Corrected confidence sets for sequentially designed experiments

Consider a linear model, yk=x'kθ+ek, k=1,2,..., in which the current design variable xk may be a function of the previous responses y1,..., yk-1 and auxiliary randomization. Here the x's and θ are p-dimensional, denotes transpose, and the errors ek are taken to be i.i.d standard normal variables. The goal is to construct confidence sets for θ which are asymptotically valid to a high order. This is accomplished by obtaining very weak asymptotic expansions for the distributions of an appropriate pivotal quantity. The accuracy of the approximation is assessed by simulation experiments for two sequential tests proposed by Siegmund (1980, 1993).