Lower Bounds on Non-Bayesian Parameter Estimation Errors under Reparameterization

This paper introduces a comprehensive approach for evaluating non-Bayesian lower bounds on the mean-squared-error in unbiased estimation of a parameter vector, for the special case where the probability density function of the measurements is given as a function of another parameter vector, such that a defined functional relation exists between the two vectors. We study two variations of these bounds and pinpoint the conditions governing the existence of each version. Subsequently, we establish connections between the bounds, showing that when both exist, one is tighter than the other. We also compare them with the Cramér-Rao bound, which could have been directly derived, given the availability of the appropriate probability density function. The paper concludes by presenting specific examples relevant to the multidimensional statistical signal processing community. The paper's results help in choosing the tightest possible bound for a given application.