Critical fault-detecting time evaluation in software with discrete compound Poisson models

Compound Poisson population models are particular conditional branching process models. A formula for the transition probabilities of the backward process for general compound Poisson models is verified. Symmetric compound Poisson models are defined in terms of a parameter θ ∈ (0, ∞) and a power series φ with positive radius r of convergence. It is shown that the asymptotic behaviour of symmetric compound Poisson models is mainly determined by the characteristic value θrφ′(r−). If θrφ′(r−)≥1, then the model is in the domain of attraction of the Kingman coalescent. If θrφ′(r−) < 1, then under mild regularity conditions a condensation phenomenon occurs which forces the model to be in the domain of attraction of a discrete-time Dirac coalescent. The proofs are partly based on the analytic saddle point method. They draw heavily from local limit theorems and from results of S. Janson on simply generated trees, conditioned Galton-Watson trees, random allocations and condensation. Several examples of compound Poisson models are provided and analysed.

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RUIN PROBABILITY UNDER COMPOUND POISSON MODELS WITH RANDOM DISCOUNT FACTOR

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964804181047 ◽

2004 ◽

Vol 18 (1) ◽

pp. 55-70 ◽

Cited By ~ 6

Author(s):

Kai W. Ng ◽

Hailiang Yang ◽

Lihong Zhang

Keyword(s):

Ruin Probability ◽

Risk Model ◽

Poisson Model ◽

Convergence Result ◽

Discount Factor ◽

Insurance Risk ◽

Compound Poisson ◽

Analysis Techniques ◽

Surplus Process ◽

Poisson Models

In this article, we consider a compound Poisson insurance risk model with a random discount factor. This model is also known as the compound filtered Poisson model. By using some stochastic analysis techniques, a convergence result for the discounted surplus process, an expression for the ruin probability, and the upper bounds for the ruin probability are obtained.

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COMPOUND POISSON CLAIMS RESERVING MODELS: EXTENSIONS AND INFERENCE

Astin Bulletin ◽

10.1017/asb.2018.12 ◽

2018 ◽

Vol 48 (3) ◽

pp. 1137-1156 ◽

Cited By ~ 1

Author(s):

Shengwang Meng ◽

Guangyuan Gao

Keyword(s):

Linear Models ◽

Mean Squared Error ◽

Predictive Distribution ◽

Compound Poisson ◽

Poisson Models ◽

Squared Error ◽

Run Off ◽

Mean And Variance ◽

The Individual ◽

Over Dispersion

AbstractWe consider compound Poisson claims reserving models applied to the paid claims and to the number of payments run-off triangles. We extend the standard Poisson-gamma assumption to account for over-dispersion in the payment counts and to account for various mean and variance structures in the individual payments. Two generalized linear models are applied consecutively to predict the unpaid claims. A bootstrap is used to estimate the mean squared error of prediction and to simulate the predictive distribution of the unpaid claims. We show that the extended compound Poisson models make reasonable predictions of the unpaid claims.

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Model Uncertainty in Claims Reserving within Tweedie's Compound Poisson Models

Astin Bulletin ◽

10.2143/ast.39.1.2038054 ◽

2009 ◽

Vol 39 (1) ◽

pp. 1-33 ◽

Cited By ~ 42

Author(s):

Gareth W. Peters ◽

Pavel V. Shevchenko ◽

Mario V. Wüthrich

Keyword(s):

Monte Carlo Simulation ◽

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Model Uncertainty ◽

Model Averaging ◽

Poisson Model ◽

Compound Poisson ◽

Poisson Models ◽

Model Selection Problem

AbstractIn this paper we examine the claims reserving problem using Tweedie's compound Poisson model. We develop the maximum likelihood and Bayesian Markov chain Monte Carlo simulation approaches to fit the model and then compare the estimated models under different scenarios. The key point we demonstrate relates to the comparison of reserving quantities with and without model uncertainty incorporated into the prediction. We consider both the model selection problem and the model averaging solutions for the predicted reserves. As a part of this process we also consider the sub problem of variable selection to obtain a parsimonious representation of the model being fitted.

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Compound Poisson Models for Financial Networks

SSRN Electronic Journal ◽

10.2139/ssrn.3401059 ◽

2019 ◽

Author(s):

Axel Gandy ◽

Luitgard A. M. Veraart

Keyword(s):

Financial Networks ◽

Compound Poisson ◽

Poisson Models

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Flexible Purchase Frequency Modeling

Journal of Marketing Research ◽

10.1177/002224379603300109 ◽

1996 ◽

Vol 33 (1) ◽

pp. 94-107 ◽

Cited By ~ 1

Author(s):

Patrick L. Brockett ◽

Linda L. Golden ◽

Harry H. Panjer

Keyword(s):

Panel Data ◽

General Framework ◽

Data Sets ◽

Compound Poisson ◽

Frequency Distributions ◽

Purchase Frequency ◽

Individual Level ◽

Poisson Models ◽

Marketing Models ◽

Second Period

The authors present a general framework for purchase frequency modeling that enables flexible fitting and convenient computation. Their easily described purchase frequency distributions subsume many previous models and provide a connection between mixed Poisson marketing models and the conceptually distinct compound Poisson models. These distributions provide simple parametric equations for individual-level prediction of second-period purchase frequency based on observed first-period purchase frequencies. The results are applied to four marketing panel data sets.

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