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. Inference based on a conditioned estimate, can be much more precise than on an unconditioned estimate. Here we give expansions in powers of $n^{-1/2}$} for the conditional density and distribution of a 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. So they have potential application to a range of practical problems. We also show that a conditional estimate is not a standard estimate, so that Edgeworth-Cornish-Fisher expansions cannot be applied directly.