Understanding the effect of policyholders' risk profile on the number and the amount of claims, as well as the dependence among different types of claims, are critical to insurance ratemaking and IBNR-type reserving. To accurately quantify such features, it is essential to develop a regression model which is flexible, interpretable and statistically tractable. In this thesis, we first propose a highly flexible nonlinear regression model, namely the logit-weighted reduced mixture of experts (LRMoE) models, for multivariate claim frequencies or severities distributions. The LRMoE model is interpretable as it has two components: Gating functions to classify policyholders into various latent sub-classes and Expert functions to govern the distributional properties of the claims. The model is also flexible to fit any types of claim data accurately and hence minimize the issue of model selection. Model implementation is then illustrated in two ways using a real automobile insurance dataset from a major European insurance company. We first fit the multivariate claim frequencies using an Erlang count expert function. Apart from showing excellent fitting results, we can interpret the fitted model in an insurance perspective and visualize the relationship between policyholders' information and their risk level. We further demonstrate how the fitted model may be useful for insurance ratemaking. The second illustration deals with insurance loss severity data that often exhibits heavy-tail behavior. Using a Transformed Gamma expert function, our model is applicable to fit the severity and reporting delay components of the dataset, which is ultimately shown to be useful and crucial for an adequate prediction of IBNR reserve. After that, we further extend the fitting algorithm to efficiently fit the LRMoE to random censored and truncated regression data. Such an extended algorithm is then found useful and important for broader actuarial applications such as unbiased claim reporting delay modeling and deductible ratemaking.

In the Property and Casualty (P&C) ratemaking process, it is critical to understand the effect of policyholders' risk profile to the number and amount of claims, the dependence among various business lines and the claim distributions. To include all the above features, it is essential to develop a regression model which is flexible and theoretically justified. Motivated by the issues above, we propose a class of logit-weighted reduced mixture of experts (LRMoE) models for multivariate claim frequencies or severities distributions. LRMoE is interpretable, as it has two components: Gating functions, which classify policyholders into various latent sub-classes; and Expert functions, which govern the distributional properties of the claims. Also, upon the development of denseness theory in regression setting, we show that LRMoE can be fully flexible to capture any distributional, dependence and regression structures. Further, the mathematical tractability of the LRMoE is guaranteed since it satisfies various marginalization and moment properties. Finally, we discuss some special choices of expert functions that make the corresponding LRMoE fully flexible. In the subsequent paper (Fung et al. 2018a), we will focus on the estimation and application aspects of the LRMoE.

This paper focuses on the estimation and application aspects of the Erlang Count Logit-weighted Reduced Mixture of Experts model (EC-LRMoE), which is a fully flexible multivariate insurance claim frequency regression model proposed in Fung et al. (2018a). We first prove the identifiability property of the proposed model to ensure that it is a suitable candidate for statistical inference. An Expectation-Conditional-Maximization (ECM) algorithm is developed for efficient model calibrations. Three simulation studies are performed so that the effectiveness of the proposed ECM algorithm and the versatility of the proposed model can be examined. The applicability of the EC-LRMoE is shown through fitting an European automobile insurance dataset. Since the dataset contains several complex features, we find it necessary to adopt such a flexible model. Apart from showing excellent fitting results, we are able to interpret the fitted model in an insurance perspective and to visualize the relationship between policyholders' information and their risk level. Finally, we demonstrate how the fitted model may be useful for insurance ratemaking.

Non-life insurance pricing is the art of setting the price of an insurance policy, taking into consideration varoius properties of the insured object and the policy holder. Introduced by British actuaries generalized linear models (GLMs) have become today a the standard aproach for tariff analysis. The book focuses on methods based on GLMs that have been found useful in actuarial practice and provides a set of tools for a tariff analysis. Basic theory of GLMs in a tariff analysis setting is presented with useful extensions of standarde GLM theory that are not in common use. The book meets the European Core Syllabus for actuarial education and is written for actuarial students as well as practicing actuaries. To support reader real data of some complexity are provided at www.math.su.se/GLMbook.

This book provides a comprehensive introduction to actuarial mathematics, covering both deterministic and stochastic models of life contingencies, as well as more advanced topics such as risk theory, credibility theory and multi-state models. This new edition includes additional material on credibility theory, continuous time multi-state models, more complex types of contingent insurances, flexible contracts such as universal life, the risk measures VaR and TVaR. Key Features: Covers much of the syllabus material on the modeling examinations of the Society of Actuaries, Canadian Institute of Actuaries and the Casualty Actuarial Society. (SOA-CIA exams MLC and C, CSA exams 3L and 4.) Extensively revised and updated with new material. Orders the topics specifically to facilitate learning. Provides a streamlined approach to actuarial notation. Employs modern computational methods. Contains a variety of exercises, both computational and theoretical, together with answers, enabling use for self-study. An ideal text for students planning for a professional career as actuaries, providing a solid preparation for the modeling examinations of the major North American actuarial associations. Furthermore, this book is highly suitable reference for those wanting a sound introduction to the subject, and for those working in insurance, annuities and pensions.

Developed from celebrated Harvard statistics lectures, Introduction to Probability provides essential language and tools for understanding statistics, randomness, and uncertainty. The book explores a wide variety of applications and examples, ranging from coincidences and paradoxes to Google PageRank and Markov chain Monte Carlo (MCMC). Additional application areas explored include genetics, medicine, computer science, and information theory. The print book version includes a code that provides free access to an eBook version. The authors present the material in an accessible style and motivate concepts using real-world examples. Throughout, they use stories to uncover connections between the fundamental distributions in statistics and conditioning to reduce complicated problems to manageable pieces. The book includes many intuitive explanations, diagrams, and practice problems. Each chapter ends with a section showing how to perform relevant simulations and calculations in R, a free statistical software environment.