Does a Gibbs sampler approach to spatial Poisson regression models outperform a single site MH sampler?

In this paper we present and evaluate a Gibbs sampler for a Poisson regression model including spatial e ects. The approach is based on Frühwirth-Schnatter and Wagner (2004b) who show that by data augmentation using the introduction of two sequences of latent variables a Poisson regression model can be transformed into an approximate normal linear model. We show how this methodology can be extended to spatial Poisson regression models and give details of the resulting Gibbs sampler. In particular, the influence of model parameterisation and di erent update strategies on the mixing of the MCMC chains is discussed. The developed Gibbs samplers are analysed in two simulation studies and applied to model the expected number of claims for policyholders of a German car insurance company. The mixing of the Gibbs samplers depends crucially on the model parameterisation and the update schemes. The best mixing is achieved when collapsed algorithms are used, reasonable low autocorrelations for the spatial e ects are obtained in this case. For the regression e ects however, autocorrelations are rather high, especially for data with very low heterogeneity. For comparison a single component Metropolis Hastings algorithms is applied which displays very good mixing for all components. Although the Metropolis Hastings sampler requires a higher computational e ort, it outperforms the Gibbs samplers which would have to be run considerably longer in order to obtain the same precision of the parameters

Introducing and evaluating a Gibbs sampler for spatial Poisson regression models

In this paper we present a Gibbs sampler for a Poisson model including spatial effects. Frühwirth-Schnatter und Wagner (2004b) show that by data augmentation via the introduction of two sequences of latent variables a Poisson regression model can be transformed into a normal linear model. We show how this methodology can be extended to spatial Poisson regression models and give details of the resulting Gibbs sampler. In particular, the influence of model parameterisation and different update strategies on the mixing of the MCMC chains are discussed. The developed Gibbs samplers are analysed in two simulation studies and appliedto model the expected number of claims for policyholders of a German car insurance data set. In general, both large and small simulated spatial effects are estimated accurately by the Gibbs samplers and reasonable low autocorrelations are obtained when the data variability is rather large. However, for data with very low heterogeneity, the autocorrelations resulting from the Gibbs samplers are very high, withdrawing the computational advantage over a Metropolis Hastings independence sampler which exhibits very low autocorrelations in all settings

Spatial modelling of claim frequency and claim size in insurance

In this paper models for claim frequency and claim size in non-life insurance are considered. Both covariates and spatial random e ects are included allowing the modelling of a spatial dependency pattern. We assume a Poisson model for the number of claims, while claim size is modelled using a Gamma distribution. However, in contrast to the usual compound Poisson model going back to Lundberg (1903), we allow for dependencies between claim size and claim frequency. Both models for the individual and average claim sizes of a policyholder are considered. A fully Bayesian approach is followed, parameters are estimated using Markov Chain Monte Carlo (MCMC). The issue of model comparison is thoroughly addressed. Besides the deviance information criterion suggested by Spiegelhalter et al. (2002), the predictive model choice criterion (Gelfand and Ghosh (1998)) and proper scoring rules (Gneiting and Raftery (2005)) based on the posterior predictive distribution are investigated. We give an application to a comprehensive data set from a German car insurance company. The inclusion of spatial e ects significantly improves the models for both claim frequency and claim size and also leads to more accurate predictions of the total claim sizes. Further we quantify the significant number of claims e ects on claim size

Modelling count data with overdispersion and spatial effects

In this paper we consider regression models for count data allowing for overdispersion in a Bayesian framework. We account for unobserved heterogeneity in the data in two ways. On the one hand, we consider more flexible models than a common Poisson model allowing for overdispersion in different ways. In particular, the negative binomial and the generalized Poisson distribution are addressed where overdispersion is modelled by an additional model parameter. Further, zero-inflated models in which overdispersion is assumed to be caused by an excessive number of zeros are discussed. On the other hand, extra spatial variability in the data is taken into account by adding spatial random effects to the models. This approach allows for an underlying spatial dependency structure which is modelled using a conditional autoregressive prior based on Pettitt et al. (2002). In an application the presented models are used to analyse the number of invasive meningococcal disease cases in Germany in the year 2004. Models are compared according to the deviance information criterion (DIC) suggested by Spiegelhalter et al. (2002) and using proper scoring rules, see for example Gneiting and Raftery (2004). We observe a rather high degree of overdispersion in the data which is captured best by the GP model when spatial effects are neglected. While the addition of spatial effects to the models allowing for overdispersion gives no or only little improvement, a spatial Poisson model is to be preferred over all other models according to the considered criteria

