An essential resource for constructing and analyzing advanced
actuarial models
Loss Models: Further Topics presents extended coverage of
modeling through the use of tools related to risk theory, loss
distributions, and survival models. The book uses these methods to
construct and evaluate actuarial models in the fields of insurance
and business. Providing an advanced study of actuarial methods, the
book features extended discussions of risk modeling and risk
measures, including Tail-Value-at-Risk. Loss Models: Further
Topics contains additional material to accompany the Fourth
Edition of Loss Models: From Data to Decisions, such as:
* Extreme value distributions
* Coxian and related distributions
* Mixed Erlang distributions
* Computational and analytical methods for aggregate claim
models
* Counting processes
* Compound distributions with time-dependent claim amounts
* Copula models
* Continuous time ruin models
* Interpolation and smoothing
The book is an essential reference for practicing actuaries and
actuarial researchers who want to go beyond the material required
for actuarial qualification. Loss Models: Further Topics is
also an excellent resource for graduate students in the actuarial
field.
Autorentext
STUART A. KLUGMAN, PhD, is Staff Fellow (Education) at the Society of Actuaries and Principal Financial Group Distinguished Professor Emeritus of Actuarial Science at Drake University. Dr. Klugman is a two-time recipient of the Society of Actuaries' Presidential Award.
HARRY H. PANJER, PhD, is Distinguished Professor Emeritus in the Department of Statistics and Actuarial Science at the University of Waterloo, Canada. Dr. Panjer was previously president of the Canadian Institute of Actuaries and the Society of Actuaries.
GORDON E. WILLMOT, PhD, is Munich Re Chair in Insurance and Professor in the Department of Statistics and Actuarial Science at the University of Waterloo, Canada. Dr. Willmot has authored more than eighty-five articles in the areas of risk theory, queuing theory, distribution theory, and stochastic modeling in insurance.
Klappentext
An essential resource for constructing and analyzing advanced actuarial models
Loss Models: Further Topics presents extended coverage of modeling through the use of tools related to risk theory, loss distributions, and survival models. The book uses these methods to construct and evaluate actuarial models in the fields of insurance and business. Providing an advanced study of actuarial methods, the book features extended discussions of risk modeling and risk measures, including Tail-Value-at-Risk. Loss Models: Further Topics contains additional material to accompany the Fourth Edition of Loss Models: From Data to Decisions, such as:
- Extreme value distributions
- Coxian and related distributions
- Mixed Erlang distributions
- Computational and analytical methods for aggregate claim models
- Counting processes
- Compound distributions with time-dependent claim amounts
- Copula models
- Continuous time ruin models
- Interpolation and smoothing
The book is an essential reference for practicing actuaries and actuarial researchers who want to go beyond the material required for actuarial qualification. Loss Models: Further Topics is also an excellent resource for graduate students in the actuarial field.
Inhalt
Preface xi
1 Introduction 1
2 Coxian and related distributions 3
2.1 Introduction 3
2.2 Combinations of exponentials 4
2.3 Coxian-2 distributions 7
3 Mixed Erlang distributions 11
3.1 Introduction 11
3.2 Members of the mixed Erlang class 12
3.3 Distributional properties 18
3.4 Mixed Erlang claim severity models 22
4 Extreme value distributions 23
4.1 Introduction 23
4.2 Distribution of the maximum 25
4.2.1 From a fixed number of losses 25
4.2.2 From a random number of losses 27
4.3 Stability of the maximum of the extreme value distribution 29
4.4 The Fisher-Tippett theorem 30
4.5 Maximum domain of attraction 32
4.6 Generalized Pareto distributions 34
4.7 Stability of excesses of the generalized Pareto 36
4.8 Limiting distributions of excesses 37
4.9 Parameter estimation 39
4.9.1 Maximum likelihood estimation from the extreme value distribution 39
4.9.2 Maximum likelihood estimation for the generalized Pareto distribution 42
4.9.3 Estimating the Pareto shape parameter 44
4.9.4 Estimating extreme probabilities 47
4.9.5 Mean excess plots 49
4.9.6 Further reading 49
4.9.7 Exercises 49
5 Analytic and related methods for aggregate claim models 51
5.1 Introduction 51
5.2 Elementary approaches 53
5.3 Discrete analogues 58
5.4 Right-tail asymptotics for aggregate losses 63
5.4.1 Exercises 71
6 Computational methods for aggregate models 73
6.1 Recursive techniques for compound distributions 73
6.2 Inversion methods 75
6.2.1 Fast Fourier transform 75
6.2.2 Direct numerical inversion 78
6.3 Calculations with approximate distributions 80
6.3.1 Arithmetic distributions 80
6.3.2 Empirical distributions 83
6.3.3 Piecewise linear cdf 84
6.3.4 Exercises 85
6.4 Comparison of methods 86
6.5 The individual risk model 87
6.5.1 Definition and notation 87
6.5.2 Direct calculation 88
6.5.3 Recursive calculation 89
7 Counting Processes 97
7.1 Nonhomogeneous birth processes 97
7.1.1 Exercises 112
7.2 Mixed Poisson processes 112
7.2.1 Exercises 116
8 Discrete Claim Count Models 119
8.1 Unification of the (a, b, 1) and mixed Poisson classes 119
8.2 A class of discrete generalized tail-based distributions 127
8.3 Higher order generalized tail-based distributions 134
8.4 Mixed Poisson properties of generalized tail-based distributions 139
8.5 Compound geometric properties of generalized tail-based distributions 146
8.5.1 Exercises 156
9 Compound distributions with time dependent claim amounts 159
9.1 Introduction 159
9.2 A model for inflation 163
9.3 A model for claim payment delays 173
10 Copula models 187
10.1 Introduction 187
10.2 Sklar's theorem and copulas 188
10.3 Measures of dependency 189
10.3.1 Spearman's rho 190
10.3.2 Kendall's tau 190
10.4 Tail dependence 191
10.5 Archimedean copulas 192
10.5.1 Exercise 197
10.6 Elliptical copulas 197
10.6.1 Exercise 199
10.7 Extreme value copulas 200
10.7.1 Exercises 202
10.8 Archimax copulas 203
10.9 Estimation of parameters 203
10.9.1 Introduction 203
10.9.2 Maximum likelihood estimation 204
10.9.3 Semiparametric estimation 206
10.9.4 The role of deductibles 206
10.9.5 Goodness-of-fit testing 208
10.9.6 An example 209
10.9.7 Exercise 210
10.10 Simulation from Copula Models 211
10.10.1 Simulating from the Gaussian copula 213
10.10.2 Simulating from the t copula 213
11 Continuous-time ruin models 215
11.1 Introduction 215
11.1.1 The Poisson process 215
11.1.2 The continuous-time problem 216
11.2 The adjustment coefficient and Lundberg's inequality 217
11.2.1 The adjustment coefficient 217
11.2.2 Lundberg's inequality 221
11.2.3 Exercis…