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 >= t) <= 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 <= 1.