Two-Stage Selection of the Best Signal-To-Noise Ratio with Related Approximations

We consider the problem of calibrating k (≥ 2) channels or signal processors and select the best one, that is the one having the largest signal-to-noise ratio (SNR). Assuming a Gaussian distribution for the back-ground white noise or clutter, the problem reduces to selecting from k (≥ 2) normal populations the one with the largest value of the absolute mean. Assuming unknown means and a common unknown variance, we proceed under Bechhofer's 1954 indifference-zone formulation. Since the common variance is unknown, a single-stage procedure guaranteeing a preassigned level of the minimum probability of correct selection (PCS) would not exist. We propose a two-stage selection methodology whose minimum PCS meets the minimum PCS requirement up to the second-order term with respect to the difference of the absolute means. The roles of the proposed approximation are examined both via exact computations and simulations. A realistic example on passive acoustic sonar detection of marine mammals (Abraham 2004) is included with data analyses.