On the deflection of a quadratic-linear test statistic

The deflection of a bounded quadratic-linear test statistic is considered for the following binary detection problem. Hypothesis
H_{1}
--received waveform is a sample function from a random process with known covariance and mean functions, but unknown probability distributions, versus
H_{2}
--received waveform is a sample function from a Gaussian process (noise) having known covariance and mean functions. Sample functions are assumed to belong to a real and separable Hilbert space. The test statistic is assumed to be the sum of a bounded quadratic operation and a bounded linear operation on the data. Necessary and sufficient conditions for the deflection to be bounded over all non-null bounded quadratic-linear operations are given, and additional results are obtained under the assumption that the deflection is bounded. Several relations are shown to exist between the deflection problem and the optimum discrimination problem when both processes are Gaussian. In particular, it is shown that nonsingular discrimination occurs if and only if a generalized deflection is bounded, and that in some cases the problem of realizing the log-likelihood ratio is equivalent to the problem of attaining the least upper bound for the deflection.