Applications of Pascal Mixture Models to Insurance and Risk Management

This thesis studies the applications of Pascal mixture models in three closely related topics in insurance and risk management. The first topic is on the modeling of correlated frequencies of operational risk (OR) losses from financial institutions. We propose a copula-free approach for modeling correlated frequencies using an Erlang-based multivariate mixed Poisson distribution. Many properties possessed by this class of distributions are investigated and a tailor-made generalized expectation-maximization (EM) algorithm is derived for fitting purposes. The applicability of the proposed distribution is illustrated in an OR management context, where this class is used to model the OR loss. The accuracy of the proposed approach is analyzed using a modified real operational loss data set. The second topic is about multivariate count regression with application in modeling correlated claim frequencies. We propose a multivariate Pascal mixture regression model as an alternative to understand the association between multivariate count response variables and their covariates. We examine the many properties possessed by this class of regression. A generalized EM algorithm is derived for fitting purposes, which also provides the standard errors of the regression coefficients which are useful for inference. Its applicability is demonstrated by fitting an automobile insurance claim count data set. The third topic is about modeling and predicting the number of incurred but not reported (IBNR) claims in Property Casualty (P) insurance. We model the claim arrival process together with the reporting delays as a marked Cox process whose intensity function is governed by a hidden Markov chain. The associated reported claim process and IBNR claim process remain to be marked Cox processes with easily convertible intensity functions and marking distributions. Closed-form expressions for both the autocorrelation function (ACF) and the distributions of the numbers of reported claims and IBNR claims are derived. A generalized EM algorithm is obtained to estimate the model parameters. The proposed model is examined through simulation studies and is also applied to a real insurance claim data set. We compare the predictive distributions of our model with those of the over-dispersed Poisson model (ODP), a stochastic model that underpins the widely used chain-ladder method.Ph.D