Mixture of bivariate Poisson regression models with an application to insurance

In a recent paper Bermúdez [2009] used bivariate Poisson regression models for ratemaking in car insurance, and included zero-inflated models to account for the excess of zeros and the overdispersion in the data set. In the present paper, we revisit this model in order to consider alternatives. We propose a 2-finite mixture of bivariate Poisson regression models to demonstrate that the overdispersion in the data requires more structure if it is to be taken into account, and that a simple zero-inflated bivariate Poisson model does not suffice. At the same time, we show that a finite mixture of bivariate Poisson regression models embraces zero-inflated bivariate Poisson regression models as a special case. Additionally, we describe a model in which the mixing proportions are dependent on covariates when modelling the way in which each individual belongs to a separate cluster. Finally, an EM algorithm is provided in order to ensure the models’ ease-of-fit. These models are applied to the same automobile insurance claims data set as used in Bermúdez [2009] and it is shown that the modelling of the data set can be improved considerably.Zero-inflation, Overdispersion, EM algorithm, Automobile insurance, A priori ratemaking.

Extended Poisson Models for Count Data With Inflated Frequencies

Count data often exhibits inflated counts for zero. There are numerous papers in the literature that show how to fit Poisson regression models that account for the zero inflation. However, in many situations the frequencies of zero and of some other value k tends to be higher than the Poisson model can fit appropriately. Recently, Sheth-Chandra (2011), Lin and Tsai (2012) introduced a mixture model to account for the inflated frequencies of zero and k. In this dissertation, we study basic properties of this mixture model and parameter estimation for grouped and ungrouped data. Using stochastic representation we show how the EM algorithm can be adapted to obtain maximum likelihood estimates of the parameters. We derive the observed information matrix which yields standard errors of the EM estimates using ideas from Louis (1982). We also derive asymptotic distributions to test significance of the inflation points. We use real life examples to illustrate the procedure of fitting our model via EM algorithm. The second part of this dissertation deals with a generalization of this mixture model where the one parameter Poisson distribution is replaced by a two parameter Conway-Maxwell-Poisson (CMP) distribution, which unlike the Poisson distribution accounts for both over and under dispersion in the count data. The CMP distribution has recently gained popularity, and a CMP model for zero inflated count data was introduced by Sellers and Raim (2016). We discuss properties of the CMP distribution and propose a new mixture distribution, namely zero and k inflated Conway-Maxwell-Poisson (ZkICMP) to address inflated counts with over and under dispersions. We develop regression models based on ZkICMP and discuss parameter estimation using analytical and numerical methods. Finally, we compare goodness of fit of inflated and standard models on simulated and real life data examples

A finite mixture of bivariate Poisson regression models with an application to insurance ratemaking

Bivariate Poisson regression models for ratemaking in car insurance have been previously used. They included zero-inflated models to account for the excess of zeros and the overdispersion in the data set. These models are now revisited in order to consider alternatives. A 2-finite mixture of bivariate Poisson regression models is used to demonstrate that the overdispersion in the data requires more structure if it is to be taken into account, and that a simple zero-inflated bivariate Poisson model does not suffice. At the same time, it is shown that a finite mixture of bivariate Poisson regression models embraces zero-inflated bivariate Poisson regression models as a special case. Finally, an EM algorithm is provided in order to ensure the models' ease-of-fit. These models are applied to an automobile insurance claims data set and it is shown that the modeling of the data set can be improved considerably

A Flexible Zero-Inflated Poisson Regression Model

A practical problem often encountered with observed count data is the presence of excess zeros. Zero-inflation in count data can easily be handled by zero-inflated models, which is a two-component mixture of a point mass at zero and a discrete distribution for the count data. In the presence of predictors, zero-inflated Poisson (ZIP) regression models are, perhaps, the most commonly used. However, the fully parametric ZIP regression model could sometimes be restrictive, especially with respect to the mixing proportions. Taking inspiration from some of the recent literature on semiparametric mixtures of regressions models for flexible mixture modeling, we propose a semiparametric ZIP regression model. We present an EM-like algorithm for estimation and a summary of asymptotic properties of the estimators. The proposed semiparametric models are then applied to a data set involving clandestine methamphetamine laboratories and Alzheimer\u27s disease

EM Estimation for Zero- and \u3ci\u3ek\u3c/i\u3e-Inflated Poisson Regression Model

Count data with excessive zeros are ubiquitous in healthcare, medical, and scientific studies. There are numerous articles that show how to fit Poisson and other models which account for the excessive zeros. However, in many situations, besides zero, the frequency of another count k tends to be higher in the data. The zero- and k-inflated Poisson distribution model (ZkIP) is appropriate in such situations The ZkIP distribution essentially is a mixture distribution of Poisson and degenerate distributions at points zero and k. In this article, we study the fundamental properties of this mixture distribution. Using stochastic representation, we provide details for obtaining parameter estimates of the ZkIP regression model using the Expectation-Maximization (EM) algorithm for a given data. We derive the standard errors of the EM estimates by computing the complete, missing, and observed data information matrices. We present the analysis of two real-life data using the methods outlined in the paper

The Doubly Inflated Poisson and Related Regression Models

Most real life count data consists of some values that are more frequent than allowed by the common parametric families of distributions. For data consisting of only excess zeros, in a seminal paper Lambert (1992) introduced Zero-Inflated Poisson (ZIP) model, which is a mixture model that accounts for the inflated zeros. In this thesis, two Doubly Inflated Poisson (DIP) probability models, DIP (p, λ) and DIP ( p1, p2, λ), are discussed for situations where there is another inflated value k \u3e 0 besides the inflated zeros. The distributional properties such as identifiability, moments, and conditional probabilities are also discussed for both probability models. For the data consisting of raw counts as well as grouped frequencies, we have considered parameter estimation using maximum likelihood (ML) and method of moments techniques. Efficiencies show that the ML estimators perform far better than the moment estimators. An application to DIP models is illustrated using data on patients\u27 length of stay in a hospital. Parameter estimation of DIP regression models using maximum likelihood approach is also discussed using data on dental cavities. Finally, we conclude with a brief introduction to two Doubly Inflated Negative Binomial (DINB) distributions and their related regression models

State-Space Models for Binomial Time Series with Excess Zeros

Count time series with excess zeros are frequently encountered in practice. In characterizing a time series of counts with excess zeros, two types of models are commonplace: models that assume a Poisson mixture distribution, and models that assume a binomial mixture distribution. Extensive work has been published dealing with modeling frameworks based on Poisson-type approaches, yet little has concentrated on binomial-type methods. To handle such data, we propose two general classes of time series models: a class of observation-driven ZIB (ODZIB) models, and a class of parameter-driven ZIB (PDZIB) models. The ODZIB model is formulated in the partial likelihood framework, which facilitates model fitting using standard statistical software for ZIB regression models. The PDZIB model is conveniently formulated in the state-space framework. For parameter estimation, we devise a Monte Carlo Expectation Maximization (MCEM) algorithm, with particle filtering and particle smoothing methods employed to approximate the intractable conditional expectations in the E-step of the algorithm. We investigate the efficacy of the proposed methodology in a simulation study, which compares the performance of the proposed ZIB models to their counterpart zero-inflated Poisson (ZIP) models in characterizing zero-inflated count time series. We also present a practical application pertaining to disease coding