Stein's method and point process approximation

Barbour, A. D.; Brown, Timothy C.

doi:10.1016/0304-4149(92)90073-y

Cited by 107 publications

(161 citation statements)

References 8 publications

Supporting

Mentioning

159

Contrasting

Order By: Relevance

“…The method is particularly interesting since results in the complex setting of dependent random variables are often not much more difficult to obtain than results for independent variables. Barbour et al (1992b) details how the method can be applied to Poisson approximations, Ehm (1991) and Loh (1992), respectively, apply the method to binomial and multinomial approximations, Barbour, Chen, and Loh (1992a) and Barbour and Chryssaphinou (2001) to compound Poisson approximation, Barbour and Brown (1992) to Poisson process approximation, Peköz (1996) to geometric approximation, and discussion of the many other distributions and settings the technique can be applied can be found in, for example, Barbour and Chen (2005) and Reinert (2005). An elementary introduction to Stein's method can be found in Chapter 2 of Ross and Peköz (2007).…”

Section: Introductionmentioning

confidence: 99%

A Three-Parameter Binomial Approximation

Peköz

Röllin²,

Čekanavičius

et al. 2009

Journal of Applied Probability

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

A Three-Parameter Binomial Approximation

Peköz

Röllin²,

Čekanavičius

et al. 2009

Journal of Applied Probability

View full text Add to dashboard Cite

show abstract

“…For examples of coupling arguments, see [3,4,5,11,23]. A related paper to these types of gradient bounds is [56], where the author used a variant of (3.20) for the fluid model of a flexible-server queueing system as a Lyapunov function.…”

Section: Moment Boundsmentioning

confidence: 99%

Stein’s Method for Steady-state Diffusion Approximations: An Introduction through the Erlang-A and Erlang-C Models

2016

View full text Add to dashboard Cite

This paper provides an introduction to the Stein method framework in the context of steady-state diffusion approximations. The framework consists of three components: the Poisson equation and gradient bounds, generator coupling, and moment bounds. Working in the setting of the Erlang-A and Erlang-C models, we prove that both Wasserstein and Kolmogorov distances between the stationary distribution of a normalized customer count process, and that of an appropriately defined diffusion process decrease at a rate of 1/ √ R, where R is the offered load. Futhermore, these error bounds are universal, valid in any load condition from lightly loaded to heavily loaded.

show abstract

“…We will exploit a probabilistic interpretation of g f 1 by constructing a coupling of {Z i (t) : t ≥ 0} for all i ∈ Z + as follows [cf Barbour (1988) and Barbour & Brown (1992)]. Consider a particle system on the site space Z, the set of all integers.…”

Section: The Proof Of (15)mentioning

confidence: 99%

“…Using Lemma 2.2, we have 0 ≤ g f 1 (3) ≤ g f 1 (2), which ensures 2α(α + β + 1) 2(β + 1)(α + β) + α 2 ≤ IEπ ≤ α(α + 2β + 2) β(α + 2β + 2) + 2α . (2.11) Therefore,…”

mentioning

confidence: 99%

Polynomial Birth–Death Distribution Approximation in the Wasserstein Distance

Xia

Zhang

2009

J Theor Probab

View full text Add to dashboard Cite

The polynomial birth-death distribution (abbr. as PBD) on I = {0, 1, 2, ...} or I = {0, 1, 2, ..., m} for some finite m introduced in Brown & Xia (2001) is the equilibrium distribution of the birth-death process with birth rates {α i } and death rates {β i }, where α i ≥ 0 and β i ≥ 0 are polynomial functions of i ∈ I. The family includes Poisson, negative binomial, binomial and hypergeometric distributions. In this paper, we give probabilistic proofs of various Stein's factors for the PBD approximation with α i = a and β i = i + bi(i − 1) in terms of the Wasserstein distance. The paper complements the work of Brown & Xia (2001) and generalizes the work of Barbour & Xia (2006) where Poisson approximation (b = 0) in the Wasserstein distance is investigated. As an application, we establish an upper bound for the Wasserstein distance between the PBD and Poisson binomial distribution and show that the PBD approximation to the Poisson binomial distribution is much more precise than the approximation by the Poisson or shifted Poisson distributions.

show abstract

Stein's method and point process approximation

Cited by 107 publications

References 8 publications

A Three-Parameter Binomial Approximation

A Three-Parameter Binomial Approximation

Stein’s Method for Steady-state Diffusion Approximations: An Introduction through the Erlang-A and Erlang-C Models

Polynomial Birth–Death Distribution Approximation in the Wasserstein Distance

Contact Info

Product

Resources

About