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E-raamat: Statistical Foundations of Actuarial Learning and its Applications

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  • Formaat: EPUB+DRM
  • Sari: Springer Actuarial
  • Ilmumisaeg: 22-Nov-2022
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783031124099
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Statistical Foundations of Actuarial Learning and its ApplicationsSuurem pilt
  • Formaat: EPUB+DRM
  • Sari: Springer Actuarial
  • Ilmumisaeg: 22-Nov-2022
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783031124099
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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.



1 Introduction
1(12)
1.1 The Statistical Modeling Cycle
1(2)
1.2 Preliminaries on Probability Theory
3(4)
1.3 Lab: Exploratory Data Analysis
7(2)
1.4 Outline of This Book
9(4)
2 Exponential Dispersion Family
13(36)
2.1 Exponential Family
13(15)
2.1.1 Definition and Properties
13(5)
2.1.2 Single-Parameter Linear EF: Count Variable Examples
18(2)
2.1.3 Vector-Valued Parameter EF: Absolutely Continuous Examples
20(7)
2.1.4 Vector-Valued Parameter EF: Count Variable Example
27(1)
2.2 Exponential Dispersion Family
28(12)
2.2.1 Definition and Properties
28(3)
2.2.2 Exponential Dispersion Family Examples
31(3)
2.2.3 Tweedie's Distributions
34(3)
2.2.4 Steepness of the Cumulant Function
37(1)
2.2.5 Lab: Large Claims Modeling
38(2)
2.3 Information Geometry in Exponential Families
40(9)
2.3.1 Kullback--Leibler Divergence
40(2)
2.3.2 Unit Deviance and Bregman Divergence
42(7)
3 Estimation Theory
49(26)
3.1 Introduction to Decision Theory
49(2)
3.2 Parameter Estimation
51(5)
3.3 Unbiased Estimators
56(11)
3.3.1 Cramer--Rao Information Bound
56(6)
3.3.2 Information Bound in the Exponential Family Case
62(5)
3.4 Asymptotic Behavior of Estimators
67(8)
3.4.1 Consistency
67(2)
3.4.2 Asymptotic Normality
69(6)
4 Predictive Modeling and Forecast Evaluation
75(36)
4.1 Generalization Loss
75(20)
4.1.1 Mean Squared Error of Prediction
76(3)
4.1.2 Unit Deviances and Deviance Generalization Loss
79(9)
4.1.3 A Decision-Theoretic Approach to Forecast Evaluation
88(7)
4.2 Cross-Validation
95(11)
4.2.1 In-Sample and Out-of-Sample Losses
95(3)
4.2.2 Cross-Validation Techniques
98(5)
4.2.3 Akaike's Information Criterion
103(3)
4.3 Bootstrap
106(5)
4.3.1 Non-parametric Bootstrap Simulation
106(3)
4.3.2 Parametric Bootstrap Simulation
109(2)
5 Generalized Linear Models
111(96)
5.1 Generalized Linear Models and Log-Likelihoods
112(14)
5.1.1 Regression Modeling
112(1)
5.1.2 Definition of Generalized Linear Models
113(2)
5.1.3 Link Functions and Feature Engineering
115(1)
5.1.4 Log-Likelihood Function and Maximum Likelihood Estimation
116(6)
5.1.5 Balance Property Under the Canonical Link Choice
122(1)
5.1.6 Asymptotic Normality
123(1)
5.1.7 Maximum Likelihood Estimation and Unit Deviances
124(2)
5.2 Actuarial Applications of Generalized Linear Models
126(15)
5.2.1 Selection of a Generalized Linear Model
126(1)
5.2.2 Feature Engineering
127(5)
5.2.3 Offsets
132(1)
5.2.4 Lab: Poisson GLM for Car Insurance Frequencies
133(8)
5.3 Model Validation
141(39)
5.3.1 Residuals and Dispersion
141(4)
5.3.2 Hypothesis Testing
145(2)
5.3.3 Analysis of Variance
147(3)
5.3.4 Lab: Poisson GLM for Car Insurance Frequencies, Revisited
150(5)
5.3.5 Over-Dispersion in Claim Counts Modeling
155(7)
5.3.6 Zero-Inflated Poisson Model
162(5)
5.3.7 Lab: Gamma GLM for Claim Sizes
167(6)
5.3.8 Lab: Inverse Gaussian GLM for Claim Sizes
173(3)
5.3.9 Log-Normal Model for Claim Sizes: A Short Discussion
176(4)
5.4 Quasi-Likelihoods
180(2)
