Efficient Estimation in the Tails of Gaussian Copulas

TL;DR

This paper develops and analyzes importance sampling estimators for efficiently estimating rare event probabilities in the tails of Gaussian copulas, focusing on the local structure around dominating points.

Contribution

It introduces three novel estimators exploiting local structure for bounded or polynomial efficiency, and a fourth estimator for the NORTA setting with no prior information.

Findings

The full-information estimator achieves bounded relative error in Gaussian settings.

Partial-information estimators attain polynomial efficiency without full set knowledge.

Numerical experiments confirm theoretical efficiency and effectiveness.

Abstract

We consider the question of efficient estimation in the tails of Gaussian copulas. Our special focus is estimating expectations over multi-dimensional constrained sets that have a small implied measure under the Gaussian copula. We propose three estimators, all of which rely on a simple idea: identify certain \emph{dominating} point(s) of the feasible set, and appropriately shift and scale an exponential distribution for subsequent use within an importance sampling measure. As we show, the efficiency of such estimators depends crucially on the local structure of the feasible set around the dominating points. The first of our proposed estimators $\estOpt$ is the "full-information" estimator that actively exploits such local structure to achieve bounded relative error in Gaussian settings. The second and third estimators $\estExp$, $\estLap$ are "partial-information" estimators, for use when complete information about the constraint set is not available, they do not exhibit bounded relative error but are shown to achieve polynomial efficiency. We provide sharp asymptotics for all three estimators. For the NORTA setting where no ready information about the dominating points or the feasible set structure is assumed, we construct a multinomial mixture of the partial-information estimator $\estLap$ resulting in a fourth estimator $\estNt$ with polynomial efficiency, and implementable through the ecoNORTA algorithm. Numerical results on various example problems are remarkable, and consistent with theory.