Recursions for Convolutions and Compound Distributions with Insurance Applications (eBook)

Bjoern Sundt, Raluca Vernic (Autoren)

eBook Download: PDF

2009
XV, 345 Seiten
Springer Berlin (Verlag)
978-3-540-92900-0 (ISBN)

Infos

Since 1980, methods for recursive evaluation of aggregate claims distributions have received extensive attention in the actuarial literature.

This book gives a unified survey of the theory and is intended to be self-contained to a large extent. As the methodology is applicable also outside the actuarial field, it is presented in a general setting, but actuarial applications are used for motivation.

The book is divided into two parts. Part I is devoted to univariate distributions, whereas in Part II, the methodology is extended to multivariate settings.

Primarily intended as a monograph, this book can also be used as text for courses on the graduate level. Suggested outlines for such courses are given.

The book is of interest for actuaries and statisticians working within the insurance and finance industry, as well as for people in other fields like operations research and reliability theory.

Inhalt

Contents 6
Preface 12
Course Outline 16
Univariate Distributions 17
Introduction 18
Summary 18
Aggregate Claims Distributions 18
Some Notation and Conventions 21
Classes of Distributions and Functions 22
Convolutions 23
Mixed Distributions 24
Compound Distributions and Functions 26
Some Useful Transforms 27
Definitions and General Results 27
Convolutions 29
Discrete Distributions 30
Compound Distributions 31
Extension to Functions 32
Some Useful Operators 32
Stop Loss Premiums 35
Convergence of Infinite Series with Positive Terms 40
Further Remarks and References 43
Counting Distributions with Recursion of Order One 44
Summary 44
Geometric Distribution 44
Poisson Distribution 45
General Recursion 45
Application of Generating Functions 46
The Panjer Class 51
Panjer Recursions 51
Subclasses 54
An Alternative Recursion 58
Convolutions of a Distribution 60
The Sundt-Jewell Class 63
Characterisation 63
Recursions 65
Subclasses 66
Higher Order Panjer Classes 69
Characterisation 69
Shifted Counting Distribution 69
Counting Distribution with Range Bounded from Above 70
Extension to Severity Distributions in P10 71
Recursions 71
Thinning 73
Conversion to Severity Distributions in P11 74
Further Remarks and References 74
Compound Mixed Poisson Distributions 79
Summary 79
Mixed Poisson Distributions 79
Gamma Mixing Distribution 81
General Recursion 82
Finite Mixtures 83
Willmot Mixing Distribution 83
The Counting Distribution 86
Invariance Properties in the Willmot Class 89
Introduction 89
Scaling 89
Shifting 90
Truncating 91
Power Transform 91
Special Classes of Mixing Distributions 93
Shifted Pareto Distribution 93
Pareto Distribution 93
Truncated Normal Distribution 94
Inverse Gauss Distribution 95
Transformed Gamma Distribution 96
Further Remarks and References 97
Infinite Divisibility 99
Summary 99
Definitions and Properties 99
Characterisation 101
Mixed Poisson Distributions 103
Infinitely Divisible Mixing Distribution 103
Mixing Distribution in P10 105
De Pril Transforms of Infinitely Divisible Distributions in P10 106
Definition and Properties 106
Characterisation of Infinitely Divisible Distributions in Terms of De Pril Transforms 111
Further Remarks and References 112
Counting Distributions with Recursion of Higher Order 114
Summary 114
Compound Distributions 114
Convolutions of the Severity Distribution 120
The Rk Classes 121
Definitions and Characterisation 121
Compound Distributions 124
Distributions in P10 on the Range { 0,1,2,…,k} 125
Convolutions 125
Cumulants 132
Counting Distributions with Rational Generating Function 135
Further Remarks and References 136
De Pril Transforms of Distributions in P10 139
Summary 139
General Results 139
The Rk Classes 142
Further Remarks and References 145
Individual Models 146
Summary 146
De Pril's Methods 146
Dhaene-Vandebroek's Method 148
De Pril's Individual Model 149
Collective Approximations 151
Further Remarks and References 156
Cumulative Functions and Tails 160
Summary 160
General Results 160
Convolutions 164
Compound Distributions 165
De Pril Transforms 166
The Special Case b0 167
Further Remarks and References 171
Moments 172
Summary 172
Convolutions of a Distribution 172
Ordinary Moments 172
The Normal Distribution 176
