Poisson Inverse Gaussian Distribution Research Articles

Analysis of Count Time Series: A Bayesian GARMA(p, q) Approach

Extensions of the Autoregressive Moving Average, ARMA(p, q), class for modeling non-Gaussian time series have been proposed in the literature in recent years, being applied in phenomena such as counts and rates. One of them is the Generalized Autoregressive Moving Average, GARMA(p, q), that is supported by the Generalized Linear Models theory and has been studied under the Bayesian perspective. This paper aimed to study models for time series of counts using the Poisson, Negative binomial and Poisson inverse Gaussian distributions, and adopting the Bayesian framework. To do so, we carried out a simulation study and, in addition, we showed a practical application and evaluation of these models by using a set of real data, corresponding to the number of vehicle thefts in Brazil.

The Poisson Inverse Gaussian Regression Model in the Analysis of Clustered Counts Data

We explore the possibility of modeling clustered count data using the Poisson Inverse Gaussian distribution. We develop a regression model, which relates the number of mastitis cases in a sample of dairy farms in Ontario, Canada, to various farm level covariates, to illustrate the method ology. Residual plots are constructed to explore the quality of the fit. We compare the results with a negative binomial regression model using max imum likelihood estimation, and to the generalized linear mixed regression model fitted in SAS.

Discrimination between Some Over Dispersed Count Distributions

The Poisson inverse Gaussian and generalized Poisson distributions are widely used in modelling overdispersed count data which are commonly found in healthcare, insurance, engineering, econometric and ecology. The inverse trinomial distribution is a relatively new count distribution arising from a one-dimensional random walk model (Shimizu & Yanagimoto, 1991). The Poisson inverse Gaussian distribution is a popular count model that has been proposed as an alternative to the negative binomial distribution. The inverse trinomial and generalized Poisson models possess a common characteristic of having a cubic variance function, while the Poisson inverse Gaussian has a quadratic variance function. The nature of the variance function seems to be an important property in modelling overdispersed count data. Hence it is of interest to be able to select among the three models in practical applications. This paper considers discrimination of three models based on the likelihood ratio statistic and computes via Monte Carlo simulation the probability of correct selection.

EM Estimation for the Poisson-Inverse Gamma Regression Model with Varying Dispersion: An Application to Insurance Ratemaking

This article presents the Poisson-Inverse Gamma regression model with varying dispersion for approximating heavy-tailed and overdispersed claim counts. Our main contribution is that we develop an Expectation-Maximization (EM) type algorithm for maximum likelihood (ML) estimation of the Poisson-Inverse Gamma regression model with varying dispersion. The empirical analysis examines a portfolio of motor insurance data in order to investigate the efficiency of the proposed algorithm. Finally, both the a priori and a posteriori, or Bonus-Malus, premium rates that are determined by the Poisson-Inverse Gamma model are compared to those that result from the classic Negative Binomial Type I and the Poisson-Inverse Gaussian distributions with regression structures for their mean and dispersion parameters.

Modelling heavy-tailedness in count time series

Count data frequently exhibit overdispersion, zero inflation and even heavy-tailedness (the tail probabilities are non-negligible or decrease very slowly) in practical applications. Many models have been proposed for modelling count data with overdispersion and zero inflation, but heavy-tailedness is less considered. The proposed model, a new integer-valued autoregressive process with generalized Poisson-inverse Gaussian innovations, is capable of capturing these features. The generalized Poisson-inverse Gaussian family is very flexible, which includes Poisson distribution, Poisson inverse Gaussian distribution, discrete stable distribution and so on. Stationarity and ergodicity of this model are investigated and the expressions of marginal mean and variance are provided. Conditional maximum likelihood is used for estimating the parameters, and consistency and asymptotic normality for the estimators are presented. Further, we consider the h-step forecast and diagnostics for the proposed model. The proposed model is applied to three real data examples. In the first example, we consider the monthly number of cases of Polio, which validates that the proposed model can take into account count data with excessive zeros. Then, we illustrate the use of the proposed model through an application to the numbers of National Science Foundation fundings. Finally, we apply the proposed model to the numbers of transactions in 5-min intervals for the stock traded at Empire District Electric Company. The second and third examples show that the proposed model has a good performance in modelling heavy-tailed count data.

