Risk modelling in general insurance by roger j gray

Risk Modelling in General Insurance
Knowledge of risk models and the assessment of risk is a fundamental part of the
training of actuaries and all who are involved in financial, pensions and insurance
mathematics. This book provides students and others with a firm foundation in a wide
range of statistical and probabilistic methods for the modelling of risk, including short
term risk modelling, model based pricing, risk sharing, ruin theory and credibility.
It covers much of the international syllabuses for professional actuarial
examinations in risk models, but goes into further depth, with numerous worked
examples and exercises (answers to many are included in an appendix). A key feature
is the inclusion of three detailed case studies that bring together a number of concepts
and applications from different parts of the book and illustrate how they are used in
practice. Computation plays an integral part: the authors use the statistical package
R to demonstrate how simple code and functions can be used profitably in an actuarial
context.
The authors’ engaging and pragmatic approach, balancing rigour and intuition, and
developed over many years of teaching the subject, makes this book ideal for
self-study or for students taking courses in risk modelling.
ro g e r j . g r ay was a Senior Lecturer in the School of Mathematical and Computer
Sciences at Heriot-Watt University, Edinburgh, until his death in 2011.
s u s a n m . pi t t s is a Senior Lecturer in the Statistical Laboratory at the University of
Cambridge.


I N T E R NAT I O NA L S E R I E S O N AC T UA R I A L S C I E N C E
Editorial Board
Christopher Daykin (Independent Consultant and Actuary)
Angus Macdonald (Heriot-Watt University)
The International Series on Actuarial Science, published by Cambridge University Press in conjunction with the Institute and Faculty of Actuaries, contains textbooks for students taking courses
in or related to actuarial science, as well as more advanced works designed for continuing professional development or for describing and synthesising research. The series is a vehicle for

publishing books that reflect changes and developments in the curriculum, that encourage the
introduction of courses on actuarial science in universities, and that show how actuarial science
can be used in all areas where there is long-term financial risk.
A complete list of books in the series can be found at www.cambridge.org/statistics. Recent titles
include the following:
Solutions Manual for Actuarial Mathematics for Life Contingent Risks
David C.M. Dickson, Mary R. Hardy & Howard R. Waters
Financial Enterprise Risk Management
Paul Sweeting
Regression Modeling with Actuarial and Financial Applications
Edward W. Frees
Actuarial Mathematics for Life Contingent Risks
David C.M. Dickson, Mary R. Hardy & Howard R. Waters
Nonlife Actuarial Models
Yiu-Kuen Tse
Generalized Linear Models for Insurance Data
Piet De Jong & Gillian Z. Heller
Market-Valuation Methods in Life and Pension Insurance
Thomas Møller & Mogens Steffensen
Insurance Risk and Ruin
David C.M. Dickson


RISK MODELLING IN GENERAL
INSURANCE
From Principles to Practice
RO G E R J . G R AY
Heriot-Watt University, Edinburgh
SUSAN M. PITTS
University of Cambridge

cambridge university press
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Cambridge University Press
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Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521863940
c Roger J. Gray and Susan M. Pitts 2012
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2012
Printed in the United Kingdom at the University Press, Cambridge
A catalogue record for this publication is available from the British Library
Library of Congress Cataloguing in Publication data
Gray, Roger J.
Risk modelling in general insurance : from principles to practice /
Roger J. Gray, Susan M. Pitts.
p. cm.
ISBN 978-0-521-86394-0 (hardback)
1. Risk (Insurance) – Mathematical models. I. Pitts, Susan M. II. Title.
HG8054.5.G735 2012
368 .01–dc23
2012010344
ISBN 978-0-521-86394-0 Hardback
Additional resources for this publication at www.cambridge.org/9780521863940

Cambridge University Press has no responsibility for the persistence or
accuracy of URLs for external or third-party internet websites referred to
in this publication, and does not guarantee that any content on such
websites is, or will remain, accurate or appropriate.


