On the Cramér-Rao Bound Under a Linear Transformation of the Parameter

This paper deals with the case where the Crarnér-Rao bound (CRB) on the estimation error of a deterministic parameter vector is given, and it is desired to evaluate a bound on a linear transformation of it. The literature suggests two versions of the bound on the estimation error of the reparameterized vector, denoted by
$B_{1}$
and
$B_{2}$
. We first identify the conditions under which either of these bounds exist, focusing on the case where they co-exist, Under such scenario, we find the relations between
$B_{1}, B_{2}$
, and
$CRB$
. The results are demonstrated on two practical applications of multi-sensor signal processing.