Moderate deviations for importance sampling estimators of risk measures

TL;DR

This paper establishes moderate deviation principles for importance sampling estimators of risk measures like quantiles and Expected Shortfall, enhancing understanding of their tail behavior and efficiency.

Contribution

It extends existing moderate deviation results to importance sampling estimators of risk measures using advanced large deviation techniques.

Findings

Moderate deviation principles are proven for importance sampling estimators of quantiles.

Moderate deviation principles are proven for importance sampling estimators of Expected Shortfall.

The results improve understanding of the tail behavior and efficiency of importance sampling in risk measurement.

Abstract

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.