Canonically optimum threshold detection

A general canonical theory is developed for the systematic approximation of optimum, or Bayes, detection procedures in the critical limiting threshold mode of operation. The approximations to Bayes detectors introduced here are called locally optimum Bayes detectors (LOBD's) and are defined by the condition that they produce the same value of average risk and its derivative for vanishingly small input signals <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(θ → 0)</tex> as do the corresponding Bayes systems. The LOBD, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x̂</tex> , in the binary (i.e., two-alternative cases: <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H₁</tex> , signal and noise, vs. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H₀</tex> , noise alone) is found to be the expansion of the logarithm of the optimum or likelihood ratio (functional) form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Λ</tex> , including a suitable bias term, e.g., <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x̂ = log μ + B̂ₙ(θ)+θ ( {δ log Λₙ}{δ θ} )θ = 0 (1)</tex> Suitable choices of bias <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</tex> can frequently be made so that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x̂</tex> is asymptotically as efficient as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x^{}</tex> , the corresponding Bayes detector. The principal advantages of the LOBD are that 1) its comparative simplicity vis-à-vis the Bayes system, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x^{}; 2</tex> ) its structure can always be found, even when the structure of the Bayes system cannot be obtained explicitly; and 3) expected performance (average risk and error probabilities) can often be determined for the LOBD when such is not possible for the Bayes system. In addition, the LOBD is canonically derived, i.e., the form of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x̂</tex> , (1), is independent of the particular noise statistics and signal, structure. New results include 1) the fact that the logarithm of the likelihood ratio is the function of the received data to be approximated, 2) correlated samples (including continuous sampling on an interval), 3) structures analogous to (1) which hold for the LOBD in binary sequential detection, and 4) similar expansions of the logarithm of the likelihood ratios, etc., which are required for the LOBD in multiple alternative cases. Moreover, in the binary situation of testing for signals <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(S₁)</tex> of one class against signals <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(S₂)</tex> of another in noise, the LOBD is seen to be a linearly weighted sum of characteristics like (1). To account for the effects of correlated samples and limiting distributions that do not obey the Central Limit Theorem, i.e., are not asymptotically normal (as illustrated in several of the examples), a criterion of asymptotic relative efficiency (ARE) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">_{θ ≥ 0</tex> is developed. Sufficient, and necessary and sufficient conditions on the bias and on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\θ</tex> itself are established, which when satisfied insure that the LOBD is at least asymptotically as efficient, or equivalent to, the corresponding Bayes detector in the limit.