On single-sample robust detection of known signals with additive unknown-mean amplitude-bounded random interference--II: The randomized decision rule solution (Corresp.)

A randomized decision rule is derived and proved to be the saddlepoint solution of the robust detection problem for known signals in independent unknown-mean amplitude-bounded noise. The saddlepoint solution\phi^{0}uses an equaUy likely mixed strategy to chose one ofNBayesian single-threshold decision rules\phi_{i}^{0}, i = 1,\cdots , Nhaving been obtained previously by the author. These decision rules are also all optimal against the maximin (least-favorable) nonrandomized noise probability densityf_{0}, wheref_{0}is a picket fence function withNpickets on its domain. Thee pair(\phi^{0}, f_{0})is shown to satisfy the saddlepoint condition for probability of error, i.e.,P_{e}(\phi^{0} , f) \leq P_{e}(\phi^{0} , f_{0}) \leq P_{e}(\phi, f_{0})holds for allfand\phi. The decision rule\phi^{0}is also shown to be an eqoaliir rule, i.e.,P_{e}(\phi^{0}, f ) = P_{e}(\phi^{0},f_{0}), for allf, with4^{-1} \leq P_{e}(\phi^{0},f_{0})=2^{-1}(1-N^{-1})\leq2^{-1} , N \geq 2. Thus nature can force the communicator to use an {\em optimal} randomized decision rule that generates a large probability of error and does not improve when less pernicious conditions prevail.