Sequential Joint Detection and Estimation: Optimal Procedures and Asymptotic Results

This thesis details the design and analysis of sequential procedures for the joint inference problem associated with hypothesis testing and parameter estimation in the context of sequentially observed data. The goal achieved is to minimize the average number of samples required to meet predefined detection and estimation error levels; thus fast inference with guaranteed performance. The first half of the thesis is devoted to the design of strictly optimal procedures, i.e., procedures that use, on average, as few samples as possible and fulfill constraints on the detection and estimation error levels. The design problem is formulated as a constrained optimization problem. The selected approach is to convert the problem to an unconstrained optimization problem and subsequently, to an optimal stopping problem. The solution of the optimal stopping problem is characterized by recursively defined non-linear integral equations that are parameterized by a set of cost coefficients. It is shown that the partial derivatives of the cost function, with respect to the cost coefficients, are equal to the detection/estimation errors up to a constant scaling factor. Based on this property, the choice of the coefficients that lead to an optimal solution is formulated as a convex optimization problem. Two numerical algorithms are provided to solve this optimization problem. The first one converts the optimization problem to a linear program. The second one solves it directly via a projected quasi-Newton method. Numerical examples are given that verify the proposed design procedure. In the second part of this dissertation, asymptotically optimal procedures for sequential joint detection and estimation are detailed. To avoid the computationally expensive solutions associated with optimality, an alternative procedure is proposed, which becomes optimal when the constraints on the detection and estimation errors approach zero. After a formal definition of asymptotic optimality, an asymptotically optimal stopping rule is detailed. This stopping rule is implemented by thresholding the instantaneous cost that is parameterized by a set of cost coefficients. It is shown, asymptotically, and similarly to the strictly optimal procedure, that the partial derivatives of the solution of the optimal stopping problem and the detection/estimation errors, differ only by a constant scaling factor. Using this result, it is shown how the projected quasi-Newton method, derived for the design of optimal procedures, can be adapted to choose the cost coefficients such that the constraints on the detection and estimation errors are fulfilled. The proposed asymptotically optimal procedures are applied to example problems that are motivated by real-world applications. By means of a numerical example, it is shown that the asymptotically optimal procedure uses, on average, only slightly more samples than the strictly optimal one while requiring significantly less computational resources to be implemented. This thesis provides a coherent framework for the design and analysis of strictly optimal, as well as asymptotically optimal procedures, for the problem of sequential joint detection and estimation