Inference about a secondary process following a sequential trial

We consider the following sequential testing problem.A group-sequential or fully-sequential test is carried out for a primary parameter, using a score process or an effective score process to eliminate nuisance parameters. After stopping, the possibility of additional parameters is considered, and appropriate tests and estimators are desired that recognise the sequential stopping rule. We formulate an asymptotic multi-dimensional Gaussian process form of such problems, and then construct tests and confidence procedures. Optimality conditions are given, and an example is summarised.