Large sample properties of separable nonlinear least squares estimators

In this paper, the large sample properties of the separable nonlinear least squares algorithm are investigated. Unlike the previous results in the literature, the data are assumed to be complex valued, and the whiteness assumption on the measurement noise sequence has been relaxed. Convergence properties of the parameter estimates are established. Asymptotic accuracy analysis has been carried out, in which the assumptions used are relatively weaker than the assumptions in the previous related works. It is shown under quite general conditions that the parameter estimates are asymptotically circular. Conditions for asymptotic complex normality are also established. Next, a bound on the deviation of the asymptotic covariance matrix from the Crame´r-Rao bound (CRB) is derived. Finally, a sufficient condition for the nonlinear least squares estimate to achieve the Crame´r-Rao lower bound is established. The results presented in this paper are general and can be applied to any specific application where separable nonlinear least squares is employed.