Open Access 2023 | Open Access | Buch

# Statistical Foundations of Actuarial Learning and its Applications

verfasst von: Mario V. Wüthrich, Michael Merz

Verlag: Springer International Publishing

Buchreihe : Springer Actuarial

Open Access 2023 | Open Access | Buch

verfasst von: Mario V. Wüthrich, Michael Merz

Verlag: Springer International Publishing

Buchreihe : Springer Actuarial

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.

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Open Access

Abstract

This chapter presents an introduction to statistics and statistical modeling in insurance and actuarial science. We discuss the statistical modeling cycle, we introduce the basic tools from probability theory and statistics, and we conclude with an exploratory data analysis of insurance claim sizes.

Open Access

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This chapter introduces and discusses the exponential family (EF) and the exponential dispersion family (EDF). The EF and the EDF are by far the most important classes of distribution functions for regression modeling. They include, among others, the Gaussian, the binomial, the Poisson, the gamma, the inverse Gaussian distributions, as well as Tweedie’s models. We introduce these families of distribution functions, discuss their properties and provide several examples. Moreover, we introduce the Kullback–Leibler (KL) divergence and the Bregman divergence, which are important tools in model evaluation and model selection.

Open Access

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This chapter is on classical statistical decision theory. It is an important chapter for historical reasons, but it also provides the right mathematical grounding and intuition for more modern statistical tools from data science and machine learning. In particular, we discuss maximum likelihood estimation (MLE), unbiasedness, consistency and asymptotic normality of MLEs in this chapter.

Open Access

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This chapter is the core theoretical chapter on predictive modeling, forecast evaluation and model selection. The main problem in actuarial modeling is to forecast and price future claims. For this, we build predictive models, and this chapter deals with assessing and ranking these predictive models. We therefore introduce the mean squared error of prediction (MSEP) and, more generally, the expected generalization loss (GL) to assess predictive models. This chapter is complemented by a more decision-theoretic approach to forecast evaluation, it discusses deviance losses, proper scoring, elicitability, forecast dominance, cross-validation, Akaike’s information criterion (AIC) and we give an introduction to the bootstrap simulation method.

Open Access

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This chapter discusses state-of-the-art statistical modeling in insurance and actuarial science, which is the generalized linear model (GLM). We discuss GLMs in the light of claim count and claim size modeling, we present feature engineering, model fitting, model selection, over-dispersion, zero-inflated claim counts problems, double GLMs, and insurance-specific issues such as the balance property for having unbiasedness.

Open Access

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This chapter summarizes some techniques that use Bayes’ theorem. These are classical Bayesian statistical models using, e.g., the Markov chain Monte Carlo (MCMC) method for model fitting. We discuss regularization of regression models such as ridge and LASSO regularization, which has a Bayesian interpretation, and we consider the Expectation-Maximization (EM) algorithm. The EM algorithm is a general purpose tool that can handle incomplete data settings. We illustrate this for different examples coming from mixture distributions, censored and truncated claims data.

Open Access

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The core of this book are deep learning methods and neural networks. This chapter considers deep feed-forward neural (FN) networks. We introduce the generic architecture of deep FN networks, and we discuss universality theorems of FN networks. We present network fitting, back-propagation, embedding layers for categorical variables and insurance-specific issues such as the balance property in network fitting, as well as network ensembling to reduce model uncertainty. This chapter is complemented by many examples on non-life insurance pricing, but also on mortality modeling, as well as tools that help to explain deep FN network regression results.

Open Access

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This chapter considers recurrent neural (RN) networks. These are special network architectures that are useful for time-series modeling, e.g., applied to time-series forecasting. We study the most popular RN networks which are the long short-term memory (LSTM) networks and the gated recurrent unit (GRU) networks. We apply these networks to mortality forecasting.

Open Access

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This chapter considers convolutional neural (CN) networks. These are special network architectures that are useful for time-series and spatial data modeling, e.g., applied to image recognition problems. Time-series and images have a natural topology, and CN networks try to benefit from this additional structure (over tabular data). We introduce these network architectures and provide insurance-relevant examples related to telematics data and mortality forecasting.

Open Access

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This chapter discusses natural language processing (NLP) which deals with regression modeling of non-tabular or unstructured text data. We explain how words can be embedded into low-dimension spaces that serve as numerical word encodings. These can then be used for text recognition, either using RN networks or attention layers. We give an example where we aim at predicting claim perils from claim descriptions.

Open Access

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This chapter presents a selection of different topics. We discuss forecasting under model uncertainty, deep quantile regression, deep composite regression and the LocalGLMnet which is an interpretable FN network architecture. Moreover, we provide a bootstrap example to assess prediction uncertainty, we discuss mixture density networks, and we give an outlook to studying variational inference.

Open Access

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This chapter is a technical chapter that discusses universality theorems for networks and sieve estimators, which are useful for studying asymptotic normality within a network architecture. In particular, this chapter is a useful outlook for studying variable selection in the framework of networks.

Open Access

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This chapter illustrates the data used in this book. These are a French motor third party liability (MTPL) claims data set, a Swedish motorcycle claims data set, a Wisconsin Local Government Property Insurance Fund data set, and a Swiss compulsory accident insurance data set.