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Relative efficiency (RE), the Pitman asymptotic relative efficiency (ARE) andefficacy are important relative performance measures of signal detectiontechniques. These measures allow comparing two detectors in terms of therelative sample sizes they require to achieve the same prescribed level offalse alarm and detection probabilities. While the finite-sample-size measureRE is useful to analyze the small sample behavior of detectors, in practice itis difficult to compute. In the limit as the signal strength approaches zero atan appropriate rate, the RE converges to an asymptotic limit only for verylarge sample sizes. This limiting ratio of the number of samples is the ARE,which lends analytical tractability, but does not provide insights into thefinite sample behavior of detectors which is important for practicalapplications. This led researchers to study the convergence of RE to ARE, andhas been well-reported for the problem of constant signal detection inadditive, independent, and identically distributed noise. When the signal to bedetected is random ({\ie}, unknown), such a convergence analysis is lacking inthe literature and is the focus of this paper. A relationship between RE andARE for random signal detection is established. We use the higher-order termsin the Taylor series expansion of the mean of the test statistic under thealternative hypothesis to derive this formula. We present preliminary remarkson the convergence of RE to ARE for random signals in comparison to that forconstant signal detection.

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