To the memory of
Roger J. Gray
1946–2011



Contents

Preface

page xiii

1

Introduction
1.1
The aim of this book
1.2
Notation and prerequisites
1.2.1 Probability
1.2.2 Statistics
1.2.3 Simulation
1.2.4 The statistical software package R

2

Models for claim numbers and claim sizes
2.1
Distributions for claim numbers
2.1.1 Poisson distribution
2.1.2 Negative binomial distribution
2.1.3 Geometric distribution
2.1.4 Binomial distribution
2.1.5 A summary note on R
2.2
Distributions for claim sizes
2.2.1 A further summary note on R
2.2.2 Normal (Gaussian) distribution
2.2.3 Exponential distribution
2.2.4 Gamma distribution
2.2.5 Fat-tailed distributions
2.2.6 Lognormal distribution
2.2.7 Pareto distribution
2.2.8 Weibull distribution
2.2.9 Burr distribution
2.2.10 Loggamma distribution
2.3
Mixture distributions
vii

1
1
2
2

9
9
9
11
12
13
16
18
20
22
23
24
24
25
28
31
35
40
45
48
51
54


viii

3

4

Contents
2.4

Fitting models to claim-number and claim-size data
2.4.1 Fitting models to claim numbers
2.4.2 Fitting models to claim sizes
Exercises

58
60
65
83

Short term risk models
3.1
The mean and variance of a compound distribution
3.2
The distribution of a random sum
3.2.1 Convolution series formula for a compound
distribution
3.2.2 Moment generating function of a compound
distribution
3.3
Finite mixture distributions
3.4
Special compound distributions
3.4.1 Compound Poisson distributions
3.4.2 Compound mixed Poisson distributions
3.4.3 Compound negative binomial distributions
3.4.4 Compound binomial distributions

3.5
Numerical methods for compound distributions
3.5.1 Panjer recursion algorithm
3.5.2 The fast Fourier transform algorithm
3.6
Approximations for compound distributions
3.6.1 Approximations based on a few moments
3.6.2 Asymptotic approximations
3.7
Statistics for compound distributions
3.8
The individual risk model
3.8.1 The mean and variance for the individual risk
model
3.8.2 The distribution function and moment generating function for the individual risk
model
3.8.3 Approximations for the individual risk model
Exercises

90
91
93

Model based pricing – setting premiums
4.1
Premium calculation principles
4.1.1 The expected value principle (EVP)
4.1.2 The standard deviation principle (SDP)
4.1.3 The variance principle (VP)
4.1.4 The quantile principle (QP)

4.1.5 The zero utility principle (ZUP)

95
98
100
103
103
108
110
114
115
116
119
124
125
126
128
134
136

137
139
140
147
148
148
149
149
149
150

Contents
4.1.6
4.1.7

The exponential premium principle (EPP)
Some desirable properties of premium
calculation principles
4.1.8 Other premium calculation principles
4.2
Maximum and minimum premiums
4.3
Introduction to credibility theory
4.4
Bayesian estimation
4.4.1 The posterior distribution
4.4.2 The wider context of decision theory
4.4.3 The binomial/beta model
4.4.4 The Poisson/gamma model
4.4.5 The normal/normal model
4.5
Bayesian credibility theory
4.5.1 Bayesian credibility estimates under the
Poisson/gamma model
4.5.2 Bayesian credibility premiums under the
normal/normal model
4.6
Empirical Bayesian credibility theory: Model 1 – the
Bühlmann model

4.7
Empirical Bayesian credibility theory: Model 2 – the
Bühlmann–Straub model
Exercises
5

Risk sharing – reinsurance and deductibles
5.1
Excess of loss reinsurance
5.1.1 Reinsurance claims
5.1.2 Simulation results
5.1.3 Aggregate claims model with excess of loss
reinsurance
5.2
Proportional reinsurance
5.3
Deductibles (policy excesses)
5.4
Retention levels and reinsurance costs
5.5
Optimising the reinsurance contract
5.6
Optimising reinsurance contracts based on maximising
expected utility
5.6.1 Excess of loss reinsurance
5.6.2 Proportional reinsurance
5.7
Optimising reinsurance contracts based on minimising
the variance of aggregate claims
5.7.1 Minimising Var[S I ] subject to fixed E[S I ]

ix
150
152
154
155
156
157
158
159
161
163
165
169
170
172
176
185
196
205
206
210
212
213
221
223
226
228
228
229

