UMP invariance in adaptive detection: kernels that preserve monotone likelihood ratio

We consider the question of optimality for the adaptive coherence estimator (ACE), which is an adaptive detection statistic for the problem in which noise in the training data is not constrained to have same power level as noise in the test data. Having previously shown that ACE is a maximal invariant statistic, we complete a proof that a threshold test on ACE is uniformly-most-powerful (UMP) invariant. This requires a second result, that the statistic possesses a monotone likelihood ratio (MLR). We establish the MLR property by relating it to the property of the density being a "totally positive" kernel. By repeatedly applying a basic composition formula for such kernels, we show that the density for ACE is totally positive. Thus the density has MLR, and a simple threshold test on ACE has the strict optimality property of being UMP-invariant.