New Bounds and Truncation Boundaries for Importance Sampling

TL;DR

This paper introduces new theoretical bounds for importance sampling estimators, demonstrating their tightness and proposing truncation strategies that improve convergence rates in practical applications like finance and machine learning.

Contribution

The work establishes tight polynomial concentration bounds for classical importance sampling likelihood ratio estimators and proposes new truncation boundaries with exponential convergence guarantees.

Findings

Tightness of polynomial concentration bounds for IS likelihood ratio estimators.

New truncation boundaries improve estimator convergence rates.

Simulation results confirm effectiveness in finance and machine learning examples.

Abstract

Importance sampling (IS) is a technique that enables statistical estimation of output performance at multiple input distributions from a single nominal input distribution. IS is commonly used in Monte Carlo simulation for variance reduction and in machine learning applications for reusing historical data, but its effectiveness can be challenging to quantify. In this work, we establish a new result showing the tightness of polynomial concentration bounds for classical IS likelihood ratio (LR) estimators in certain settings. Then, to address a practical statistical challenge that IS faces regarding potentially high variance, we propose new truncation boundaries when using a truncated LR estimator, for which we establish upper concentration bounds that imply an exponential convergence rate. Simulation experiments illustrate the contrasting convergence rates of the various LR estimators and the effectiveness of the newly proposed truncation-boundary LR estimators for examples from finance and machine learning.