An Intrinsic Modified Cramér-Rao Bound on Lie Groups

The Modified Cramér-Rao Bound (MCRB) proves to be of significant importance in non-standard estimation scenarios, when in addition to unknown deterministic parameters to be estimated, observations also depend on random nuisance parameters. Given the interest of applications that involve estimation on Lie Groups (LGs), as well as the relevance of nonstandard estimation problems in many practical scenarios, the main concern in this communication is to derive an intrinsic MCRB on LGs (LG-MCRB). For this purpose, a modified unbiasedness constraint must be defined, yielding a modified Barankin Bound. A closed-form formula of the LG-MCRB is then provided for a LG Gaussian model on $S O(2)$, representing $2 D$ rotation matrices, while considering non-Gaussian random nuisance parameters. The validity of this expression is then assessed through numerical simulations, and compared with the intrinsic CRB on LGs for a simplified illustrative scenario, involving a concentrated Gaussian prior distribution on the random nuisance parameters.