On the Relationship Between Measures of Relative Efficiency for Random Signal Detection

Abstract: Relative efficiency (RE), the Pitman asymptotic relative efficiency (ARE) and efficacy are important relative performance measures of signal detection techniques. These measures allow comparing two detectors in terms of the relative sample sizes they require to achieve the same prescribed level of false alarm and detection probabilities. While the finite-sample-size measure RE is useful to analyze the small sample behavior of detectors, in practice it is difficult to compute. In the limit as the signal strength approaches zero at an appropriate rate, the RE converges to an asymptotic limit only for very large sample sizes. This limiting ratio of the number of samples is the ARE, which lends analytical tractability, but does not provide insights into the finite sample behavior of detectors which is important for practical applications. This led researchers to study the convergence of RE to ARE, and has been well-reported for the problem of constant signal detection in additive, independent, and identically distributed noise. When the signal to be detected is random ({\ie}, unknown), such a convergence analysis is lacking in the literature and is the focus of this paper. A relationship between RE and ARE for random signal detection is established. We use the higher-order terms in the Taylor series expansion of the mean of the test statistic under the alternative hypothesis to derive this formula. We present preliminary remarks on the convergence of RE to ARE for random signals in comparison to that for constant signal detection.