Minimax Quantization for Distributed Maximum Likelihood Estimation

We consider the design of quantizers for the distributed estimation of a deterministic parameter, when the fusion center uses a Maximum-Likelihood estimator. We define a new metric of performance, which is to minimize the maximum ratio between the Fisher Information of the unquantized and quantized observations. Since the estimator is M-L, the criterion is equivalent to the minimizing the maximum asymptotic relative efficiency due to quantization. We propose an algorithm to obtain the quantizer that optimizes the metric and prove its convergence. Through simulations, we illustrate that the quantizer performance is close to the best possible Fisher Information as number of quantization bits increases. Furthermore, under certain conditions, the quantizer structure is found to belong to the class of score-function quantizers, hich maximize Fisher Information for a given value of the parameter.