Let \({\hat \omega}\) be an estimate of an unknown parameter \(\omega\) in \(R^ p\) with distribution determined by \(\omega\). Suppose that for \(r\geq 1\) the r-th order cross-cumulants of \({\hat \omega}\) have magnitude \(n^{1-r}\) and can be expanded in powers of \(n^{-1}\), where n is known. Let t(\(\omega)\) be a real function on \(R^ p\) with finite derivatives. We show how to obtain confidence intervals for t(\(\omega)\) of level \(1-\alpha +O(n^{-j/2})\) for any given j. We call this a j-th order C.I.