  Importance sampling has become an important tool for the computation of
tail-based risk measures. Since such quantities are often determined mainly by
rare events standard Monte Carlo can be inefficient and importance sampling
provides a way to speed up computations. This paper considers moderate
deviations for the weighted empirical process, the process analogue of the
weighted empirical measure, arising in importance sampling. The moderate
deviation principle is established as an extension of existing results. Using a
delta method for large deviations established by Gao and Zhao (Ann. Statist.,
2011) together with classical large deviation techniques, the moderate
deviation principle for the weighted empirical process is extended to
functionals of the weighted empirical process which correspond to risk
measures. The main results are moderate deviation principles for importance
sampling estimators of the quantile function of a distribution and Expected
Shortfall.
