Asymptotic expansions and rate of convergence in signal detection

The number of samples required for signal detection is considered as a function of the error probabilities. This problem is treated in the context of detecting a constant signal in additive, independent, and identically distributed noise. Detectors that base their decisions on the comparison with a threshold of accumulated, nonlinearly transformed observations are treated. Asymptotic expressions are derived for the relationship between sample size and error probabilities for this model in two situations: that in which the nonlinearity has a partially absolutely continuous output distribution; and that in which it has a lattice output distribution. Traditional analyses of such problems have involved only the lowest order terms of such relationships (i.e. central limit theorem results), leading to performance indices such as the Pitman asymptotic relative efficiency (ARE). Such indices are known to be of limited accuracy in predicting performance for more moderate sample sizes. Here, the behavior of sample size as a function of error probabilities is considered in more detail, leading to more accurate indices of relative efficiencies for such detection problems. Several specific examples are examined in detail, and numerical results are included to illustrate the significantly improved performance estimation afforded thereby for even small sample sizes.