Compound Poisson Models Research Articles

Compound Conway-Maxwell Poisson Gamma Distribution: Properties and Estimation

The distribution of a random sum of random events is called a compound distribution. It involves a counting (discrete) distribution to model the number of occurrences of the random event in a fixed time period and a continuous distribution to model the outcome of the random event. It has applications in the fields of actuarial sciences, meteorology etc. For example, in modelling insurance loss amounts through compound distributions, the number of claims and the claim amounts are used to calculate the total claim amount of a portfolio. The number of claims is modelled through a discrete distribution and the claim amounts are modelled through continuous distributions. Generally, Poisson distribution is used in compound models as the discrete distribution and such models are known as compound Poisson models. However, the equi-dispersion property of the Poisson distribution hinders its application in scenarios where the underlying count data is either over- or under-dispersed. In this paper, a two-parameter Poisson distribution, namely, Conway-Maxwell Poisson (CMP) distribution, which handles both over- and under-dispersed data, is considered as the counting distribution, and the corresponding compound CMP distribution is developed. Some mathematical properties of the distribution are derived and a methodology to estimate the parameters using the likelihood approach is proposed. A numerical illustration of the proposed methodology is given through a simulation study. An application of the compound CMP model is illustrated through transportation security administration (TSA) insurance claim data. Also, the estimation of the risk measures associated with the TSA claim data is discussed.

Tweedie Compound Poisson Models with Covariate-Dependent Random Effects for Multilevel Semicontinuous Data.

Multilevel semicontinuous data occur frequently in medical, environmental, insurance and financial studies. Such data are often measured with covariates at different levels; however, these data have traditionally been modelled with covariate-independent random effects. Ignoring dependence of cluster-specific random effects and cluster-specific covariates in these traditional approaches may lead to ecological fallacy and result in misleading results. In this paper, we propose Tweedie compound Poisson model with covariate-dependent random effects to analyze multilevel semicontinuous data where covariates at different levels are incorporated at relevant levels. The estimation of our models has been developed based on the orthodox best linear unbiased predictor of random effect. Explicit expressions of random effects predictors facilitate computation and interpretation of our models. Our approach is illustrated through the analysis of the basic symptoms inventory study data where 409 adolescents from 269 families were observed at varying number of times from 1 to 17 times. The performance of the proposed methodology was also examined through the simulation studies.

Nagging Predictors

We define the nagging predictor, which, instead of using bootstrapping to produce a series of i.i.d. predictors, exploits the randomness of neural network calibrations to provide a more stable and accurate predictor than is available from a single neural network run. Convergence results for the family of Tweedie’s compound Poisson models, which are usually used for general insurance pricing, are provided. In the context of a French motor third-party liability insurance example, the nagging predictor achieves stability at portfolio level after about 20 runs. At an insurance policy level, we show that for some policies up to 400 neural network runs are required to achieve stability. Since working with 400 neural networks is impractical, we calibrate two meta models to the nagging predictor, one unweighted, and one using the coefficient of variation of the nagging predictor as a weight, finding that these latter meta networks can approximate the nagging predictor well, only with a small loss of accuracy.

Compound Poisson models for weighted networks with applications in finance

We develop a modelling framework for estimating and predicting weighted network data. The edge weights in weighted networks often arise from aggregating some individual relationships between the nodes. Motivated by this, we introduce a modelling framework for weighted networks based on the compound Poisson distribution. To allow for heterogeneity between the nodes, we use a regression approach for the model parameters. We test the new modelling framework on two types of financial networks: a network of financial institutions in which the edge weights represent exposures from trading Credit Default Swaps and a network of countries in which the edge weights represent cross-border lending. The compound Poisson Gamma distributions with regression fit the data well in both situations. We illustrate how this modelling framework can be used for predicting unobserved edges and their weights in an only partially observed network. This is for example relevant for assessing systemic risk in financial networks.

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

Software developers predict their product’s failure rate using reliability growth models that are typically based on nonhomogeneous Poisson (NHP) processes. In this article, we extend that practice to a nonhomogeneous discrete-compound Poisson process that allows for multiple faults of a system at the same time point. Along with traditional reliability metrics such as average number of failures in a time interval, we propose an alternative reliability index called critical fault-detecting time in order to provide more information for software managers making software quality evaluation and critical market policy decisions. We illustrate the significant potential for improved analysis using wireless failure data as well as simulated data.

