Asymptotic Efficiency of the PHD in Multitarget/Multisensor Estimation

Tracking an unknown number of objects is challenging, and often requires looking beyond classical statistical tools. When many sensors are available the estimation accuracy can reasonably be expected to improve, but there is a concomitant rise in the complexity of the inference task. Nowadays, several practical algorithms are available for multitarget/multisensor estimation and tracking. In terms of current research activity one of the most popular is the probability hypothesis density, commonly referred to as the PHD, in which the goal is estimation of object locations (unlabeled estimation) without concern for object identity (which is which). While it is relatively well understood in terms of its implementation, little is known about its performance and ultimate limits. This paper is focused on the characterization of PHD estimation performance for the static multitarget case, in the limiting regime where the number of sensors goes to infinity. It is found that the PHD asymptotically behaves as a mixture of Gaussian components, whose number is the true number of targets, and whose peaks collapse in the neighborhood of the classical maximum likelihood estimates, with a spread ruled by the Fisher information. Similar findings are obtained with reference to a naive, two-step algorithm which first detects the number of targets, and then estimates their positions.