231
234
235


x

Contents
Minimising Var[S R ] subject to fixed Var[S I ]
Comparing stop loss and equivalent
proportional reinsurance arrangements
5.7.4 Minimising Var[S I ] + Var[S R ]
5.7.5 Minimising the sum of variances when two
independent risks are shared between two
insurers
5.8
Optimising reinsurance contracts for a group of independent risks based on minimising the variance of the
direct insurer’s net profit – finding the optimal relative
retentions
5.8.1 Optimal relative retentions in the case of
excess of loss reinsurance
5.8.2 Optimal relative retentions in the case of
proportional reinsurance
Exercises

236

Ruin theory for the classical risk model
6.1
The classical risk model

6.1.1 The relative safety loading
6.1.2 Ruin probabilities
6.2
Lundberg’s inequality and the adjustment coefficient
6.2.1 Properties of the adjustment coefficient
6.2.2 Proof of Lundberg’s inequality
6.2.3 When does the adjustment coefficient exist?
6.3
Equations for ψ(u) and ϕ(u): the ruin probability and
the survival probability
6.4
Compound geometric representations for ψ(u) and
ϕ(u): the ruin probability and the survival probability
6.5
Asymptotics for the probability of ruin
6.6
Numerical methods for ruin quantities
6.6.1 Numerical calculation of the adjustment
coefficient
6.6.2 Numerical calculation of the probability of ruin
6.7
Statistics for ruin quantities
Exercises

267
267
269
270
272
272

276
279

Case studies
7.1
Case study 1: comparing premium setting principles
7.1.1 Case 1 – in the presence of an assumed model

316
316
316

5.7.2
5.7.3

6

7

237
238

239

247
247
251
253

282

291
296
303
303
305
308
310


Contents
7.1.2

7.2

7.3

Case 2 – without model assumptions, using
bootstrap resampling
Case study 2: shared liabilities – who pays what?
7.2.1 Case 1 – exponential losses
7.2.2 Case 2 – Pareto losses
7.2.3 Case 3 – lognormal losses
Case study 3: reinsurance and ruin
7.3.1 Introduction
7.3.2 Proportional reinsurance
7.3.3 Proportional reinsurance with exponential
claim sizes
7.3.4 Excess of loss reinsurance in a layer
7.3.5 Excess of loss reinsurance in a layer with
exponential claim sizes

xi
322
332
333
338
344
348
348
351
353
356
360

Appendix A

Utility theory

368

Appendix B

Answers to exercises

380

References
Index

386

389



Preface

My co-author Roger died in March 2011. His tragic death was a terrible shock,
and he is, and will be, greatly missed by me and, I am sure, by all who knew
him.
The original plan for writing this book was that Roger and I would each write
our own chapters separately. We then planned to go through the whole book
together, chapter by chapter, and make various changes as necessary when
we had each read what the other had written. Unfortunately, and very sadly,
Roger died before this process was completed. At the time of his death, the
draft versions of Chapters 2 to 7 and Appendix A were written, and we had
a very preliminary sketch of Chapter 1. However, only two chapters had been
discussed in detail by both of us together. Fred Gray (Roger’s brother), David
Tranah (Cambridge University Press) and I were unanimous that Roger would
have wanted the book to be completed, and so I began to put Roger’s and my
draft chapters together, to complete Chapter 1, and to edit the whole book in
order to unify our two approaches, to fill obvious gaps, and to avoid too much
repetition. My aim was that the result would be in line with what Roger would
have wanted, and I very much hope that the finished book stands as a fitting
tribute to his memory.
There are many people to thank for their help during the production of this
book. First and foremost, thanks are due to everyone at Cambridge University
Press. Special thanks go to David Tranah, who has been most helpful, with
great patience and kindness at every stage. Thanks also go to Irene Pizzie for
her careful and efficient copy-editing.
During our discussions Roger told me that he had a long list of people to

