Bounds for target tracking accuracy with probability of detection smaller than one

Recently several new results for Cramer-Rao lower bounds (CRLB´s) in dynamical systems have been obtained. Several different approaches and approximations have been presented. For the general case of target tracking with a detection probability smaller than one and possibly in the presence of false measurements, two main approaches have been presented. One is the so called information reduction factor (IRF) approach and the other the enumeration (ENUM) approach, also referred to as conditioning approach. It has been shown that the ENUM approach leads to a strictly larger covariance matrix than the IRF approach, still being a lower bound of on the performance however. Thus, the ENUM approach provides a strictly tighter bound on the attainable performance. It has been conjectured that these bounds converge to one another in the limit or equivalently after an initial transition stage. In this paper we show, using some recent results on the so called modified Riccati (MR) equation and by means of counter examples, that this conjecture does not hold true in general. We also prove that it does hold true in the special case of deterministic target motion. Furthermore, we show that the detection probability has an influence on the limiting behaviors of the bounds. The various results are illustrated by means of representative examples.