Topic

# Compound Poisson distribution

About: Compound Poisson distribution is a research topic. Over the lifetime, 1824 publications have been published within this topic receiving 39938 citations.

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TL;DR: In this article, regression-based tests for mean-variance equality were proposed in a very general setting, which requires specification of only the mean variance relationship under the alternative, rather than the complete distribution whose choice is usually arbitrary.

Abstract: A property of the Poisson regression model is mean-variance equality, conditional on explanatory variables. ‘Regression-based’ tests for this property are proposed in a very general setting. Unlike classical statistical tests, these tests require specification of only the mean-variance relationship under the alternative, rather than the complete distribution whose choice is usually arbitrary. The optimal regression-based test is easily computed as the t-test from an auxiliary regression. If a distribution under the alternative hypothesis is in fact specified and is in the Katz system of distributions or is Cox's local approximation to the Poisson, the score test for the Poisson distribution is equivalent to the optimal regression-based test.

949 citations

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TL;DR: In this article, a simple and relatively efficient method for simulating one-dimensional and two-dimensional nonhomogeneous Poisson processes is presented, which is applicable for any rate function and is based on controlled deletion of points in a Poisson process whose rate function dominates the given rate function.

Abstract: : A simple and relatively efficient method for simulating one- dimensional and two-dimensional nonhomogeneous Poisson processes is presented. The method is applicable for any rate function and is based on controlled deletion of points in a Poisson process whose rate function dominates the given rate function. In its simplest implementation, the method obviates the need for numerical integration of the rate function, for ordering of points, and for generation of Poisson variates.

805 citations

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TL;DR: In this paper, a general method of obtaining and bounding the error in approximating the distribution of the dependent Bernoulli random variables by the Poisson distribution is presented, which is similar to that of Charles Stein (1970) in his paper on normal approximation for dependent random variables.

Abstract: Let $X_1, \cdots, X_n$ be an arbitrary sequence of dependent Bernoulli random variables with $P(X_i = 1) = 1 - P(X_i = 0) = p_i.$ This paper establishes a general method of obtaining and bounding the error in approximating the distribution of $\sum^n_{i=1} X_i$ by the Poisson distribution. A few approximation theorems are proved under the mixing condition of Ibragimov (1959), (1962). One of them yields, as a special case and with some improvement, an approximation theorem of Le Cam (1960) for the Poisson binomial distribution. The possibility of an asymptotic expansion is also discussed and a refinement in the independent case obtained. The method is similar to that of Charles Stein (1970) in his paper on the normal approximation for dependent random variables.

624 citations

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TL;DR: In this paper, a generalization of the Poisson distribution with two parameters λ1 and λ2 is obtained as a limiting form of the generalized negative binomial distribution, where the variance of the distribution is greater than, equal to or smaller than the mean according as λ 2 is positive, zero or negative.

Abstract: A new generalization of the Poisson distribution, with two parameters λ1 and λ2, is obtained as a limiting form of the generalized negative binomial distribution. The variance of the distribution is greater than, equal to or smaller than the mean according as λ2 is positive, zero or negative. The distribution gives a very close fit to supposedly binomial, Poisson and negative-binomial data and provides with a model suitable to most unimodel or reverse J-shaped distributions. Diagrams showing the variations in the form of the distribution for different values of λ1 and λ2 are given.

474 citations

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TL;DR: In this article, the spectral representations for arbitrary discrete and continuous infinitely divisible processes were obtained using a polar factorization of an arbitrary Levy measure on a separable Hilbert space and the Wiener-type stochastic integrals of non-random functions relative to arbitrary "infinitely divisible noise".

Abstract: The spectral representations for arbitrary discrete parameter infinitely divisible processes as well as for (centered) continuous parameter infinitely divisible processes, which are separable in probability, are obtained. The main tools used for the proofs are (i) a “polar-factorization” of an arbitrary Levy measure on a separable Hilbert space, and (ii) the Wiener-type stochastic integrals of non-random functions relative to arbitrary “infinitely divisible noise”.

462 citations