Symmetric and centered binomial approximation of sums of locally dependent random variables

Roellin, Adrian

doi:10.1214/ejp.v13-503

Cited by 26 publications

(32 citation statements)

References 27 publications

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“…Define the following "smoothness" measure of an integer valued distribution, S l (L(W )) := sup h: h 1 |E∆ l h(W )|, l 1, (1.2) where ∆ denotes the first difference operator ∆g(k) := g(k + 1) − g(k). Variations of the smoothing terms (1.2) frequently appear in integer supported distributional approximation results; see, for example, Barbour (1999), Goldstein and Xia (2006), Röllin (2008) and Fang (2014). The next result shows the typical smoothness expected for approximately discretized normal distributions.…”

Section: Translated Poisson Distributionmentioning

confidence: 52%

Error bounds in local limit theorems using Stein’s method

Barbour¹,

Röllin²,

Ross³

2019

Bernoulli

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We provide a general result for bounding the difference between point probabilities of integer supported distributions and the translated Poisson distribution, a convenient alternative to the discretized normal. We illustrate our theorem in the context of the Hoeffding combinatorial central limit theorem with integer valued summands, of the number of isolated vertices in an Erdős-Rényi random graph, and of the Curie-Weiss model of magnetism, where we provide optimal or near optimal rates of convergence in the local limit metric. In the Hoeffding example, even the discrete normal approximation bounds seem to be new. The general result follows from Stein's method, and requires a new bound on the Stein solution for the Poisson distribution, which is of general interest.

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Section: Translated Poisson Distributionmentioning

confidence: 52%

Error bounds in local limit theorems using Stein’s method

Barbour¹,

Röllin²,

Ross³

2019

Bernoulli

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“…A difficulty in pushing the results of [18] through to the stronger total variation metric is that the support of the distribution to be approximated may not match the support of the geometric distribution well enough. This issue is typical in bounding the total variation distance between integer-valued random variables and can be handled by introducing a term into the bound that quantifies the "smoothness" of the distribution of interest; see, for example, [2,21,22]. To illustrate this point, we apply our abstract formulation to obtain total variation error bounds in two of the examples treated in [18].…”

Section: Introductionmentioning

confidence: 99%

Total variation error bounds for geometric approximation

2013

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We develop a new formulation of Stein's method to obtain computable upper bounds on the total variation distance between the geometric distribution and a distribution of interest. Our framework reduces the problem to the construction of a coupling between the original distribution and the "discrete equilibrium" distribution from renewal theory. We illustrate the approach in four non-trivial examples: the geometric sum of independent, non-negative, integer-valued random variables having common mean, the generation size of the critical Galton-Watson process conditioned on non-extinction, the in-degree of a randomly chosen node in the uniform attachment random graph model and the total degree of both a fixed and randomly chosen node in the preferential attachment random graph model.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ406 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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“…Thus, we obtain a significant improvement over O(n −1/2 ) which can be obtained by twoparametric binomial approximation (1.3) or by a shifted Bi(n, 1/2) distribution as in Röllin (2008).…”

Section: Corollary 22 Under the Conditions Of Theorem 21 We Havementioning

confidence: 78%

A Three-Parameter Binomial Approximation

Peköz

Röllin²,

Čekanavičius

et al. 2009

Journal of Applied Probability

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We approximate the distribution of the sum of independent but not necessarily identically distributed Bernoulli random variables using a shifted binomial distribution where the three parameters (the number of trials, the probability of success, and the shift amount) are chosen to match up the first three moments of the two distributions. We give a bound on the approximation error in terms of the total variation metric using Stein's method. A numerical study is discussed that shows shifted binomial approximations typically are more accurate than Poisson or standard binomial approximations. The application of the approximation to solving a problem arising in Bayesian hierarchical modeling is also discussed.

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Symmetric and centered binomial approximation of sums of locally dependent random variables

Cited by 26 publications

References 27 publications

Error bounds in local limit theorems using Stein’s method

Error bounds in local limit theorems using Stein’s method

Total variation error bounds for geometric approximation

A Three-Parameter Binomial Approximation

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