Over-Dispersed Age-Period-Cohort Models

ABSTRACTWe consider inference and forecasting for aggregate data organized in a two-way table with age and cohort as indices, but without measures of exposure. This is modeled using a Poisson likelihood with an age-period-cohort structure for the mean while allowing for over-dispersion. We propose a repetitive structure that keeps the dimension of the table fixed while increasing the latent exposure. For this, we use a class of infinitely divisible distributions which include a variety of compound Poisson models and Poisson mixture models. This results in asymptotic F inference and t forecast distributions.

COMPOUND POISSON CLAIMS RESERVING MODELS: EXTENSIONS AND INFERENCE

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.

Double Generalized Linear Compound Poisson models to Insurance Claims Data

This paper describes the specification, estimation and comparison of double generalized linear compound Poisson models based on the likelihood paradigm. The models are motivated by insurance applications, where the distribution of the response variable is composed by a degenerate distribution at the origin and a continuous distribution on the positive real line. We present maximum likelihood and restricted maximum likelihood algorithms for parameter estimation, with emphasis to the analysis of insurance data. Simulation studies are employed to evaluate the bias and consistency of the estimators in a finite sample framework. The simulation studies are also used to validate the fitting algorithms and check the computational implementation. Furthermore, we investigate the impact of an unsuitable choice for the response variable distribution on both mean and dispersion parameter estimates. We provide R implementation and illustrate the application of double generalized linear compound Poisson models using a data set about car insurances.

Insurance Premium Prediction via Gradient Tree-Boosted Tweedie Compound Poisson Models

The Tweedie GLM is a widely used method for predicting insurance premiums. However, the structure of the logarithmic mean is restricted to a linear form in the Tweedie GLM, which can be too rigid for many applications. As a better alternative, we propose a gradient tree-boosting algorithm and apply it to Tweedie compound Poisson models for pure premiums. We use a profile likelihood approach to estimate the index and dispersion parameters. Our method is capable of fitting a flexible nonlinear Tweedie model and capturing complex interactions among predictors. A simulation study confirms the excellent prediction performance of our method. As an application, we apply our method to an auto-insurance claim data and show that the new method is superior to the existing methods in the sense that it generates more accurate premium predictions, thus helping solve the adverse selection issue. We have implemented our method in a user-friendly R package that also includes a nice visualization tool for interpreting the fitted model.

Compound Poisson Claims Reserving Models: Extensions and Inference

We 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.

Compound geometric and Poisson models

Compound geometric and Poisson models

Cramér–Von Mises distance estimation for some positive infinitely divisible parametric families with actuarial applications

Cramér–Von Mises (CVM) inference techniques are developed for some positive flexible infinitely divisible parametric families generalizing the compound Poisson family. These larger families appear to be useful for parametric inference for positive data. The methods are based on inverting the characteristic functions. They are numerically implementable whenever the characteristic function has a closed form. In general, likelihood methods based on density functions are more difficult to implement. CVM methods also lead to model testing, with test statistics asymptotically following a chi-square distribution. The methods are for continuous models, but they can also handle models with a discontinuity point at the origin such as the case of compound Poisson models. Simulation studies seem to suggest that CVM estimators are more efficient than moment estimators for the common range of the compound Poisson gamma family. Actuarial applications include estimation of the stop loss premium, and estimation of the present value of cash flows when interest rates are assumed to be driven by a corresponding Lévy process.

Asymptotics of Symmetric Compound Poisson Population 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 radiusrof 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.

Population Genetics Models with Skewed Fertilities: A Forward and Backward Analysis

Discrete population genetics models with unequal (skewed) fertilities are considered, with an emphasis on skewed versions of Cannings models, conditional branching process models in the spirit of Karlin and McGregor, and compound Poisson models. Three particular classes of models with skewed fertilities are investigated, the Wright–Fisher model, the Dirichlet model, and the Kimura model. For each class the asymptotic behavior as the total population size N tends to infinity is investigated for power law fertilities and for geometric fertilities. This class of models can exhibit a rich variety of sub-linear or even constant effective population sizes. Therefore, the models are not necessarily in the domain of attraction of the Kingman coalescent. For a substantial range of the parameters, discrete-time coalescent processes with simultaneous multiple collisions arise in the limit.

