On variational bounds in the compound Poisson approximation of the individual risk model

Roos, Bero

doi:10.1016/j.insmatheco.2006.06.003

Cited by 10 publications

(10 citation statements)

References 28 publications

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“…Proof of Corollary 2.7: The proof follows arguments very similar to those used in the proofs of Theorems 1 and 2 in Roos (2007), where a comparable result was shown, generalizing (1.11) and (1.12). The idea here is a standard approximation procedure: In the first step, construct a new set of distributions Q 1 , .…”

Section: Proof Of Theorem 23mentioning

confidence: 72%

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Refined total variation bounds in the multivariate and compound Poisson approximation

Roos¹

2017

ALEA

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We consider the approximation of a convolution of possibly different probability measures by (compound) Poisson distributions and also by related signed measures of higher order. We present new total variation bounds having a better structure than those from the literature. A numerical example illustrates the usefulness of the bounds, and an application in the Poisson process approximation is given. The proofs use arguments from Kerstan (1964) and Roos (1999b) in combination with new smoothness inequalities, which could be of independent interest.2000 Mathematics Subject Classification. Primary 60F05; secondary 60G50, 62E17.

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Section: Proof Of Theorem 23mentioning

confidence: 72%

“…In this paper, we show how some bounds from Roos (1999b) can be further substantially improved. We also indicate how these improved bounds in combination with ideas in Roos (2007) can be useful in the compound Poisson approximation.…”

Section: Introductionmentioning

confidence: 99%

Refined total variation bounds in the multivariate and compound Poisson approximation

Roos¹

2017

ALEA

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“…The compound Poisson approximation is frequently used to approximate aggregate losses in risk models (see, for example, [5,8,9,12,14,21]); however, in those models it is usually assumed that rvs are independent of time period n ∈ N. The compound Poisson approximation to sums of Markov dependent rvs was investigated in [6]. Numerous papers were devoted to Markov Binomial distribution, see [1,3,4,7,10,18,19], and the references therein.…”

Section: Known Resultsmentioning

confidence: 99%

Asymptotics for the sum of three state Markov dependent random variables

Liaudanskaitė

Čekanavičius

2018

Modern Stochastics: Theory and Applications

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The insurance model when the amount of claims depends on the state of the insured person (healthy, ill, or dead) and claims are connected in a Markov chain is investigated. The signed compound Poisson approximation is applied to the aggregate claims distribution after n ∈ N periods. The accuracy of order O(n −1 ) and O(n −1/2 ) is obtained for the local and uniform norms, respectively. In a particular case, the accuracy of estimates in total variation and non-uniform estimates are shown to be at least of order O(n −1 ). The characteristic function method is used. The results can be applied to estimate the probable loss of an insurer to optimize an insurance premium.

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“…We start with a brief historical account of Poisson approximation, focusing particular on the evolution of the total variation distance; a more detailed, technical discussion will be given in Section 6. For other surveys, see [38,9,4,22,72]. Fourier Table 1: Some results of the form d TV := d TV (L (S n ), P(λ)) cθ.…”

Section: A Historical Account With Brief Review Of Resultsmentioning

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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On variational bounds in the compound Poisson approximation of the individual risk model

Cited by 10 publications

References 28 publications

Refined total variation bounds in the multivariate and compound Poisson approximation

Refined total variation bounds in the multivariate and compound Poisson approximation

Asymptotics for the sum of three state Markov dependent random variables

A Charlier–Parseval approach to poisson approximation and its applications

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