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By: Bjoern Sundt, Raluca Vernic

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Paperback | 3 April 2009

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364 Pages

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23.5 x 15.88 x 1.27

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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.

Industry Reviews

From the reviews: "The goal of this book is to give a unified survey of the existing literature on this topic. ... Each chapter starts with a summary and ends with a section of further remarks and references. ... this book will surely become an important reference source for both researchers ... and practitioners working in the insurance and finance industry. This book can be used for a course ... on the topic of recursions. ... In sum, this is a worthwhile book on an important topic." (Qihe Tang, Mathematical Reviews, Issue 2010 h)

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You Can Find This Book In

Non-Fiction Computing & I.T.Computer Science Computer Architecture & Logic Design Mathematics Probability & Statistics Applied Mathematics Optimisation Game Theory Accounting & Finance Finance Banking

Univariate Distributions
Introduction	p. 3
Aggregate Claims Distributions	p. 3
Some Notation and Conventions	p. 6
Classes of Distributions and Functions	p. 7
Convolutions	p. 8
Mixed Distributions	p. 9
Compound Distributions and Functions	p. 11
Some Useful Transforms	p. 12
Definitions and General Results	p. 12
Convolutions	p. 14
Discrete Distributions	p. 15
Compound Distributions	p. 16
Extension to Functions	p. 17
Some Useful Operators	p. 17
Stop Loss Premiums	p. 20
Convergence of Infinite Series with Positive Terms	p. 25
Counting Distributions with Recursion of Order One	p. 29
Geometric Distribution	p. 29
Poisson Distribution	p. 30
General Recursion	p. 30
Application of Generating Functions	p. 31
The Panjer Class	p. 36
Panjer Recursions	p. 36
Subclasses	p. 39
An Alternative Recursion	p. 43
Convolutions of a Distribution	p. 45
The Sundt-Jewell Class	p. 48
Characterisation	p. 48
Recursions	p. 50
Subclasses	p. 51
Higher Order Panjer Classes	p. 54
Characterisation	p. 54
Shifted Counting Distribution	p. 54
Counting Distribution with Range Bounded from Above	p. 55
Extension to Severity Distributions in P10	p. 56
Recursions	p. 56
Thinning	p. 58
Conversion to Severity Distributions in P11	p. 59
Compound Mixed Poisson Distributions	p. 65
Mixed Poisson Distributions	p. 65
Gamma Mixing Distributions	p. 67
General Recursion	p. 68
Finite Mixtures	p. 69
Willmot Mixing Distribution	p. 69
The Counting Distribution	p. 72
Invariance Properties in the Willmot Class	p. 75
Introduction	p. 75
Scaling	p. 75
Shifting	p. 76
Truncating	p. 77
Power Transform	p. 77
Special Classes of Mixing Distributions	p. 79
Shifted Pareto Distribution	p. 79
Pareto Distribution	p. 79
Truncated Normal Distribution	p. 80
Inverse Gauss Distribution	p. 81
Transformed Gamma Distribution	p. 82
Infinite Divisibility	p. 85
Definitions and Properties	p. 85
Characterisation	p. 87
Mixed Poisson Distributions	p. 89
Infinitely Divisible Mixing Distribution	p. 89
Mixing Distribution in P10	p. 91
De Pril Transforms of Infinitely Divisible Distributions in P10	p. 92
Definition and Properties	p. 92
Characterisation of Infinitely Divisible Distributions in Terms of De Pril Transforms	p. 97
Counting Distributions with Recursion of Higher Order	p. 101
Compound Distributions	p. 101
Convolutions of the Severity Distribution	p. 107
The Rk Classes	p. 108
Definitions and Characterisation	p. 108
Compound Distributions	p. 111
Distributions in P10 on the Range {0, 1, 2, ... ,k}	p. 112
Convolutions	p. 112
Cumulants	p. 119
Counting Distributions with Rational Generating Function	p. 122
De Pril Transforms of Distributions in P10	p. 127
General Results	p. 127
The Rk Classes	p. 130
Individual Models	p. 135
De Pril's Methods	p. 135
Dhaene-Vandebroek's Method	p. 137
