Abstract :
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 ??0 uses an equally-likely, mixed strategy to choose one of N Bayesian, single-threshold decision rules ??i 0, i = 1,...., N obtained previously by Morris [4]. These decision rules are also all optimal against the maximin (least favorable), nonrandomized, noise probability density f0, where f0 is a picket-fence function with N pickets on its domain. The pair (??0, f0) are shown to satisfy the saddlepoint condition for probability of error, i.e., Pe(??0, f) ?? Pe(??0, f0) ?? Pe(??, f0), Vf, V?? for this problem. The decision rule ??0 is shown also to be an equalizer rule, i.e., Pe(??0, f)= Pe(??0, f0), Vf, with 4-1 ?? Pe(??0, f0) = 2-1 (1-N-1) ?? 2-1, N ?? 2. We conclude that nature can force the communicator to use an optimal randomized decision rule that generates large probability of error and does not improve when less pernicious conditions prevail.
