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 of
N
Bayesian single-threshold decision rules
\phi_{i}^{0}, i = 1,\cdots , N
having been obtained previously by the author. These decision rules are also all optimal against the maximin (least-favorable) nonrandomized noise probability density
f_{0}
, where
f_{0}
is a picket fence function with
N
pickets 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 all
f
and
\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 all
f
, with
4^{-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.