thank in connection with the book, but unfortunately the conversation moved
on without any names being mentioned. I know that David Wilkie, Iain Currie
and Edward Kinley would have been on Roger’s list, and I would like to take
this opportunity to thank them. I would also like to thank everyone else who
xiii


xiv

Preface

was helpful to Roger in the writing of his parts of the book, but whose names
are unknown to me.
For my own part, I have been fortunate in having had excellent teachers,
co-workers and students over the years, and my understanding of the subject
matter of the book, and of effective ways to teach it, would not have been
possible without them. I would like to thank them all. In addition, my thanks
go to all those who were so supportive of my efforts to complete the book after
Roger’s death. Among these, I am especially grateful to David Tranah (whose
wise advice and generous practical help were invaluable), Alan and Brenda
Cole, Brigitte Snell and Rita McLoughlin. Finally, but most importantly of all,
I thank my husband, Andrew, for his unfailingly good-humoured support and
encouragement throughout the writing of this book.
Susan M. Pitts


1
Introduction

1.1 The aim of this book

Knowledge of risk models and the assessment of risk will be of great importance to actuaries as they apply their skills and expertise today and in the future.
The title of this book “Risk Modelling in General Insurance: From Principles
to Practice” reflects our intention to present a wide range of statistical and
probabilistic topics relevant to actuarial methodology in general insurance. Our
aim is to achieve this in a focused and coherent manner, which will appeal to
actuarial students and others interested in the topics we cover.
We believe that the material is suitable for advanced undergraduates and students taking master’s degree courses in actuarial science, and also those taking
mathematics and statistics courses with some insurance mathematics content.
In addition, students with a strong quantitative/mathematical background taking economics and business courses should also find much of interest in the
book. Prerequisites for readers to benefit fully from the book include first
undergraduate-level courses in calculus, probability and statistics. We do not
assume measure theory.
Our aim is that readers who master the content will extend their knowledge
effectively and will build a firm foundation in the statistical and actuarial concepts and their applications covered. We hope that the approach and content
will engage readers and encourage them to develop and extend their critical
and comparative skills. In particular, our aim has been to provide opportunities for readers to improve their higher-order skills of analysis and synthesis of
ideas across topics.
A key feature of our approach is the inclusion of a large number of worked
examples and extensive sets of exercises, which we think readers will find
stimulating. In addition, we include three case studies, each of which brings
1


2

Introduction

together a number of concepts and applications from different parts of the
book.
While the book covers much of the international syllabuses for professional

actuarial examinations in risk models, it goes further and deeper in places.
The book includes appropriate references to the open source (free and easily downloadable) statistical software package R throughout, giving readers
opportunities to learn how simple code and functions can be used profitably in
an actuarial context.

1.2 Notation and prerequisites
The tools of probability theory are crucial for the study of the risk models in
this book, and, in §1.2.1, we give an overview of the required basic concepts
of probability. This overview also serves to introduce the notation that we will
use throughout the book. In §1.2.2 and §1.2.3, we indicate the assumed prerequisites in statistics and simulation, and finally in §1.2.4 we give information
about the statistical software package R.

1.2.1 Probability
We start with definitions and notation for basic quantities related to a random
variable X. Our first such quantity is the distribution function (or cumulative
distribution function) F X of X, given by
F X (x) = Pr(X ≤ x),

x ∈ R.

