The binary detection problem of Chapter 4 was concerned with distinguishing between two possible hypotheses, H0 and H1. When these were both simple hypotheses, we were concerned with determining whether Θ = Θ0 or Θ = Θ1 was true. In this case, the optimal detector essentially determined our best guess of true θ corresponding to a given observation y. On the other hand, in the general case of composite hypothesis, testing the optimal detection does not bother about what the exact value of θ could be for a given observation y. Instead, the optimal detectors only attempt to come up with a best guess as to which set Θj the θ corresponding to the observation may belong to. Hence, detection problem is concerned only about distinguishing among a set of finite number of choices.

However, in some situations, we may not be satisfied by knowing whether a parameter θ associated with an observation belongs to a certain set of values. We may want to know the exact value itself of the parameter θ corresponding to the given observation. For example, recall the case of θ being the fixed signal amplitude in which hypothesis H0 corresponds to θ =0 whereas hypothesis H1 corresponds to θ > 0. An optimal detector for H0 versus H1 only determines whether θ = 0 or θ0. When θ0, it does not tell us what the exact value of θ is for a given observation. Indeed, for a signal in Gaussian noise, the UMP Neyman–Pearson test does not require the knowledge ...

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