5.5 Double Generalized Linear Model
182(8)
5.5.1 The Dispersion Submodel
182(1)
5.5.2 Saddlepoint Approximation
183(3)
5.5.3 Residual Maximum Likelihood Estimation
186(1)
5.5.4 Lab: Double GLM Algorithm for Gamma Claim Sizes
187(2)
5.5.5 Tweedie's Compound Poisson GLM
189(1)
5.6 Diagnostic Tools
190(5)
5.6.1 The Hat Matrix
190(2)
5.6.2 Case Deletion and Generalized Cross-Validation
192(3)
5.7 Generalized Linear Models with Categorical Responses
195(3)
5.7.1 Logistic Categorical Generalized Linear Model
195(1)
5.7.2 Maximum Likelihood Estimation in Categorical Models
196(2)
5.8 Further Topics of Regression Modeling
198(9)
5.8.1 Longitudinal Data and Random Effects
198(1)
5.8.2 Regression Models Beyond the GLM Framework
199(3)
5.8.3 Quantile Regression
202(5)
6 Bayesian Methods, Regularization and Expectation-Maximization
207(60)
6.1 Bayesian Parameter Estimation
207(3)
6.2 Regularization
210(20)
6.2.1 Maximal a Posterior Estimator
210(2)
6.2.2 Ridge vs. LASSO Regularization
212(3)
6.2.3 Ridge Regression
215(2)
6.2.4 LASSO Regularization
217(9)
6.2.5 Group LASSO Regularization
226(4)
6.3 Expectation-Maximization Algorithm
230(18)
6.3.1 Mixture Distributions
230(2)
6.3.2 Incomplete and Complete Log-Likelihoods
232(1)
6.3.3 Expectation-Maximization Algorithm for Mixtures
233(7)
6.3.4 Lab: Mixture Distribution Applications
240(8)
6.4 Truncated and Censored Data
248(19)
6.4.1 Lower-Truncation and Right-Censoring
248(2)
6.4.2 Parameter Estimation Under Right-Censoring
250(4)
6.4.3 Parameter Estimation Under Lower-Truncation
254(10)
6.4.4 Composite Models
264(3)
7 Deep Learning
267(114)
7.1 Deep Learning and Representation Learning
267(2)
7.2 Generic Feed-Forward Neural Networks
269(24)
7.2.1 Construction of Feed-Forward Neural Networks
269(5)
7.2.2 Universality Theorems
274(4)
7.2.3 Gradient Descent Methods
278(15)
7.3 Feed-Forward Neural Network Examples
293(5)
7.3.1 Feature Pre-processing
293(2)
7.3.2 Lab: Poisson FN Network for Car Insurance Frequencies
295(3)
7.4 Special Features in Networks
298(44)
7.4.1 Special Purpose Layers
298(7)
7.4.2 The Balance Property in Neural Networks
305(10)
7.4.3 Boosting Regression Models with Network Features
315(4)
7.4.4 Network Ensemble Learning
319(21)
7.4.5 Identifiability in Feed-Forward Neural Networks
340(2)
7.5 Auto-encoders
342(15)
7.5.1 Standardization of the Data Matrix
343(1)
7.5.2 Introduction to Auto-encoders
343(1)
7.5.3 Principal Components Analysis
344(3)
7.5.4 Lab: Lee-Carter Mortality Model
347(4)
7.5.5 Bottleneck Neural Network
351(6)
7.6 Model-Agnostic Tools
357(19)
7.6.1 Variable Permutation Importance
357(2)
7.6.2 Partial Dependence Plots
359(6)
7.6.3 Interaction Strength
365(1)
7.6.4 Local Model-Agnostic Methods
366(1)
7.6.5 Marginal Attribution by Conditioning on Quantiles
366(10)
7.7 Lab: Analysis of the Fitted Networks
376(5)
8 Recurrent Neural Networks
381(26)
8.1 Motivation for Recurrent Neural Networks
381(2)
8.2 Plain-Vanilla Recurrent Neural Network
383(7)
8.2.1 Recurrent Neural Network Layer
383(2)
8.2.2 Deep Recurrent Neural Network Architectures
385(2)
8.2.3 Designing the Network Output
387(1)
8.2.4 Time-Distributed Layer
388(2)
8.3 Special Recurrent Neural Networks
390(4)
8.3.1 Long Short-Term Memory Network
390(2)
8.3.2 Gated Recurrent Unit Network
392(2)
8.4 Lab: Mortality Forecasting with RN Networks
394(13)
8.4.1 Lee--Carter Model, Revisited
394(8)
8.4.2 Direct LSTM Mortality Forecasting
402(5)
9 Convolutional Neural Networks
407(18)
9.1 Plain-Vanilla Convolutional Neural Network Layer
407(6)
9.1.1 Input Tensors and Channels
408(1)
9.1.2 Generic Convolutional Neural Network Layer
408(3)
9.1.3 Example: Time-Series Analysis and Image Recognition
