Table Of ContentLecture Notes in Statistics 156
Edited by P. Bickel, P. Diggle, S. Fienberg, K. Krickeberg,
I. Olkin, N. Wermuth, S. Zeger
Springer Science+Business Media, LLC
Gordon E. Willmot
x. Sheldon Lin
Lundberg Approximations for Compound
Distributions with Insurance Applications
, Springer
Gordon E. Willmot X. Sheldon Lin
Department of Statistics and Department of Statistics and
Actuarial Science Actuarial Science
University of Waterloo University of Iowa
Waterloo, Ontario N2L 3Gl Iowa City, IA 52242-1409
Canada USA
[email protected] [email protected]
Library ofCongress Cataloging-in-Publication Data
Willmot, Gordon E., 1957-
Lundberg approximations for compound distributions with insurance
applications / Gordon E. Willmot, X. Sheldon Lin.
p. cm.--{Lecture notes in statistics; 156)
Includes bibliographical references and indexes.
ISBN 978-0-387-95135-5 ISBN 978-1-4613-0111-0 (eBook)
DOI 10.1007/978-1-4613-0111-0
1. Insurance-Statistical methods. 2. Distribution (Probability theory) 1. Lin,
X. Sheldon. II. Title. III. Lecture notes in statistics (Springer-Verlag); v. 156.
HG8781 .W55 2000
368'.01--dc21 00-061264
Printed on acid-free paper.
1C2001 Springer Science+Business Media New York
Origina11y published by Springer-Verlag New York, Ine in2001
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ISBN 978-0-387-95135-5
Preface
These notes represent our summary of much of the recent research that has
been done in recent years on approximations and bounds that have been
developed for compound distributions and related quantities which are of
interest in insurance and other areas of application in applied probability.
The basic technique employed in the derivation of many bounds is induc
tive, an approach that is motivated by arguments used by Sparre-Andersen
(1957) in connection with a renewal risk model in insurance. This technique
is both simple and powerful, and yields quite general results. The bounds
themselves are motivated by the classical Lundberg exponential bounds
which apply to ruin probabilities, and the connection to compound dis
tributions is through the interpretation of the ruin probability as the tail
probability of a compound geometric distribution. The initial exponential
bounds were given in Willmot and Lin (1994), followed by the nonexpo
nential generalization in Willmot (1994).
Other related work on approximations for compound distributions and
applications to various problems in insurance in particular and applied
probability in general is also discussed in subsequent chapters. The results
obtained or the arguments employed in these situations are similar to those
for the compound distributions, and thus we felt it useful to include them
in the notes. In many cases we have included exact results, since these
are useful in conjunction with the bounds and approximations developed.
These exact results, which normally involve mixtures of Erlang distribu
tions or combinations of exponentials, provide a means of comparison of
the approximate results with the exact results.
vi Preface
The results depend quite heavily on monotonicity ideas which are sum
marized in notions from reliability theory. As such, chapter 2 is devoted to
a discussion of those ideas which are relevant in the main part of the notes.
Chapter 3 considers properties of mixed Poisson distributions, an impor
tant modeling component of many insurance models. The Lundberg bounds
are the subject matter of chapters 4 through 6, and the important special
cases involving compound geometric and negative binomial distributions
are considered in chapter 7. An approximation due to Tijms (1986) in a
ruin or queueing theoretic context is discussed and generalized in chapter
8, where a close connection to the Lundberg bounds of the earlier chapters
is established. Many quantities of interest in insurance and other applied
probability models are known to satisfy defective renewal equations. The
close connection to compound geometric distributions allows for the ap
plication of the results from earlier chapters to these situations. Defective
renewal equations are thus discussed in chapter 9, including an important
general equation in insurance modeling due to Gerber and Shiu (1998). The
severity of ruin is considered in chapter 10, where a mixture representation
is given which allows for the derivation of various useful results. Finally,
the renewal risk model of Sparre-Andersen (1957) is discussed in chapter
II.
Three other aspects of the notes deserve mention as well. First, many new
results which have not been published elsewhere appear here. Second, the
treatment of the topics included is not meant to be comprehensive. That
is, we have concentrated on particular aspects of the models, and more
complete treatments of the subject may be found in the cited references.
Third, an alternative to the use of mathematical induction in the derivation
of the Lundberg bounds is the use of Wald-type martingale arguments. We
believe that the induction arguments are useful in their own right due
to their power and simplicity, and provide much insight into the types of
results which may be obtained. On the other hand, martingale arguments
often provide insight in complex situations.
