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

@article{Wiroonsri2022ConcentrationIU,
  title={Concentration inequalities using approximate zero bias couplings with applications to Hoeffding’s statistic under the Ewens distribution},
  author={Nathakhun Wiroonsri},
  journal={Communications in Statistics - Theory and Methods},
  year={2022}
}
  • Nathakhun Wiroonsri
  • Published 2 February 2022
  • Mathematics
  • Communications in Statistics - Theory and Methods
We prove concentration inequalities of the form P (Y ≥ t) ≤ exp(−B(t)) for a random variable Y with mean zero and variance σ 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 θ > 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… 

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References

SHOWING 1-10 OF 30 REFERENCES

Stein's method using approximate zero bias couplings with applications to combinatorial central limit theorems under the Ewens distribution

We generalize the well-known zero bias distribution and the $\lambda$-Stein pair to an approximate zero bias distribution and an approximate $\lambda,R$-Stein pair, respectively. Berry Esseen type

Stein’s method for the Poisson–Dirichlet distribution and the Ewens sampling formula, with applications to Wright–Fisher models

We provide a general theorem bounding the error in the approximation of a random measure of interest--for example, the empirical population measure of types in a Wright-Fisher model--and a Dirichlet

Stein's method and the zero bias transformation with application to simple random sampling

Let W be a random variable with mean zero and variance 2 . The distribution of a variate W , satisfying EWf(W) = 2 Ef 0 (W ) for smooth functions f, exists uniquely and defines the zero bias

Bounded size bias coupling: a Gamma function bound, and universal Dickman-function behavior

Under the assumption that the distribution of a nonnegative random variable $$X$$X admits a bounded coupling with its size biased version, we prove simple and strong concentration bounds. In

Random derangements and the Ewens Sampling Formula

We study derangements of $\{1,2,\ldots,n\}$ under the Ewens distribution with parameter $\theta$. We give the moments and marginal distributions of the cycle counts, the number of cycles, and

A Non-Uniform Concentration Inequality for a Random Permutation Sum

The purpose of this article is to give a non-uniform concentration inequality of a random permutation sum, , where π = (π(1), π(2),…, π(n)) is a uniformly distributed random permutation of 1, 2,…, n

A bound for the error in the normal approximation to the distribution of a sum of dependent random variables

This paper has two aims, one fairly concrete and the other more abstract. In Section 3, bounds are obtained under certain conditions for the departure of the distribution of the sum of n terms of a

Normal Approximation by Stein's Method

Preface.- 1.Introduction.- 2.Fundamentals of Stein's Method.- 3.Berry-Esseen Bounds for Independent Random Variables.- 4.L^1 Bounds.- 5.L^1 by Bounded Couplings.- 6 L^1: Applications.- 7.Non-uniform

A Non-uniform Concentration Inequality for Randomized Orthogonal Array Sampling Designs

Let $ f : [0 ; 1]^3 \rightarrow R$ be a measurable function. In many computer experiments, we estimate the value of $\int _{[0,1]^3} f (x) dx$ , which is the mean $ \mu = E ( f \circ X ), where X is

Concentration of measures via size-biased couplings

AbstractLet Y be a nonnegative random variable with mean μ and finite positive variance σ2, and let Ys, defined on the same space as Y, have the Y size-biased distribution, characterized by $$