The function F X is non-decreasing and right-continuous. It satisfies 0 ≤
F X (x) ≤ 1 for all x in R, lim F X (x) = 1 and lim F X (x) = 0. Most of the
x→∞
x→−∞
random variables in this book are non-negative, i.e. they take values in [0, ∞).
If V is a non-negative random variable, then we assume without comment that
FV (v) = 0 for v < 0. For a non-negative random variable V, the tail of FV is
Pr(V > v) = 1 − FV (v) for v ≥ 0.
A continuous random variable Y has a probability density function fY , which
∞

is a non-negative function fY , with −∞ fY (y)dy = 1, such that the distribution
function of Y is
y

FY (y) =

−∞

fY (t)dt,

y ∈ R.

This means that FY is a continuous function. The probability that Y is in a set
A is Pr(Y ∈ A) = A fY (y)dy. (For those readers who are familiar with measure


1.2 Notation and prerequisites

3

theory, note that we will tacitly assume the word “measurable” where necessary. Those readers who are not familiar with measure theory may ignore this
remark, but may like to note that a rigorous treatment of probability theory
requires more careful definitions and statements than appear in introductory
courses and in this overview.)
Let N be a discrete random variable that takes values in N = {0, 1, 2, . . .}.
Then Pr(N = x), x ∈ R, is the probability mass function of N. We see that
Pr(N = x) = 0 for x N, so that, for a discrete random variable concentrated
on N, the probability mass function is specified by Pr(N = k) for k ∈ N. We
then have ∞
k=0 Pr(N = k) = 1. The distribution function of N is

F N (x) =

Pr(N = k),

x ∈ R,

{k:k≤x}

and the graph of F N is a non-decreasing step function, with an upward jump of
size Pr(N = k) at k for all k ∈ N. The probability that N is in a set A is
Pr(N = k).

Pr(N ∈ A) =
{k:k∈A}

We use the notation E[X] for the expected value (or expectation, or mean) of
a random variable X. The expectation of the continuous random variable Y is
E[Y] =

∞
−∞

y fY (y)dy,

while for the discrete random variable N taking values in N, the expectation is
∞

E[N] =

k Pr(N = k).

k=0

We note that there are various possibilities for the expectation: it may be finite,
it may take the value +∞ or −∞, or it may not be defined. The expectation of
a non-negative random variable is either a finite non-negative value or +∞.
For a real-valued function h on R and a continuous random variable Y, the
expectation of h(Y) is
E h(Y) =

∞
−∞

h(y) fY (y)dy,

whenever the integral is defined, and for a discrete random variable N taking
values in N, the expectation of h(N) is
∞

E h(N) =

h(k) Pr(N = k).
k=0


4

Introduction

For r ≥ 0, the rth moment of X is E[X r ], when it is defined. The rth moment
of a continuous random variable Y is

∞
−∞

yr fY (y)dy,

and the rth moment of the discrete random variable N taking values in N is
∞

kr Pr(N = k).
k=0

Recall that if E[|X| ] is finite for some r > 0, then E[|X| s ] is finite for all
0 ≤ s ≤ r. Throughout the book, when we write down a particular moment
such as E[N 3 ], then, unless otherwise stated, we assume that this moment is
finite.
The rth central moment of a random variable X is E[(X−E[X])r ]. The second
central moment of X is called the variance of X, and is denoted by Var[X]. The
variance of X is given by
r

Var[X] = E (X − E[X])2 = E X 2 − E[X] 2 .
√
The standard deviation of X is SD[X] = Var[X]. We define the skewness of X
to be the third central moment, E[(X − E[X])3 ], and the coefficient of skewness
to be given by
E[(X − E[X])3 ]/ (SD[X])3 .

(1.1)

We define the coefficient of kurtosis of X to be

E[(X − E[X])4 ]/ (SD[X])4 ,

(1.2)

but note that various definitions are given in the literature; see the discussion
in §2.2.5.
The covariance of random variables X and W is given by
Cov[X, W] = E (X − E[X])(W − E[W]) = E[XW] − E[X]E[W].
The correlation between random variables X and W (with Var[X] > 0 and
Var[W] > 0) is given by
Cov[X, W]
.
Corr[X, W] = √
Var[X] Var[W]
For random variables X1 , . . . , Xn we have
n

