Poisson approximations for the distribution and moments of ordered m-spacings

1994 ◽  
Vol 31 (A) ◽  
pp. 271-281 ◽  
Author(s):  
Joseph Glaz ◽  
Joseph Naus ◽  
Malgorzata Roos ◽  
Sylvan Wallenstein
Keyword(s):  
Simulation Study ◽  
Numerical Procedure ◽  
Poisson Approximation ◽  
Accurate Approximation ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Poisson Approximations

This article investigates the accuracy of approximations for the distribution of ordered m-spacings for i.i.d. uniform observations in the interval (0, 1). Several Poisson approximations and a compound Poisson approximation are studied. The result of a simulation study is included to assess the accuracy of these approximations. A numerical procedure for evaluating the moments of the ordered m-spacings is developed and evaluated for the most accurate approximation.

Poisson approximations for the distribution and moments of ordered m-spacings

1994 ◽  
Vol 31 (A) ◽  
pp. 271-281 ◽  
Author(s):  
Joseph Glaz ◽  
Joseph Naus ◽  
Malgorzata Roos ◽  
Sylvan Wallenstein
Keyword(s):  
Simulation Study ◽  
Numerical Procedure ◽  
Poisson Approximation ◽  
Accurate Approximation ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Poisson Approximations

This article investigates the accuracy of approximations for the distribution of ordered m-spacings for i.i.d. uniform observations in the interval (0, 1). Several Poisson approximations and a compound Poisson approximation are studied. The result of a simulation study is included to assess the accuracy of these approximations. A numerical procedure for evaluating the moments of the ordered m-spacings is developed and evaluated for the most accurate approximation.

scholarly journals Improved compound Poisson approximation for the number of occurrences of any rare word family in a stationary markov chain

2007 ◽  
Vol 39 (01) ◽  
pp. 128-140 ◽  
Author(s):  
Etienne Roquain ◽  
Sophie Schbath
Keyword(s):  
Markov Chain ◽  
Poisson Distribution ◽  
Approximation Error ◽  
Rare Event ◽  
Poisson Approximation ◽  
Rare Word ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Stein Method ◽  
Poisson Approximations

We derive a new compound Poisson distribution with explicit parameters to approximate the number of overlapping occurrences of any set of words in a Markovian sequence. Using the Chen-Stein method, we provide a bound for the approximation error. This error converges to 0 under the rare event condition, even for overlapping families, which improves previous results. As a consequence, we also propose Poisson approximations for the declumped count and the number of competing renewals.

scholarly journals Compound Poisson approximation of subgraph counts in stochastic block models with multiple edges

2018 ◽  
Vol 50 (3) ◽  
pp. 759-782
Author(s):  
Matthew Coulson ◽  
Robert E. Gaunt ◽  
Gesine Reinert
Keyword(s):  
Poisson Approximation ◽  
Simple Graph ◽  
Block Model ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Stochastic Block Model ◽  
Multiple Edges ◽  
Block Models ◽  
Balanced Graphs ◽  
Poisson Approximations

Abstract We use the Stein‒Chen method to obtain compound Poisson approximations for the distribution of the number of subgraphs in a generalised stochastic block model which are isomorphic to some fixed graph. This model generalises the classical stochastic block model to allow for the possibility of multiple edges between vertices. We treat the case that the fixed graph is a simple graph and that it has multiple edges. The former results apply when the fixed graph is a member of the class of strictly balanced graphs and the latter results apply to a suitable generalisation of this class to graphs with multiple edges. We also consider a further generalisation of the model to pseudo-graphs, which may include self-loops as well as multiple edges, and establish a parameter regime in the multiple edge stochastic block model in which Poisson approximations are valid. The results are applied to obtain Poisson and compound Poisson approximations (in different regimes) for subgraph counts in the Poisson stochastic block model and degree corrected stochastic block model of Karrer and Newman (2011).

scholarly journals Improved compound Poisson approximation for the number of occurrences of any rare word family in a stationary markov chain

2007 ◽  
Vol 39 (1) ◽  
pp. 128-140 ◽  
Author(s):  
Etienne Roquain ◽  
Sophie Schbath
Keyword(s):  
Markov Chain ◽  
Poisson Distribution ◽  
Approximation Error ◽  
Rare Event ◽  
Poisson Approximation ◽  
Rare Word ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Stein Method ◽  
Poisson Approximations

We derive a new compound Poisson distribution with explicit parameters to approximate the number of overlapping occurrences of any set of words in a Markovian sequence. Using the Chen-Stein method, we provide a bound for the approximation error. This error converges to 0 under the rare event condition, even for overlapping families, which improves previous results. As a consequence, we also propose Poisson approximations for the declumped count and the number of competing renewals.

scholarly journals Compound Poisson approximation for the distribution of extremes

2002 ◽  
Vol 34 (1) ◽  
pp. 223-240 ◽  
Author(s):  
A. D. Barbour ◽  
S. Y. Novak ◽  
A. Xia
Keyword(s):  
Point Processes ◽  
Poisson Approximation ◽  
Value Theory ◽  
Wasserstein Metric ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Limiting Behaviour ◽  
Explicit Bounds ◽  
Empirical Point ◽  
Poisson Cluster

Empirical point processes of exceedances play an important role in extreme value theory, and their limiting behaviour has been extensively studied. Here, we provide explicit bounds on the accuracy of approximating an exceedance process by a compound Poisson or Poisson cluster process, in terms of a Wasserstein metric that is generally more suitable for the purpose than the total variation metric. The bounds only involve properties of the finite, empirical sequence that is under consideration, and not of any limiting process. The argument uses Bernstein blocks and Lindeberg's method of compositions.

scholarly journals An Improved Error Bound for the Compound Poisson Approximation of a Nearly Homogeneous Portfolio

1987 ◽  
Vol 17 (2) ◽  
pp. 165-169 ◽  
Author(s):  
R. Michel
Keyword(s):  
Error Bound ◽  
Poisson Approximation ◽  
Claim Amount ◽  
Compound Poisson ◽  
Compound Poisson Approximation

AbstractFor the case of a portfolio with identical claim amount distributions, Gerber's error bound for the compound Poisson approximation is improved (in the case λ ⩾ 1). The result can also be applied to more general portfolios by partitioning them into homogeneous subportfolios.

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