De Pril's Individual Model	p. 138
Collective Approximations	p. 140
Cumulative Functions and Tails	p. 149
General Results	p. 149
Convolutions	p. 153
Compound Distributions	p. 154
De Pril Transforms	p. 155
The Special Case b $$ 0	p. 156
Moments	p. 161
Convolutions of a Distribution	p. 161
Ordinary Moments	p. 161
The Normal Distribution	p. 165
Factorial Moments	p. 167
Compound Distributions	p. 168
General Results	p. 168
Compound Panjer Distributions	p. 174
Compound Poisson Distributions	p. 178
Approximations Based on De Pril Transforms	p. 181
Introduction	p. 181
Extension of Results for Distributions	p. 183
Key Result	p. 183
Applications	p. 185
Error Bounds	p. 186
Main Result	p. 186
The Dhaene-De Pril Transform	p. 189
Corollaries to the Main Result	p. 190
Convolutions and Compound Distributions	p. 192
The Generalised De Pril Individual Model	p. 194
The De Pril Approximation	p. 197
General Counting Distribution	p. 197
Counting Distribution in R1	p. 199
De Pril's Individual Model	p. 199
The Kornya Approximation	p. 200
General Counting Distribution	p. 200
Counting Distribution in R1	p. 201
De Pril's Individual Model	p. 202
The Hipp Approximation	p. 204
General Counting Distribution	p. 204
Bernoulli Counting Distribution	p. 205
De Pril's Individual Model	p. 208
Numerical Example	p. 209
Extension to Distributions in P1	p. 217
De Pril Transforms	p. 217
Definition	p. 217
Extension of Results	p. 219
Error Bounds	p. 220
Allowing for Negative Severities	p. 223
Introduction	p. 223
Binomial Counting Distribution	p. 224
Poisson Counting Distribution	p. 226
Negative Binomial Counting Distribution	p. 228
Underflow and Overflow	p. 231
Simple Scaling	p. 231
Exponential Scaling	p. 231
Convolutions	p. 233
Multivariate Distributions
Introduction	p. 237
Aggregate Claims Distributions	p. 237
Vectors and Matrices	p. 239
Induction Proofs in a Multivariate Setting	p. 240
Classes of Distributions and Functions	p. 241
Convolutions	p. 241
Compound Distributions	p. 242
Moments	p. 244
Multivariate Compound Distributions of Type 1	p. 245
Covariances	p. 245
Counting Distribution in the Panjer Class	p. 246
Convolutions of a Distribution	p. 251
Infinite Divisibility	p. 254
Counting Distribution with Recursion of Higher Order	p. 254
Multivariate Bernoulli Severity Distribution	p. 256
De Pril Transforms	p. 259
Definitions	p. 259
The Rk Classes	p. 260
Infinite Divisibility	p. 262
Extension to Functions in Fm0	p. 263
Vectors of Independent Random Subvectors	p. 263
Vectors of Linear Combinations of Independent Random Variables	p. 267
Individual Models	p. 271
Moments	p. 275
Convolutions of a Distribution	p. 275
Moments	p. 275
The Multinormal Distribution	p. 279
Compound Distributions	p. 281
General Results	p. 281
Compound Panjer Distributions	p. 283
Approximations Based on De Pril Transforms	p. 287
Approximations	p. 287
Error Bounds	p. 287
Multivariate Compound Distributions of Type 2	p. 289
Main Framework	p. 289
Recursions of Higher Order	p. 294
Multivariate Compound Distributions of Type 3	p. 295
Compound Mixed Multivariate Poisson Distributions	p. 297
Multivariate Poisson Distributions	p. 297
Extension to Mixed Distributions	p. 300
Gamma Mixing Distribution	p. 301
Compound Distributions with Univariate Counting Distribution	p. 304
General Recursion	p. 304
Willmot Mixing Distribution	p. 305
The Univariate Mixed Counting Distribution	p. 308
Compound Distributions with Multivariate Counting Distribution	p. 311
The Multivariate Counting Distribution	p. 311
General Design	p. 311
The Special Design	p. 312
The Special Case &eq; 0	p. 318
Special Classes of Mixing Distributions	p. 320
Shifted Pareto Distribution	p. 320
Pareto Distribution	p. 321
Truncated Normal Distribution	p. 322
Inverse Gauss Distribution	p. 323
Transformed Gamma Distribution	p. 324
References	p. 327
List of Notation	p. 337
Author Index	p. 339
Subject Index	p. 343
Table of Contents provided by Ingram. All Rights Reserved.

Eaa

At a Glance

Paperback

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