Minimax discrimination for observed Poisson processes with uncertain rate functions

The problem of robust design is considered in the context of testing hypotheses concerning the rate function of an observed point process. Designs that are insensitive to uncertainty in the rate functions are developed by applying a minimax formulation to two different measures of signal-to-noise ratio. Uncertainty in the rate is modeled by using general classes of rate measures generated by Choquet 2-alternating capacities, and solutions are characterized for this case by a Radon-Nikodym type derivative between such classes. It is shown that for uncertainty within capacity classes the robust decision design developed for the signal-to-noise ratio is also robust in a weaker sense for the Chernoff upper bounds on the error probabilities. Furthermore, the use of such a test guarantees the exponential convergence of these bounds to zero with increasing length of the observation interval for all rates in the uncertainty class.