Var[X1 + · · · + Xn ] =

Var[Xi ] + 2
i=1

Cov[Xi , X j ].
i< j


1.2 Notation and prerequisites

5

Random variables X1 , . . . , Xn are independent if, for all x1 , . . . , xn in R,
Pr(X1 ≤ x1 , . . . , Xn ≤ xn ) = Pr(X1 ≤ x1 ) . . . Pr(Xn ≤ xn ).
For independent random variables X1 , . . . , Xn and functions h1 , . . . , hn , we have
E h1 (X1 ) . . . hn (Xn ) = E h1 (X1 ) . . . E hn (Xn ) .
This means that, for independent random variables X1 , . . . , Xn , we have
Var[X1 + · · · + Xn ] = Var[X1 ] + · · · + Var[Xn ],
because, for i j, the independence of Xi and X j implies that Cov[Xi , X j ] = 0.
Random variables X1 , X2 , . . . are independent if every finite subset of the Xi
is independent. We say X1 , X2 , . . . are independent and identically distributed
(iid) if they are independent and all have the same distribution.
Conditioning is one of the main tools used throughout this book, and it is
often the key to a neat approach to derivation of properties and features of
the risk models considered in later chapters. The conditional expectation of X
given W is denoted E[X | W]. The very useful conditional expectation formula
states that
E E[X | W] = E[X].

(1.3)

The conditional variance of X given W is defined to be
Var[X | W] = E X − E[X | W]

2

|W

= E X | W − E[X | W] 2 .
2

The conditional variance formula is

Var[X] = E Var[X | W] + Var E[X | W] .

(1.4)

This may be seen by considering the terms on the right-hand side of (1.4). We
have
E [Var[X | W]] = E E[X 2 | W] − (E[X | W])2
= E X 2 − E (E[X | W])2 ,
where we have used the conditional expectation formula, and
Var E[X | W] = E (E[X | W])2 − E E[X | W]

2

= E (E[X | W])2 − (E[X])2 ,
on using the conditional expectation formula again. Adding these terms it is
easy to see that the right-hand side of (1.4) is equal to the left-hand side.


6

Introduction

We assume that moment generating functions, probability generating functions and their properties are familiar to the reader. The moment generating
function of a random variable X is denoted
MX (r) = E[erX ],

(1.5)

and this may not be finite for all r in R. For every random variable X, we have
MX (0) = 1, and so the moment generating function is certainly finite at r = 0.

If MX (r) is finite for |r| < h for some h > 0, then, for any k = 1, 2, . . ., the
function MX (r) is k-times differentiable at r = 0, with
MX(k) (0) = E X k ,

(1.6)

with E |X|k finite. If random variables X and W have MX (r) = MW (r) for all
|r| < h for some h > 0, then X and W have the same distribution.
The moment generating function of a continuous random variable Y is
∞

MY (r) =

−∞

ery fY (y)dy.

The moment generating function of a discrete random variable N concentrated
on N is
∞

MN (r) =

erk Pr(N = k).
k=0

The probability generating function of N is
∞

G N (z) = E zN =

zk Pr(N = k),

(1.7)

k=0

for those z in R for which the series converges absolutely. Since the series
converges for |z| ≤ 1 (and possibly for a larger set of z-values), we see that the
radius of convergence of the series is greater than or equal to 1. If E[N] < ∞
then
E[N] = G N (1),
and if E N 2 < ∞ then
Var[N] = G N (1) + G N (1) − G N (1) 2 ,
(k)
where G(k)
N (1) = lim G N (z) if the radius of convergence of G N is 1. From (1.5)
z↑1

and (1.7) we have
G N (z) = MN log(z) and MN (r) = G N er ,


1.2 Notation and prerequisites

7

where here, and throughout the book, when we write down relationships
between generating functions, we assume the phrase “for values of the
argument for which both sides are finite”.

