Loss Models

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 8 3 Mixed Erlang distributions 11 3.1 Introduction 11 3.2 Members of the mixed Erlang class 13 3.3 Distributional properties 18 3.4 Mixed Erlang claim severity models 22 4 Extreme value distributions 25 4.1 Introduction 25 4.2 Distribution of the maximum 27 4.3 Stability of the maximum of the extreme value distribution 31 4.4 The Fisher--Tippett theorem 33 4.5 Maximum domain of attraction 34 4.6 Generalized Pareto distributions 37 4.7 Stability of excesses of the generalized Pareto 39 4.8 Limiting distributions of excesses 40 4.9 Parameter estimation 42 5 Analytic and related methods for aggregate claim models 55 5.1 Introduction 55 5.2 Elementary approaches 58 5.3 Discrete analogues 63 5.4 Right-tail asymptotics for aggregate losses 67 6 Computational methods for aggregate models 77 6.1 Recursive techniques for compound distributions 77 6.2 Inversion methods 79 6.3 Calculations with approximate distributions 84 6.4 Comparison of methods 90 6.5 The individual risk model 91 7 Counting Processes 101 7.1 Nonhomogeneous birth processes 101 7.2 Mixed Poisson processes 117 8 Discrete Claim Count Models 125 8.1 Unification of the (a, b, 1) and mixed Poisson classes 125 8.2 A class of discrete generalized tail-based distributions 133 8.3 Higher order generalized tail-based distributions 140 8.4 Mixed Poisson properties of generalized tail-based distributions 146 8.5 Compound geometric properties of generalized tail-based distributions 153 9 Compound distributions with time dependent claim amounts 165 9.1 Introduction 165 9.2 A model for inflation 169 9.3 A model for claim payment delays 180 10 Copula models 193 10.1 Introduction 193 10.2 Sklar's theorem and copulas 194 10.3 Measures of dependency 196 10.4 Tail dependence 197 10.5 Archimedean copulas 198 10.6 Elliptical copulas 203 10.7 Extreme value copulas 206 10.8 Archimax copulas 210 10.9 Estimation of parameters 210 10.10 Simulation from Copula Models 218 11 Continuous-time ruin models 223 11.1 Introduction 223 11.2 The adjustment coefficient and Lundberg's inequality 225 11.3 An integrodifferential equation 233 11.4 The maximum aggregate loss 238 11.5 Cramer's asymptotic ruin formula and Tijms' approximation 242 11.6 The Brownian motion risk process 249 11.7 Brownian motion and the probability of ruin 253 12 Interpolation and smoothing 259 12.1 Introduction 259 12.2 Interpolation with Splines 261 12.3 Extrapolating with splines 268 12.4 Smoothing with Splines 269 Appendix A: An inventory of continuous distributions 277 A.1 Introduction 277 A.2 transformed beta family 281 A.3 transformed gamma family 285 A.4 Distributions for large losses 288 A.5 Other distributions 289 A.6 Distributions with finite support 291 Appendix B: An inventory of discrete distributions 293 B.1 Introduction 293 B.2 The (a, b, 0) class 294 B.3 The (a, b, 1) class 295 B.4 The compound class 298 B.5 A hierarchy of discrete distributions 299 Appendix C: Discretization of the severity distribution 301 C.1 The method of rounding 301 C.2 Mean preserving 302 C.3 Undiscretization of a discretized distribution 302 Appendix D: Solutions to Exercises 305 D.1 Chapter 4 305 D.2 Chapter 5 307 D.3 Chapter 6 308 D.4 Chapter 7 310 D.5 Chapter 8 316 D.6 Chapter 10 321 D.7 Chapter 11 324 D.8 Chapter 12 344 References 349