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(¿,K) versus N(¿¿,K) that exhibits this ordering. Thus no known nonexhaustive sequential k-measurement selection procedure is optimal, even for jointly normal measurements.