Bounds for the total variation distance between the binomial and the Poisson distribution in case of medium-sized success probabilities

Weba, Michael

doi:10.1017/s0021900200016879

Cited by 6 publications

(5 citation statements)

References 10 publications

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“…Now it is straightforward to show that the maximum of the function |Φ( √ 1 − θ x) − Φ(x)| is reached at x = ± −θ −1 ln(1 − θ) (see also [31]). This proves (61).…”

Section: 4mentioning

confidence: 95%

Uniform asymptotics of Poisson approximation to the Poisson-binomial distribution

Hwang¹,

Zacharovas²

2010

Teor. Veroyatnost. i Primenen.

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“…Now it is straightforward to show that the maximum of the function |Φ( √ 1 − θ x) − Φ(x)| is reached at x = ± −θ −1 ln(1 − θ) (see also [31]). This proves (61).…”

Section: 4mentioning

confidence: 95%

Uniform asymptotics of Poisson approximation to the Poisson-binomial distribution

Hwang¹,

Zacharovas²

2010

Teor. Veroyatnost. i Primenen.

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“…Now it is straightforward to show that the maximum of the function jˆ. p 1 Â x/ ˆ.x/j is reached at x D˙p Â 1 log.1 Â / (see also [31]). This proves (2.51).…”

Section: Kolmogorov Distance and The Point Metricmentioning

confidence: 96%

Uniform Asymptotics of Poisson Approximation to the Poisson-Binomial Distribution

Hwang¹,

Zacharovas²

2011

Theory Probab. Appl.

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“…where ℓ ± := ⌊λ + 1/2 ± λ + 1/4⌋. Witte [86] then derived the estimate for θ < e −p * ; see also Weba [85]. Roos [69,70] gives, among several other fine estimates,…”

Section: The Kolmogorov Distancementioning

confidence: 98%

“…for θ < 1 2 e −2p * , as well as other more complicated ones. Another very different form for d TV can be found in Weba [85], which results from combining several known estimates.…”

Section: The Total Variation Distancementioning

confidence: 99%

A Charlier–Parseval approach to poisson approximation and its applications

Zacharovas

Hwang

2010

Lith Math J

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A new approach to Poisson approximation is proposed. The basic idea is very simple and based on properties of the Charlier polynomials and the Parseval identity. Such an approach quickly leads to new effective bounds for several Poisson approximation problems. A selected survey on diverse Poisson approximation results is also given.MSC 2000 Subject Classifications: Primary 62E17; secondary 60C05 60F05.The early history of Poisson approximation. Poisson distribution appeared naturally as the limit of the sum of a large number of independent trials each with very small probability of success. Such a limit form, being the most primitive version of Poisson approximation, dates back to at least de Moivre's work [32] in the early eighteenth century and Poisson's book [61] in the nineteenth century. Haight [38] writes: ". . . although Poisson (or de Moivre) discovered the mathematical expression (1.1-1) [which is e −λ λ k /k!], Bortkiewicz discovered the probability distribution (1.1-1)." And according to Good [37], "perhaps the Poisson distribution should have been named after von Bortkiewicz (1898) because he was the first to write extensively about rare events whereas Poisson added little to what de Moivre had said on the matter and was probably aware of de Moivre's work;" see also Seneta's account in [74] on Abbe's work. In addition to Bortkiewicz's book [17], another important contribution to the early history of Poisson approximation was made by Charlier [21] for his type B expansion, which will play a crucial role in our development of arguments.The next half a century or so after Bortkiewicz and Charlier then witnessed an increase of interests in the properties and applications of the Poisson distribution and Charlier's expansion. In particular, Jordan [47] proved the orthogonality of the Charlier polynomials with respect to the Poisson measure, and considered a formal expansion pair, expressing the Taylor coefficients of a given function in terms of series of Charlier polynomials and vice versa. A sufficient condition justifying the validity of such an expansion pair was later on provided by Uspensky [83]; he also derived very precise estimates for the coefficients in the case of binomial distribution. His complex-analytic approach was later on extended by Shorgin [80] to the more general Poisson-binomial distribution (each trial with a different probability; see next paragraph). Schmidt [73] then gives a sufficient and necessary condition for justifying the Charlier-Jordan expansion; see also Boas [13] and the references therein. Prohorov [65] was the first to study, using elementary arguments, the total variation distance between binomial and Poisson distributions, thus upgrading the classical limit theorem to an approximation theorem.

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Bounds for the total variation distance between the binomial and the Poisson distribution in case of medium-sized success probabilities

Cited by 6 publications

References 10 publications

Uniform asymptotics of Poisson approximation to the Poisson-binomial distribution

Uniform asymptotics of Poisson approximation to the Poisson-binomial distribution

Uniform Asymptotics of Poisson Approximation to the Poisson-Binomial Distribution

A Charlier–Parseval approach to poisson approximation and its applications

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