Average Fisher information maximisation in presence of cost-constrained measurements

An optimal estimation framework is considered in the presence of cost-constrained measurements. The aim is to maximise the average Fisher information under a constraint on the total cost of measurement devices. An optimisation problem is formulated to calculate the optimal costs of measurement devices that maximise the average Fisher information for arbitrary observation and measurement statistics. In addition, a closed-form expression is obtained in the case of Gaussian observations and measurement noise. Numerical examples are presented to explain the results.