A general method of computing mixed Poisson probabilities by Monte Carlo sampling

Mixed Poisson distributions form an important class of distributions in applications. However, the application of many of these mixed Poisson distributions is hampered by the complicated probability distributions. The paper examines Monte Carlo sampling as a general technique for computation of mixed Poisson probabilities which is applicable to any mixed Poisson distribution with arbitrary mixing distribution. The accuracy and computational speed of this method is illustrated with the Poisson–inverse Gaussian distribution. The proposed method is then applied to compute probabilities of the Poisson–lognormal distribution, a popular species abundance model. It is also shown that in the maximum likelihood estimation of Poisson–lognormal parameters by E–M algorithm, the application of the proposed Monte Carlo computation in the algorithm avoids numerical problems.

Properties and applications of the Poisson-reciprocal inverse Gaussian distribution

ABSTRACTThe barely known continuous reciprocal inverse Gaussian distribution is used in this paper to introduce the Poisson-reciprocal inverse Gaussian discrete distribution. Several of its most relevant statistical properties are examined, some of them directly inherited from the reciprocal of the inverse Gaussian distribution. Furthermore, a mixed Poisson regression model that uses the reciprocal inverse Gaussian as mixing distribution is presented. Parameters estimation in this regression model is performed via an EM type algorithm. In light of the numerical results displayed in the paper, the distributions introduced in this work are competitive with the classical negative binomial and Poisson-inverse Gaussian distributions.

Design of an Optimal Bonus-Malus System Using the Sichel Distribution as a Model of Claim Counts

This paper presents the design of an optimal Bonus-Malus System (BMS) using the Sichel distribution to model the claim frequency distribution. This system is proposed as an alternative to the optimal BMS obtained by the traditional Negative Binomial model (Lemaire, 1995). In fact the Sichel distribution has a thicker tail than the Negative Binomial distribution and it is considered as a plausible model for highly dispersed count data. We also consider the optimal BMS provided by the Poisson-Inverse Gaussian distribution, which is a special case of the Sichel distribution. In the above setup optimality is achieved by minimizing the insurer’s risk. Furthermore we develop a generalized BMS that takes into account both the a priori and a posteriori characteristics of each policyholder. For this purpose we consider to the generalized additive models for location, scale and shape (GAMLSS) in order to use all available information in the estimation of the claim frequency distribution. The GAMLSS were introduced by Rigby and Stasinopoulos (2001, 2005) and Akantziliotou et al. (2002) as a framework for fi tting regression type models where the distribution ofthe response variable does not have to belong to the exponential family and includes highly skew and kurtotic continuous and discrete distribution. The GAMLSS allows all the parameters of the distribution of the response variable to be modeled as linear/non-linear or smooth functions of the explanatory variables. Within the framework of the GAMLSS we propose the Sichel GAMLSS for assessing claim frequency as an alternative to the Negative Binomial regression model used by Dionne and Vanasse (1989, 1992). We also consider the Negative Binomial Type I and the Poisson-Inverse Gaussian GAMLSS for assessing claim frequency. With the aim of constructing an optimal BMS by updating the posterior mean claim frequency, we adopt the parametric linear formulation of these models and we allow only their mean parameter to be modeled as a function of the significant a priori rating variables for the number of claims.

Modelling heavy-tailed count data using a generalised Poisson-inverse Gaussian family

We generalise the Poisson-inverse Gaussian distribution to a three-parameter family, which includes the Poisson and discrete stable distributions as boundary cases. It is flexible in modelling count data sets with different tail heaviness. Although the family only has a closed-form probability generating function, a recursive method is developed for statistical inferences based on the likelihood. As an example, this new family is applied to data sets of citation counts of published articles.

