Optimal combined detection and estimation

We offer a novel framework for composite hypothesis testing problems for deciding among multiple hypotheses, where each hypothesis involves some unknown parameters that obtaining their estimates is also of interest. This frameworks, unlike the precedent literature, jointly solves the detection and estimation tasks in composite hypothesis testing. Moreover, it has the flexibility that one can strike any desired balance between the detection and estimation qualities. By exploiting this flexibility, depending on the application in hand this framework offers the freedom to put the appropriate emphasis on each of the detection and estimation performances. In this framework we define an estimation cost, which is a function of a parameter and its estimate, and minimize this cost subject to assuring a tolerable level of type-I and type-II detection errors. The results are provided for a binary hypothesis setup and can be easily extended to the general case of any arbitrary number of hypotheses.