On Distributed Linear Estimation With Observation Model Uncertainties

We consider distributed estimation of a Gaussian source in a heterogenous bandwidth constrained sensor network, where the source is corrupted by independent multiplicative and additive observation noises. We assume the additive observation noise is zero-mean Gaussian with known variance, however, the system designer is unaware of the distribution of multiplicative observation noise and only knows its first- and second-order moments. For multibit quantizers, we derive an accurate closed-form approximation for the mean-square error (MSE) of the linear minimum MSE) estimator at the fusion center. For both error-free and erroneous communication channels, we propose several rate allocation methods named as longest root to leaf path, greedy, integer relaxation, and individual rate allocation to minimize the MSE given a network bandwidth constraint, and minimize the required network bandwidth given a target MSE. We also derive the Bayesian Cramér–Rao lower bound (CRLB) for an arbitrarily distributed multiplicative observation noise and compare the MSE performance of our proposed methods against the CRLB. Our results corroborate that, for the low-power multiplicative observation noise and adequate network bandwidth, the gaps between the MSE of greedy and integer relaxation methods and the CRLB are negligible, while the MSE of individual rate allocation and uniform methods is not satisfactory. Through analysis and simulations, we also explore why maximum likelihood and maximum a posteriori estimators based on one-bit quantization perform poorly for the low-power additive observation noise.