A Modified Cramér-Rao Bound for Discrete-Time Markovian Dynamic Systems

It is well-known that the Modified Cramér-Rao Bound (MCRB) holds particular value in nonstandard deterministic estimation scenarios. Specifically, it proves invaluable when, in addition to estimating deterministic parameters, one needs to determine the probability density function (p.d.f) of the data through the marginalization of a joint p.d.f over random variables. In general, this process of marginalization is mathematically intractable, which restricts the utility of the conventional CRB. This limitation is especially pertinent in the context of discrete-time Markovian dynamic systems. However, we demonstrate that for such systems, the MCRB can be computed recursively with minimal computational burden, provided certain mild regularity conditions are met for the random nuisance parameters. Although this computational advantage may entail a degree of looseness in the bound, we present evidence showcasing the practical relevance of the proposed expressions in a scenario where the MCRB and CRB align.