Error Bounds for Joint Detection and Estimation of a Single Object With Random Finite Set Observation

This paper considers the performance limits for joint detection and estimation from a finite set-valued observation that is stochastically related to the state or parameter of interest. Detection refers to inference about the existence of the state, whereas estimation refers to inference about its value, when detected. Since we need to determine the existence/non-existence of the state as well as its value, the usual notion of Euclidean distance error does not jointly capture detection and estimation error in a meaningful manner. Treating the state as set, which can be either empty or singleton, admits a meaningful distance error for joint detection and estimation. We derive bounds on this distance error for a widely used class of observation models. When existence of the state is a certainty, our bounds coincide with recent results on Cramer-Rao bounds for estimation only problems.