The stability of nonlinear least squares problems and the crameér-rao bound

A number of problems of interest in signal processing can be reduced to nonlinear parameter estimation problems. The traditional approach to studying the stability of these estimation problems is to demonstrate finiteness of the Cramér-Rao bound (CRB) for a given noise distribution. We review an alternate, determinstic notion of stability for the associated nonlinear least squares (NLS) problem from the realm of nonlinear programming (i.e., that the global minimizer of the least squares problem exists and varies smoothly with the noise). Furthermore, we show that under mild conditions, identifia bill ty of the parameters along with a finite CRB for the case of Gaussian noise is equivalent to deterministic stability of the NLS problem. Finally, we demonstrate the application of our result, which is general, to the problems of multichannel blind deconvolution and sinusoid retrieval to generate new stability results for these problems with little additional effort.