# On perturbations of Stein operator

@article{Kumar2016OnPO,
title={On perturbations of Stein operator},
author={A. N. Kumar and N. S. Upadhye},
journal={Communications in Statistics - Theory and Methods},
year={2016},
volume={46},
pages={9284 - 9302}
}
• Published 2016
• Mathematics
• Communications in Statistics - Theory and Methods
ABSTRACT In this article, we obtain a Stein operator for the sum of n independent random variables (rvs) which is shown as the perturbation of the negative binomial (NB) operator. Comparing the operator with NB operator, we derive the error bounds for total variation distance by matching parameters. Also, three-parameter approximation for such a sum is considered and is shown to improve the existing bounds in the literature. Finally, an application of our results to a function of waiting time… Expand
Approximation Results for Sums of Independent Random Variables.
In this article, we consider Poisson and Poisson convoluted geometric approximation to the sums of $n$ independent random variables under moment conditions. We use Stein's method to derive theExpand
Stein operators for product distributions, with applications
• Mathematics
• 2016
We build upon recent advances on the distributional aspect of Stein's method to propose a novel and flexible technique for computing Stein operators for random variables that can be written asExpand
Bounds on Negative Binomial Approximation to Call Function
In this paper, we develop Stein’s method for negative binomial distribution using call function defined by fz(k) = (k − z) + = max{k − z, 0}, for k ≥ 0 and z ≥ 0. We obtain error bounds betweenExpand
On discrete Gibbs measure approximation to runs
• Mathematics
• 2017
In this paper, some erroneous results for a dependent setup arising from independent sequence of Bernoulli trials are corrected. Next, a Stein operator for discrete Gibbs measure is derived using PGFExpand
PR ] 2 6 Se p 20 16 Pseudo-binomial Approximation to ( k 1 , k 2 )-runs
• A. N. Kumar
• 2016
A k-dependent setup arising from independent sequence of Bernoulli trials is considered and a Stein operator for this setup is obtained which is shown as a perturbation of pseudo-binomial operator.Expand
A Non-uniform Bound on Negative Binomial Approximation via Stein’s Method and z-functions
• Mathematics
• 2020
In this article, Stein’s method and z-functions are used to determine a non-uniform bound for approximating the cumulative distribution function of a nonnegative integer-valued random variable X byExpand
PR ] 2 A ug 2 01 9 On Discrete Gibbs Measure Approximation to Runs
• 2019
Runs and patterns is an important topic in the areas related to probability and statistics, such as reliability theory, meteorology and agriculture, statistical testing and quality control among manyExpand
An algebra of Stein operators
• Mathematics
• 2016
Abstract We build upon recent advances on the distributional aspect of Stein's method to propose a novel and flexible technique for computing Stein operators for random variables that can be writtenExpand
Pseudo-binomial approximation to (k1,k2)-runs
• Mathematics
• 2016
Abstract The distribution of ( k 1 , k 2 ) -runs is well-known (Dafnis et al., 2010), under independent and identically distributed (i.i.d.) setup of Bernoulli trials but is intractable under nonExpand

#### References

SHOWING 1-10 OF 51 REFERENCES
On Stein operators for discrete approximations
• Mathematics
• 2014
In this paper, a new method based on probability generating functions is used to obtain multiple Stein operators for various random variables closely related to Poisson, binomial and negativeExpand
Compound Poisson Approximation for Nonnegative Random Variables Via Stein's Method
• Mathematics
• 1992
The aim of this paper is to extend Stein's method to a compound Poisson distribution setting. The compound Poisson distributions of concern here are those of the form POIS$(\nu)$, where $\nu$ is aExpand
Normal Approximation by Stein ’ s Method
The aim of this paper is to give an overview of Stein’s method, which has turned out to be a powerful tool for estimating the error in normal, Poisson and other approximations, especially for sums ofExpand
Approximate Computation of Expectations: the Canonical Stein Operator
• Mathematics
• 2014
We propose a canonical definition of the Stein operator and Stein class of a distribution. The resulting Stein identity highlights the unifying theme behind the literature on Stein's method. ViewingExpand
Total variation asymptotics for sums of independent integer random variables
• Mathematics
• 2002
Let W n := Σ n j=1 Zj be a sum of independent integer-valued random variables. In this paper, we derive an asymptotic expansion for the probability P[W n ∈ A] of an arbitrary subset A E Z. OurExpand
Negative Binomial Approximation with Stein's Method
• Mathematics
• 1999
Bounds on the rate of convergence to the negative binomial distribution are found, where this rate is measured by the total variation distance between probability laws. For an arbitrary discreteExpand
Stein’s density approach and information inequalities
• Mathematics, Computer Science
• 2012
A new perspective on Stein's so-called density approach is provided by introducing a new operator and characterizing class which are valid for a much wider family of probability distributions on the real line and proposing a new Stein identity which is used to derive information inequalities in terms of the "generalized Fisher information distance". Expand
Stein's method for comparison of univariate distributions
• Mathematics
• 2014
We propose a new general version of Stein's method for univariate distributions. In particular we propose a canonical definition of the Stein operator of a probability distribution {which is based onExpand
On Stein's method and perturbations
• Mathematics
• 2007
Stein's (1972) method is a very general tool for assessing the quality of approximation of the distribution of a random element by another, often sim- pler, distribution. In applications of Stein'sExpand
Local Pinsker Inequalities via Stein's Discrete Density Approach
• Computer Science, Mathematics
• IEEE Transactions on Information Theory
• 2013
This paper introduces generalized Fisher information distances and proves that these also dominate the square of the total variation distance and introduces a general discrete Stein operator for which a useful covariance identity is proved. Expand