An Asymptotically Pointwise Optimal Procedure For Sequential Joint Detection And Estimation

Abstract

We investigate the problem of jointly testing two hypotheses and estimating a random parameter based on sequentially observed data whose distribution belongs to the exponential family. The aim is to design a scheme which minimizes the expected number of used samples while limiting the detection and estimation errors to pre-set levels. This constrained problem is first converted to an unconstrained problem which is then reduced to an optimal stopping problem. To solve the optimal stopping problem, we propose an asymptotically pointwise optimal (APO) stopping rule, i.e., a stopping rule that is optimal when the tolerated detection and estimation errors tend to zero. The policy parameterizing coefficients are then chosen such that the constraints on the detection and estimation errors are fulfilled. The proposed theory is illustrated with a numerical example.