On modeling count data: a comparison of some well-known discrete distributions

In a recent paper, Joe and Zhu [2005, Generalized Poisson distribution: the property of mixture of Poisson and comparison with negative binomial distribution. Biometrical Journal, 47(2), 219–229.] compared the negative binomial and the generalized Poisson distributions. We aim at extending this comparison by including more distributions and more aspects, like tailness and kurtosis. We also aim at describing the related problem, whether from real data one may distinguish between the candidate models. To do this, we consider several aspects of three of the most commonly used mixed Poisson distributions, namely the negative binomial, the Poisson inverse Gaussian and the generalized Poisson distributions. The results show that for small mean and overdispersion, all the models are quite the same, whereas for larger means the generalized Poisson and the Poisson inverse Gaussian distributions have larger tails than the negative binomial, and the differences are much larger. In practice, it is not easy to discriminate between them for small counts and small overdispersion, but for large overdispersion discrimination is relatively easy. Applications to real data are provided to illustrate the ideas.

On Strict Arcsine Distribution

It is shown that the strict arcsine distribution is overdispersed, skewed to the right, and leptokurtic. Comparing to the negative binomial and Poisson-inverse Gaussian distributions, the strict arcsine can be seen as their alternative. Moreover, we discuss about unimodality and bimodality conditions, and Poisson mixture distribution. Application of the distribution to automobile claim frequency data is attempted and its performance is demonstrated for some data set.

The Mixed Bivariate Hofmann Distribution

AbstractIn this paper we study a class of Mixed Bivariate Poisson Distributions by extending the Hofmann Distribution from the univariate case to the bivariate case.We show how to evaluate the bivariate aggregate claims distribution and we fit some insurance portfolios given in the literature.This study typically extends the use of the Bivariate Independent Poisson Distribution, the Mixed Bivariate Negative Binomial and the Mixed Bivariate Poisson Inverse Gaussian Distribution.

A note on the mixed poisson formulation of the poisson-inverse gaussian distribution

The Poisson–inverse Gaussian distribution is expressed in terms of a mixed Poisson distribution with the inverted gamma as mixing distribution. A simple approximation by Taylor expansion is derived for the Poisson–inverse Gaussian probabilities from this mixing expression. Based on this expansion, the asymptotic behaviour is considered. Shaban's approximation (Communications in Statistics, A, 1981, pp. 1389–1399) of the Poisson-inverse Gaussian probability by its limiting case is examined, and it is shown that this approximation can be improved by using the inverted gamma mixing formulation.

Using the Poisson Inverse Gaussian in Bonus-Malus Systems

AbstractIn this paper, we will cover the bonus-malus system in automobile insurance. Bonus-malus systems are based on the distribution of the number of car accidents. Therefore, the modelling and fitting of that distribution are considered. Fitting of data is done using the Poisson inverse Gaussian distribution, which shows a good fit. Building the bonus system is done by minimizing the insurer's risk, according to Lemaire's (1985) bonus system.

Trend et systèmes de Bonus-Malus

AbstractThis paper deals with the Bonus-Malus system obtained when the claims frequency is submitted to trend. This system is specified in the two particular cases of Poisson-Gamma and Poisson-Inverse Gaussian distributions. The theoretical results are checked on data issued from automobile insurance policies observed during three years.

The Poisson-Inverse Gaussian distribution as an alternative to the negative binomial

Abstract The basic distributional properties and estimation techniques of the Poisson-Inverse Gaussian (P-IG) distribution are reviewed. Its use both as a mixed and compound claim frequency model are also discussed, as well as a review of the aggregate claims distribution when the P-IG is the claim frequency component. The many properties which are analogous to those of the negative binomial are emphasized, and the superior fit to automobile claim frequency data is demonstrated. The P-IG merits consideration as a model for claim frequency data due to its good fit to data, physical justification, and its abundance of convenient mathematical properties.