Signed Poisson Approximation: A Possible Alternative to Normal and Poisson Laws

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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“…For other results in connection with Wasserstein metrics, see Deheuvels et al [27], Hwang [43], Cekanavičius and Kruopis [20].…”

Section: The Wasserstein Distancementioning

confidence: 98%

A Charlier–Parseval approach to poisson approximation and its applications

Zacharovas

Hwang

2010

Lith Math J

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“…The formulation of the full asymptotic expansions is cumbersome. Considerable attention has been given to first-order asymptotic expansions (see, e.g., Kerstan [27], Kruopis [31],Čekanavičius & Kruopis [16]), Barbour &Čekanavičius [6], Barbour et al [4]).…”

Section: Note That Iexmentioning

confidence: 99%

“…A number of authors approximated L(S n ) by unit measures (signed measures) in order to achieve a higher rate of the accuracy of approximation (cf. [14,11,16,6]). We shall understand by unit measures only those unit measures that are not probability distributions.…”

Section: Introductionmentioning

confidence: 99%

On the accuracy of Poisson approximation

Novak

2019

Extremes

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“…Alternative proofs for this problem and its variants are given in Kolchin et al (1978, Chap. VII), Holst (1980), Mitwalli (2002, Harris (1989), and Cekanavicius et al (2000). See Smythe (2011) for an extension to the case in which a 1 , .…”

Section: Collector 2 Collected Not Collectedmentioning

confidence: 99%

Asymptotics of multivariate contingency tables with fixed marginals

Zhou

2019

Journal of Statistical Planning and Inference

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We consider the asymptotic distribution of a cell in a 2 × · · · × 2 contingency table as the fixed marginal totals tend to infinity. The asymptotic order of the cell variance is derived and a useful diagnostic is given for determining whether the cell has a Poisson limit or a Gaussian limit. There are three forms of Poisson convergence. The exact form is shown to be determined by the growth rates of the two smallest marginal totals. The results are generalized to contingency tables with arbitrary sizes and are further complemented with concrete examples.

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Signed Poisson Approximation: A Possible Alternative to Normal and Poisson Laws

Cited by 17 publications

References 31 publications

A Charlier–Parseval approach to poisson approximation and its applications

A Charlier–Parseval approach to poisson approximation and its applications

On the accuracy of Poisson approximation

Asymptotics of multivariate contingency tables with fixed marginals

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