The goal of this paper is to develop a general framework for constructing sequential fixed size confidence regions based on maximum likelihood estimates. Asymptotic properties of the sequential procedure for setting up the confidence regions are analyzed under very broad assumptions on the underlying parametric model. It is shown that the proposed sequential procedure is asymptotically optimal in the sense that it approximates the optimal fixed-sample size procedure. It is further shown that the “cost of ignorance” associated with the sequential procedure is bounded. Applications are made to estimation problems arising in prospective and retrospective studies.
"Sequential confidence regions for maximum likelihood estimates." Ann. Statist. 28 (5) 1472 - 1501, October2000. https://doi.org/10.1214/aos/1015957403