# Concentration-based confidence intervals for U-statistics

## TL;DR

This paper develops new one-sided empirical confidence intervals for U-statistics and their variances, enabling tighter two-sided CIs and improved bounds for means without requiring population variance knowledge.

## Contribution

It introduces novel one-sided empirical CIs for U-statistics and variances, enhancing the construction of tighter two-sided CIs and mean bounds.

## Key findings

- New one-sided empirical CIs for U-statistics and variances.
- Tighter two-sided CIs for U-statistics than existing methods.
- More accurate empirical CIs for the mean with fewer observations.

## Abstract

Concentration inequalities have become increasingly popular in machine learning, probability, and statistical research. Using concentration inequalities, one can construct confidence intervals (CIs) for many quantities of interest. Unfortunately, many of these CIs require the knowledge of population variances, which are generally unknown, making these CIs impractical for numerical application. However, recent results regarding the simultaneous bounding of the probabilities of quantities of interest and their variances have permitted the construction of empirical CIs, where variances are replaced by their sample estimators. Among these new results are two-sided empirical CIs for U-statistics, which are useful for the construction of CIs for a rich class of parameters. In this article, we derive a number of new one-sided empirical CIs for U-statistics and their variances. We show that our one-sided CIs can be used to construct tighter two-sided CIs for U-statistics, than those currently reported. We also demonstrate how our CIs can be used to construct new empirical CIs for the mean, which provide tighter bounds than currently known CIs for the same number of observations, under various settings.