Asymptotically optimal dividend policy for regime-switching compound Poisson models

This work develops asymptotically optimal dividend policies to maximize the expected present value of dividends until ruin. Compound Poisson processes with regime switching are used to model the surplus and the switching (a continuous-time controlled Markov chain) represents random environment and other economic conditions. Assuming the switching to be fast varying together with suitable conditions, it is shown that the system has a limit that is an average with respect to the invariant measure of a related Markov chain. Under simple conditions, the optimal policy of the limit dividend strategy is a threshold policy. Using the optimal policy of the limit system as a guide, feedback control for the original surplus is then developed. It is demonstrated that the constructed dividend policy is asymptotically optimal.

Statistical models to standardize catch rates in the multi-species trawl fishery for Patagonian grenadier ( Macruronus magellanicus) off Southern Chile

Changing targeting behavior of fishing fleets and zero-catch records of the target species are two characteristics common to many multi-species trawl fisheries. These features complicate analyses intended to produce indices of relative abundance from commercial catch rate data. Trawl catch rates of Patagonian grenadier caught off Southern Chile were used to calculate indices of relative abundance for this species. To allow for targeting behavior and zero catches, we first defined a subset of relevant tows combining a logistic regression on the species catch composition and knowledge about the persistence of Patagonian grenadier in the catches from the area where historically it has been more abundant. A standardization analysis was then conducted on this subset of the data using a generalized linear model (GLM) framework. Analysis of deviance tables and standard residuals GLM diagnostic plots were used for selecting the statistical models that best described the data. The candidate models tested were the delta-lognormal, delta-gamma, quasi-Poisson and compound Poisson. The models that best supported the Patagonian grenadier catch rate data were the delta-lognormal and compound Poisson models. Standardization models were then extended to generalized linear mixed models to include the vessel main effect and the significant interactions with the year effect as random effects. Estimations of the annual series of relative abundance were then obtained from the coefficients of the year effects of these models, both of which showed higher abundance for the years 2001--2008 than in previous years of the series.

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

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.

Novel yield model for integrated circuits with clustered defects

As wafer sizes increase, the clustering phenomenon of defects increases. Clustered defects cause the conventional Poisson yield model underestimate actual wafer yield, as defects are no longer uniformly distributed over a wafer. Although some yield models, such as negative binomial or compound Poisson models, consider the effects of defect clustering on yield prediction, these models have some drawbacks. This study presents a novel yield model that employs General Regression Neural Network (GRNN) to predict wafer yield for integrated circuits (IC) with clustered defects. The proposed method utilizes five relevant variables as input for the GRNN yield model. A simulated case is applied to demonstrate the effectiveness of the proposed model.

Ruin probabilities and overshoots for general Lévy insurance risk processes

We formulate the insurance risk process in a general Lévy process setting, and give general theorems for the ruin probability and the asymptotic distribution of the overshoot of the process above a high level, when the process drifts to −∞ a.s. and the positive tail of the Lévy measure, or of the ladder height measure, is subexponential or, more generally, convolution equivalent. Results of Asmussen and Klüppelberg [Stochastic Process. Appl. 64 (1996) 103–125] and Bertoin and Doney [Adv. in Appl. Probab. 28 (1996) 207–226] for ruin probabilities and the overshoot in random walk and compound Poisson models are shown to have analogues in the general setup. The identities we derive open the way to further investigation of general renewal-type properties of Lévy processes.

A Compound Poisson Model for Word Occurrences in DNA Sequences

Summary We present a compound Poisson model describing the occurrence process of a set of words in a random sequence of letters. The model takes into account the frequency of the words and their overlapping structure. The model is compared with a Markov chain model in terms of fit and parsimony. Special attention is given to the detection of poor or rich regions. Several applications of the model are presented and a combination of the Markov and compound Poisson models is proposed.