The notes have undergone much revision over the past two years, and
have been used in graduate courses at the University of Waterloo and at
the University of Western Ontario. We wish to thank many individuals at
these institutions for their valuable comments and input. In particular, J un
Cai, Steve Drekic, Bruce Jones, David Stanford, Ken Seng Tan, and Cary
Tsai deserve special mention. We also wish to thank Jan Grandell and two
anonymous reviewers for their valuable suggestions. Special thanks goes to
Lynda Clarke for her expert typing of the manuscript, and to John Kimmel
of Springer-Verlag for his enthusiasm and support of the project. We wish
to acknowledge the financial support of the Committee on Knowledge Ex
tension Research of the Society of Actuaries as well, and Curtis Huntington
in particular for his assistance.
Preface vii
Finally, we wish to thank our wives Deborah (for GW) and Feng (for
XL), and our children Rachel, Lauren, and Kristen (for GW), and Jason
(for XL), for their patience and tolerance of this project.
Contents
1 Introduction 1
2 Reliability background 7
2.1 The failure rate . . . 8
2.2 Equilibrium distributions ........... . 14
2.3 The residual lifetime distribution and its mean 18
2.4 Other classes of distributions ........ . 24
2.5 Discrete reliability classes .......... . 28
2.6 Bounds on ratios of discrete tail probabilities 34
3 Mixed Poisson distributions 37
3.1 Tails of mixed Poisson distributions 38
3.2 The radius of convergence. . . . . . 40
3.3 Bounds on ratios of tail probabilities 42
3.4 Asymptotic tail behaviour of mixed Poisson distributions 46
4 Compound distributions 51
4.1 Introduction and examples. 52
4.2 The general upper bound . 65
4.3 The general lower bound . . 73
4.4 A Wald-type martingale approach 78
5 Bounds based on reliability classifications 81
5.1 First order properties ........... . 81
x Contents
5.2 Bounds based on equilibrium properties 87
6 Parametric Bounds 93
6.1 Exponential bounds 94
6.2 Pareto bounds ... 97
6.3 Product based bounds 100
7 Compound geometric and related distributions 107
7.1 Compound modified geometric distributions. 108
7.2 Discrete compound geometric distributions 114
7.3 Application to ruin probabilities ...... 129
7.4 Compound negative binomial distributions. 132
8 Tijms approximations 141
8.1 The asymptotic geometric case . . . . . . . 141
8.2 The modified geometric distribution . . . . 147
8.3 Transform derivation of the approximation. 148
9 Defective renewal equations 151
9.1 Some properties of defective renewal equations ...... 152
9.2 The time of ruin and related quantities. . . . . . . . . .. 159
9.3 Convolutions involving compound geometric distributions 174
10 The severity of ruin 183
10.1 The associated defective renewal equation . . . . . . . .. 183
10.2 A mixture representation for the conditional distribution. 186
10.3 Erlang mixtures with the same scale parameter 192
10.4 General Erlang mixtures. 198
10.5 Further results . . . . . . . . . . . . . . . . . . 205
11 Renewal risk processes 209
11.1 General properties of the model. 210
11.2 The Coxian-2 case . . . . . . . . 216
11.3 The sum of two exponentials .. 224
11A Delayed and equilibrium renewal risk processes 226
Bibliography 235
Symbol Index 243
Author Index 245
Subject Index 248
1
Introd uction
One of the standard stochastic models used in various areas of applied
probability such as insurance risk theory and queueing theory is the ran
dom sum model. Random sums are defined here as a sum of independent
and identically distributed (iid) random variables, the number of which
is also random. The literature on such models is voluminous, both from
an analytic and a numerical viewpoint. The difficulty in evaluation arises
from the presence of convolutions, but such evaluation is important for
many applications.
In this work we focus on the right tail of the distribution of the ran
dom sum (referred to as a compound distribution). The complement of the
distribution function (df), referred to as the tail, has been the subject of
much study. In particular, many asymptotic results are available. These
range from the light-tailed Lundberg type asymptotics of Embrechts, Mae
jima, and Teugels (1985) to the heavy-tailed subexponential asymptotics
of Embrechts, Goldie, and Veraverbeke (1979) and include an intermedi
ate class discussed by Embrechts and Goldie (1982). In general, it appears
that the light-tailed Lundberg asymptotics appear to be more suited for
practical numerical use from the viewpoint of accuracy. We will discuss the
light-tailed Lundberg results in what follows.
Related to the Lundberg asymptotics are Lundberg type bounds on the
tail of the compound distribution. These are similar to, and both gener
alize and refine the well known Cramer-Lundberg inequality of insurance
ruin theory (see Gerber, 1979). In many cases the parameter involved in
the bound is the same as that in the Lundberg asymptotic formula, thus
providing a link between bounds and asymptotics. The classical bound
G. E. Willmot et al., Lundberg Approximations for Compound Distributions with Insurance Applications
© Springer Science+Business Media New York 2001
