Expansions for the Conditional Density and Distribution of a Standard Estimate

Conditioning is a very useful way of using correlated information to reduce the variability of an estimate. Conditioning an estimate on a correlated estimate, reduces its covariance, and so provides more precise inference than using an unconditioned estimate. Here we give expansions in powers of $n^{-1/2}$ for the conditional density and distribution of any multivariate standard estimate based on a sample of size n. Standard estimates include most estimates of interest, including smooth functions of sample means and other empirical estimates. We also show that a conditional estimate is not a standard estimate, so that Edgeworth-Cornish-Fisher expansions cannot be applied directly.