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

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 $\theta \le 1$.