Optimal Estimator for Distributed Anonymous Observers

In this paper, we consider a distributed estimation problem in which multiple observations of a signal process are combined via the maximum function for the decision making. A key result established is that, under suitable technical conditions, the optimal decision function can be implemented by means of thresholds. A natural question is how to determine the optimal threshold value. We propose here an algorithm for threshold adjustment by means of training sequences. The algorithm is a variation of the Kiefer-Wolfowitz algorithm with expanding truncations and randomized differences. A result of the paper is to establish the convergence of the algorithm if the variance of observation noises is small enough.