On the Possible Orderings in the Measurement Selection Problem

An aspect of the measurement selection problem-the existence of anomalous orderings on the probability of error obtained by selected subsets of measurements-is discussed. It is shown that for any ordering on the probability of error as a function of the subset of measurements (subject to an obvious set monotonicity condition), there exists a multivariate normal two-hypothesis problem N(μ,K) versus N(μ,K) that exhibits this ordering. Thus no known nonexhaustive sequential k-measurement selection procedure is optimal, even for jointly normal measurements.