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This paper considers the use of an auxiliary variable X* to estimate quantiles of a test statistic X; X* may be an asymptotic expansion of X, or a simplified version which ignores some of the covariance structure. The proposed estimator involves three stages. First a large sample is drawn, and X* is evaluated. Then a first subsample is drawn, X and X* are evaluated and the conditional distribution of X given X* is estimated. Then a second subsample is drawn with weighting which is optimised for this conditional distribution, and for a particular quantile. From this subsample, an importance sampling estimator of the quantile is obtained. The resulting estimator is shown to have substantially lower mean squared error than the conventional estimator, and to be reasonably robust both to errors in the model for the conditional distribution and to the quantile assumed for the optimisation. An example in genetics is given.