Distribution of the Fisher information loss due to random compressed sensing

In this work, we study the impact of compressive sampling with random matrices on Fisher information and the Cramér-Rao bound (CRB) for nonlinear parameter estimation in a complex multivariate normal measurement model. We consider the class of random compression matrices whose distribution is invariant to right-unitary transformations. For this class of random compression matrices, we show that the normalized Fisher information matrix after compression has a complex matrix-variate beta distribution, which is independent of the Fisher information matrix before compression and the values of the parameters. We also derive the distribution of CRB. Our results can be used to quantify the amount of loss in Fisher information and the increase in CRB due to random compression.