Factorial Moments 178
Compound Distributions 179
General Results 179
Compound Panjer Distributions 185
Compound Poisson Distributions 189
Further Remarks and References 190
Approximations Based on De Pril Transforms 192
Summary 192
Introduction 192
Extension of Results for Distributions 194
Key Result 194
Applications 196
Error Bounds 197
Main Result 197
The Dhaene-De Pril Transform 200
Corollaries to the Main Result 201
Convolutions and Compound Distributions 203
The Generalised De Pril Individual Model 205
The De Pril Approximation 208
General Counting Distribution 208
Counting Distribution in R1 210
De Pril's Individual Model 210
The Kornya Approximation 211
General Counting Distribution 211
Counting Distribution in R1 212
De Pril's Individual Model 213
The Hipp Approximation 215
General Counting Distribution 215
Bernoulli Counting Distribution 216
De Pril's Individual Model 219
Numerical Example 220
Further Remarks and References 225
Extension to Distributions in P1_ 227
Summary 227
De Pril Transforms 227
Definition 227
Extension of Results 229
Error Bounds 230
Further Remarks and References 231
Allowing for Negative Severities 232
Summary 232
Introduction 232
Binomial Counting Distribution 233
Poisson Counting Distribution 235
Negative Binomial Counting Distribution 237
Further Remarks and References 238
Underflow and Overflow 239
Summary 239
Simple Scaling 239
Exponential Scaling 239
Convolutions 241
Further Remarks and References 241
Multivariate Distributions 243
Introduction 244
Summary 244
Aggregate Claims Distributions 244
Vectors and Matrices 246
Induction Proofs in a Multivariate Setting 247
Classes of Distributions and Functions 248
Convolutions 248
Compound Distributions 249
Moments 251
Further Remarks and References 251
Multivariate Compound Distributions of Type 1 252
Summary 252
Covariances 252
Counting Distribution in the Panjer Class 253
Convolutions of a Distribution 258
Infinite Divisibility 261
Counting Distribution with Recursion of Higher Order 261
Multivariate Bernoulli Severity Distribution 263
Further Remarks and References 264
De Pril Transforms 266
Summary 266
Definitions 266
The Rk Classes 267
Infinite Divisibility 269
Extension to Functions in Fm0 270
Vectors of Independent Random Subvectors 270
Vectors of Linear Combinations of Independent Random Variables 274
Individual Models 278
Further Remarks and References 280
Moments 282
Summary 282
Convolutions of a Distribution 282
Moments 282
The Multinormal Distribution 286
Compound Distributions 288
General Results 288
Compound Panjer Distributions 290
Further Remarks and References 293
Approximations Based on De Pril Transforms 294
Summary 294
Approximations 294
Error Bounds 294
Further Remarks and References 295
Multivariate Compound Distributions of Type 2 296
Summary 296
Main Framework 296
Recursions of Higher Order 301
Multivariate Compound Distributions of Type 3 302
Further Remarks and References 302
Compound Mixed Multivariate Poisson Distributions 303
Summary 303
Multivariate Poisson Distributions 303
Extension to Mixed Distributions 306
Gamma Mixing Distribution 307
Compound Distributions with Univariate Counting Distribution 310
General Recursion 310
Willmot Mixing Distribution 311
The Univariate Mixed Counting Distribution 314
Compound Distributions with Multivariate Counting Distribution 317
The Multivariate Counting Distribution 317
General Design 317
The Special Design 318
The Special Case µ=0 324
Special Classes of Mixing Distributions 326
Shifted Pareto distribution 326
Pareto Distribution 327
Truncated Normal Distribution 328
Inverse Gauss Distribution 329
Transformed Gamma Distribution 330
Further Remarks and References 331
References 332
List of Notation 342
Author Index 343
Subject Index 346

Produkt-Details

• Erscheint lt. Verlag: 21.4.2009
• Reihe/Serie: EAA Series
• Reihe/Serie: EAA Series
• Zusatzinfo: XV, 345 p. 3 illus.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Technik
• Themenwelt: Wirtschaft ► Allgemeines / Lexika
• Themenwelt: Wirtschaft ► Betriebswirtschaft / Management ► Finanzierung
• Schlagworte: Approximation • convolution • Counting • Distribution • Evaluation • Field • Finance • Finite • Form • Function • Functions • Operations Research • Poisson distribution • Quantitative Finance • Recursion • Reliability
• ISBN-10: 3-540-92900-2 / 3540929002
• ISBN-13: 978-3-540-92900-0 / 9783540929000