411(2)
9.2 Special Purpose Tools for Convolutional Neural Networks
413(3)
9.2.1 Padding with Zeros
413(1)
9.2.2 Stride
414(1)
9.2.3 Dilation
414(1)
9.2.4 Pooling Layer
415(1)
9.2.5 Flatten Layer
416(1)
9.3 Convolutional Neural Network Architectures
416(9)
9.3.1 Illustrative Example of a CN Network Architecture
416(2)
9.3.2 Lab: Telematics Data
418(4)
9.3.3 Lab: Mortality Surface Modeling
422(3)
10 Natural Language Processing
425(28)
10.1 Feature Pre-processing and Bag-of-Words
425(4)
10.2 Word Embeddings
429(11)
10.2.1 Word to Vector Algorithms
430(6)
10.2.2 Global Vectors Algorithm
436(4)
10.3 Lab: Predictive Modeling Using Word Embeddings
440(5)
10.4 Lab: Deep Word Representation Learning
445(3)
10.5 Outlook: Creating Attention
448(5)
11 Selected Topics in Deep Learning
453(84)
11.1 Deep Learning Under Model Uncertainty
453(23)
11.1.1 Recap: Tweedie's Family
454(4)
11.1.2 Lab: Claim Size Modeling Under Model Uncertainty
458(8)
11.1.3 Lab: Deep Dispersion Modeling
466(6)
11.1.4 Pseudo Maximum Likelihood Estimator
472(4)
11.2 Deep Quantile Regression
476(7)
11.2.1 Deep Quantile Regression: Single Quantile
477(1)
11.2.2 Deep Quantile Regression: Multiple Quantiles
478(1)
11.2.3 Lab: Deep Quantile Regression
479(4)
11.3 Deep Composite Model Regression
483(9)
11.3.1 Joint Elicitability of Quantiles and Expected Shortfalls
483(4)
11.3.2 Lab: Deep Composite Model Regression
487(5)
11.4 Model Uncertainty: A Bootstrap Approach
492(3)
11.5 LocalGLMnet: An Interpretable Network Architecture
495(18)
11.5.1 Definition of the LocalGLMnet
495(2)
11.5.2 Variable Selection in LocalGLMnets
497(2)
11.5.3 Lab: LocalGLMnet for Claim Frequency Modeling
499(8)
11.5.4 Variable Selection Through Regularization of the LocalGLMnet
507(2)
11.5.5 Lab: LASSO Regularization of LocalGLMnet
509(4)
11.6 Selected Applications
513(24)
11.6.1 Mixture Density Networks
513(8)
11.6.2 Estimation of Conditional Expectations
521(9)
11.6.3 Bayesian Networks: An Outlook
530(7)
12 Appendix A: Technical Results on Networks
537(16)
12.1 Universality Theorems
537(3)
12.2 Consistency and Asymptotic Normality
540(6)
12.3 Functional Limit Theorem
546(3)
12.4 Hypothesis Testing
549(4)
13 Appendix B: Data and Examples
553(24)
13.1 French Motor Third Party Liability Data
553(11)
13.2 Swedish Motorcycle Data
564(6)
13.3 Wisconsin Local Government Property Insurance Fund
570(3)
13.4 Swiss Accident Insurance Data
573(4)
Bibliography 577(18)
Index 595
Mario Wüthrich is Professor in the Department of Mathematics at ETH Zurich, Honorary Visiting Professor at City, University of London (2011-2022), Honorary Professor at University College London (2013-2019), and Adjunct Professor at University of Bologna (2014-2016). He holds a Ph.D. in Mathematics from ETH Zurich (1999). From 2000 to 2005, he held an actuarial position at Winterthur Insurance, Switzerland. He is Actuary SAA (2004), served on the board of the Swiss Association of Actuaries (2006-2018), and is Editor-in-Chief of ASTIN Bulletin (since 2018).  Michael Merz has been the holder of the Chair of Mathematics and Statistics in Economics at the University of Hamburg since 2009. After completing his doctorate at the University of Tübingen on a topic from the field of risk theory, he worked from 2004 to 2006 in the actuarial department of Baloise Insurance Group and gained practical experience in the areas of quantitative risk management and Actuarial Science. He then worked until 2009 as a Juniorprofessor for statistics, risk and insurance at the University of Tübingen. Since the beginning of 2018 he is editor of ASTIN Bulletin. 
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