Concentration inequalities using approximate zero bias couplings with applications to Hoeffding's statistic under the Ewens distribution

TL;DR

This paper develops concentration inequalities for random variables using approximate zero bias couplings, with applications to Hoeffding's statistic under the Ewens distribution, supported by simulations.

Contribution

It introduces a novel coupling technique for deriving concentration bounds and applies it to Hoeffding's statistic with Ewens distribution, including empirical validation.

Findings

Bounds are effective especially for  

Simulation results confirm the bounds' accuracy

Method extends concentration inequalities to Ewens distribution

Abstract

We prove concentration inequalities of the form $P(Y \ge t) \le \exp(-B(t))$ for a random variable $Y$ with mean zero and variance $\sigma^2$ using a coupling technique from Stein's method that is so-called approximate zero bias couplings. Applications to the Hoeffding's statistic where the random permutation has the Ewens distribution with parameter $\theta>0$ are also presented. A few simulation experiments are then provided to visualize the tail probability of the Hoeffding's statistic and our bounds. Based on the simulation results, our bounds work well especially when $\theta \le 1$.