Bootstrap Confidence Intervals for a Class of Parametric Problems

We consider the following class of problems: having observed a multivariate normal data vector y with unknown mean vector η, covariance matrix the identity, find an approximate confidence interval for ø = t(η), a real-valued function of η. A simple geometric construction is given which leads to highly accurate solutions. This construction shows that the standard approximation based on maximum likelihood theory, ø7plusmn;σz(α), can be quite misleading when ø is nonlinear in η. We discuss bootstrap-based confidence intervals which remove most of the error in the standard approximation, at the expense of considerably more calculation. The bootstrap intervals are invariant under transformation of both y and η, and so they automatically produce accurate solutions in problems which can be transformed to multivariate normality, without requiring knowledge of the normalizing transformation.