point process that satisfies certain conditions, least squares procedures can be used to solve for maximum likelihood estimates of the period. Under more general conditions, Fourier analytic methods, e.g., Wiener's periodogram, can be used to solve for estimates which are approximately maximum likelihood. However, these methods break down when the data has increasing numbers of missing observations. Juxtaposed with these methods, number theoretic methods provide parameter estimations that, while not being maximum likelihood, can be used as initialization in an algorithm that achieves the Cramer-Rao bound for moderate noise levels. We describe the conditions under which the least squares procedures and Fourier analytic methods do not produce estimates close to maximum likelihood, and show that the number theoretic methods provide a reliable estimate in these cases. We also discuss the type of data for which the number theoretic methods fail to produce good estimates.