Intrinsic Slepian-Bangs Type Formula for Parameters on LGs with Unknown Measurement Noise Variance

Intrinsic lower bounds on the intrinsic mean square error are of major importance to characterize the best achievable estimation performance of any unbiased estimator on smooth manifold as Lie group (LG). When the parameter is described by a LG model and the observation noise is with unknown variance, it is also necessary to determine a bound both on the LG parameter and this variance. In this communication, we propose an intrinsic generic Fisher information matrix taking into account this problem. To achieve that, we derive an intrinsic Slepian-Bangs formula on the LG product of the unknown parameter of interest and the LG of positive scalar values in which the variance intrinsically lies. The proposed bound is validated on a Gaussian observation model for unknown parameter lying to SE (3) and variance noise on R+.