  Corrected confidence intervals are developed for the mean of the second
component of a bivariate normal process when the first component is being
monitored sequentially. This is accomplished by constructing a first
approximation to a pivotal quantity, and then using very weak expansions to
determine the correction terms. The asymptotic sampling distribution of the
renormalised pivotal quantity is established in both the case where the
covariance matrix is known and when it is unknown. The resulting approximations
have a simple form and the results of a simulation study of two well-known
sequential tests show that they are very accurate. The practical usefulness of
the approach is illustrated by a real example of bivariate data. Detailed
proofs of the main results are provided.
