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Author: Tsz Chai Fung Publisher: ISBN: Category : Languages : en Pages : 0
Book Description
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.
Author: Tsz Chai Fung Publisher: ISBN: Category : Languages : en Pages : 0
Book Description
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.
Author: Tsz Chai Fung Publisher: ISBN: Category : Languages : en Pages : 0
Book Description
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.
Author: Tsz Chai Fung Publisher: ISBN: Category : Languages : en Pages : 0
Book Description
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.
Author: Mario V. Wüthrich Publisher: Springer Nature ISBN: 303112409X Category : Mathematics Languages : en Pages : 611
Book Description
This open access book discusses the statistical modeling of insurance problems, a process which comprises data collection, data analysis and statistical model building to forecast insured events that may happen in the future. It presents the mathematical foundations behind these fundamental statistical concepts and how they can be applied in daily actuarial practice. Statistical modeling has a wide range of applications, and, depending on the application, the theoretical aspects may be weighted differently: here the main focus is on prediction rather than explanation. Starting with a presentation of state-of-the-art actuarial models, such as generalized linear models, the book then dives into modern machine learning tools such as neural networks and text recognition to improve predictive modeling with complex features. Providing practitioners with detailed guidance on how to apply machine learning methods to real-world data sets, and how to interpret the results without losing sight of the mathematical assumptions on which these methods are based, the book can serve as a modern basis for an actuarial education syllabus.
Author: Esbjörn Ohlsson Publisher: Springer Science & Business Media ISBN: 3642107915 Category : Mathematics Languages : en Pages : 181
Book Description
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.
Author: Andrew Gelman Publisher: CRC Press ISBN: 1439840954 Category : Mathematics Languages : en Pages : 677
Book Description
Now in its third edition, this classic book is widely considered the leading text on Bayesian methods, lauded for its accessible, practical approach to analyzing data and solving research problems. Bayesian Data Analysis, Third Edition continues to take an applied approach to analysis using up-to-date Bayesian methods. The authors—all leaders in the statistics community—introduce basic concepts from a data-analytic perspective before presenting advanced methods. Throughout the text, numerous worked examples drawn from real applications and research emphasize the use of Bayesian inference in practice. New to the Third Edition Four new chapters on nonparametric modeling Coverage of weakly informative priors and boundary-avoiding priors Updated discussion of cross-validation and predictive information criteria Improved convergence monitoring and effective sample size calculations for iterative simulation Presentations of Hamiltonian Monte Carlo, variational Bayes, and expectation propagation New and revised software code The book can be used in three different ways. For undergraduate students, it introduces Bayesian inference starting from first principles. For graduate students, the text presents effective current approaches to Bayesian modeling and computation in statistics and related fields. For researchers, it provides an assortment of Bayesian methods in applied statistics. Additional materials, including data sets used in the examples, solutions to selected exercises, and software instructions, are available on the book’s web page.
Author: Jan Beirlant Publisher: John Wiley & Sons ISBN: 0470012374 Category : Mathematics Languages : en Pages : 522
Book Description
Research in the statistical analysis of extreme values has flourished over the past decade: new probability models, inference and data analysis techniques have been introduced; and new application areas have been explored. Statistics of Extremes comprehensively covers a wide range of models and application areas, including risk and insurance: a major area of interest and relevance to extreme value theory. Case studies are introduced providing a good balance of theory and application of each model discussed, incorporating many illustrated examples and plots of data. The last part of the book covers some interesting advanced topics, including time series, regression, multivariate and Bayesian modelling of extremes, the use of which has huge potential.
Author: Christoph Molnar Publisher: Lulu.com ISBN: 0244768528 Category : Artificial intelligence Languages : en Pages : 320
Book Description
This book is about making machine learning models and their decisions interpretable. After exploring the concepts of interpretability, you will learn about simple, interpretable models such as decision trees, decision rules and linear regression. Later chapters focus on general model-agnostic methods for interpreting black box models like feature importance and accumulated local effects and explaining individual predictions with Shapley values and LIME. All interpretation methods are explained in depth and discussed critically. How do they work under the hood? What are their strengths and weaknesses? How can their outputs be interpreted? This book will enable you to select and correctly apply the interpretation method that is most suitable for your machine learning project.
Author: Tsz Chai Fung
Author: Tsz Chai Fung
Author: Tsz Chai Fung
Author: Tsz Chai Fung
Author: Mario V. Wüthrich
Author: Ludger Evers
Author: Esbjörn Ohlsson
Author: Institute of Electrical and Electronics Engineers
Author: Andrew Gelman
Author: Jan Beirlant
Author: Christoph Molnar