Detection and Extraction of Signals in Noise from the Point of View of Statistical Decision Theory. I

Abstract

As was pointed out in Section 2-2, extraction in the theory of signal reception is the counterpart of parameter estimation in statistics. Here the signal S is a parameter of the distribution density F(V I S) governing the occurrence of the data V; point estimation, or the direct estimation of (each component of) S is treated here, rather than estimation by confidence intervals. As before, cf. Section 2-4, we let y represent the decision to be made about S, and observe that when y is to be an estimate of S, the spaces Q and A of Fig. (2.2) have the same structure. We assume that each space contains a continuum of points and is a finite, closed region, which may, however, be taken large enough to be essentially infinite for practical purposes. The cost function 0j = C(S, y), Eq. (2.4), to be used in the risk analysis, is, as before, preassigned in accordance with the external constraints of the problem and is critical in determining the specific structure of the resulting system. A theorem of Hodges and Lehmann [4.1] states: If A is the real line and C(S, y) is a convex functiont of y for every S, then for any decision rule a there exists a nonrandomized decision rule whose risk is not greater than that of a for all S in Q.