Calculation of LTC Premiums based on direct estimates of transition probabilities

In this paper we model the life-history of LTC patients using a Markovian multi-state model in order to calculate premiums for a given LTC-plan. Instead of estimating the transition intensities in this model we use the approach suggested by Andersen et al. (2003) for a direct estimation of the transition probabilities. Based on the Aalen-Johansen estimator, an almost unbiased estimator for the transition matrix of a Markovian multi-state model, we calculate so-called pseudo-values, known from Jackknife methods. Further, we assume that the relationship between these pseudo-values and the covariates of our data are given by a GLM with the logit as link-function. Since the GLMs do not allow for correlation between successive observations we use instead the "Generalized Estimating Equations" (GEEs) to estimate the parameters of our regression model. The approach is illustrated using a representative sample from a German LTC portfolio

Modeling of transition intensities and probabilities in a German long term care portfolio with known diagnosis

In this paper a semiparametric hazard model introduced by Cox (1972) is used to model transitions intensities for a long term care (LTC) data set. The main focus is the inclusion of the diagnoses which led to LTC as explanatory variables. Modern model diagnostic techniques are applied to check the model assumptions. Fractional Polynomials proposed by Royston and Altman (1994) are used to model the functional form of continuous covariates. Time dependency is examined graphically by using scaled Schoenfeld residuals (see Grambsch and Therneau(1994)). It is shown that the inclusion of diagnoses significantly improves the estimated transition probabilities on which premiums are based. As an alternative approach a piecewise exponential model is fitted and compared to the semiparametric hazard model

The inception selection effect of diagnosis in a German long term care portfolio

In this paper we quantify the inception selection effect of diagnosis in a large German long term care (LTC) portfolio. First we are interested in modeling transition intensities, which will then be used in a multistate model set up to estimate transition probabilities. Finally we use these probability estimates as the basis for premium calculations. For the estimation of transition intensities we use semiparametric hazard models introduced by Cox (1972) allowing the inclusion of diagnosis as explanatory variable. Using modern model diagnostics we build a statistical model for the transition intensities and show that the resulting transition probability estimates including diagnosis perform better than when diagnosis is neglected. To quantify the inception selection effect of diagnosis we show how these improved transition probability estimates affect the premiums in an LTC insurance contract. In particular for younger age groups higher premiums are obtained when the diagnoses are taken into account compared to a model which disregards diagnosis. This demonstrates the actuarial need for allowing for an inception selection effect of diagnosis

Modelling count data with overdispersion and spatial effects

In this paper we consider regression models for count data allowing for overdispersion in a Bayesian framework. We account for unobserved heterogeneity in the data in two ways. On the one hand, we consider more flexible models than a common Poisson model allowing for overdispersion in different ways. In particular, the negative binomial and the generalized Poisson distribution are addressed where overdispersion is modelled by an additional model parameter. Further, zero-inflated models in which overdispersion is assumed to be caused by an excessive number of zeros are discussed. On the other hand, extra spatial variability in the data is taken into account by adding spatial random effects to the models. This approach allows for an underlying spatial dependency structure which is modelled using a conditional autoregressive prior based on Pettitt et al. (2002). In an application the presented models are used to analyse the number of invasive meningococcal disease cases in Germany in the year 2004. Models are compared according to the deviance information criterion (DIC) suggested by Spiegelhalter et al. (2002) and using proper scoring rules, see for example Gneiting and Raftery (2004). We observe a rather high degree of overdispersion in the data which is captured best by the GP model when spatial effects are neglected. While the addition of spatial effects to the models allowing for overdispersion gives no or only little improvement, a spatial Poisson model is to be preferred over all other models according to the considered criteria