Moment generating functions and probability generating functions are both
examples of transforms. Transforms are useful for calculations involving sums
of independent random variables. Let X1 , . . . , Xn be independent random variables, and let MXi be the moment generating function of Xi , i = 1, . . . , n. Then
the moment generating function of T = X1 + · · · + Xn is the product of the
moment generating functions of the Xi :
MT (r) = MX1 (r) . . . MXn (r).

(1.8)

Similarly, let N1 , . . . , Nn be independent discrete random variables taking values in N, and let G Ni be the probability generating function of Ni , i = 1, . . . , n.
Then the probability generating function of M = N1 + · · · + Nn is
G M (z) = G N1 (z) . . . G Nn (z).

(1.9)

Sums of independent random variables play an important role in the models in
this book, so transform methods will be important for us.
The cumulant generating function KX (t) of a random variable X is given by
KX (t) = log MX (t) ,
and this is discussed further in §2.2.5.
In the above discussion, we have given separate expectation formulae for
continuous random variables and for discrete random variables. We now introduce a more general notation that covers both of these cases (and other cases
as well). For a general random variable X with distribution function F X , we
write
E[X] =

xF X (dx).

(1.10)

This is a Lebesgue–Stieltjes integral. We can think of the integral as shorthand
notation for x fX (x)dx if X is continuous with density fX , and as shorthand
for ∞
k=0 k Pr(X = k) if X is discrete and takes values in {0, 1, 2, . . .}. This
notation means we can give just one formula that covers both continuous and
discrete random variables. However, it also covers more general random variables. Later in this book we will meet and use random variables which are
neither purely continuous, nor purely discrete, but which have both a discrete
part and a continuous part. To make this precise, suppose that there exist real
numbers x1 , . . . , xm and p1 , . . . , pm , where 0 ≤ pk ≤ 1 for k = 1, . . . , m, and


8

Introduction

where m
k=1 pk ≤ 1, and suppose there also exists a non-negative function f ,
∞
with −∞ f (t)dt ≤ 1, such that the distribution function of X is
F X (x) = Pr(X ≤ x) =

pk +
{k:xk ≤x}

x

f (t)dt.

(1.11)

−∞

Of course, we must have
∞

m

pk +

−∞

k=1

f (x)dx = 1.

In this case, the distribution of X consists of a discrete part, specified by the xk
and the pk (with Pr(X = xk ) = pk ), and also a continuous part, specified by f .
The distribution function F X has an upward jump of size pk at xk , k = 1, . . . , m,
and is continuous and non-decreasing (and not necessarily flat) between these
jumps. We say that the distribution of X has an atom at xk (of size pk ), for
k = 1, . . . , m. For this X, and for a set A, we have
Pr(X ∈ A) =

F X (dx) =
A

pk +

f (x)dx.

As in (1.10), the expectation of X is E[X] =
(1.11), the integral is

xF X (dx), and, with F X as in
∞

m

xF X (dx) =

kpk +
k=1

(1.12)

A

{k:xk ∈A}

−∞

x f (x)dx.

(1.13)

In general, for a function h, we have
E[h(X)] =

h(x)F X (dx),

(1.14)

and, when h(x) = erx , we find that the moment generating function of X is
MX (r) = E erX =

erx F X (dx).

With F X as in (1.11), the equations (1.14) and (1.15) become
m

E[h(X)] =

h(k)pk +
k=1

∞
−∞

h(x) f (x)dx

and
m

MX (r) =

erx F X (dx) =

erk pk +
k=1

∞
−∞

erx f (x)dx.

(1.15)


1.2 Notation and prerequisites

9

Note that a Lebesgue–Stieltjes integral over an interval (a, b], a ≤ b, is written
. . . F X (dx),
(a,b]

where . . . is to be replaced by the required function to be integrated. Finally,
we have, from (1.12),
F X (dx) = Pr X ∈ (a, b] = F X (b) − F X (a− ),
(a,b]

where F X (a− ) denotes lim− F X (x), and x → a− means that x converges to a
x→a
from the left.
In this subsection, we have given a brief overview of probability. For more
discussion and details, see, for example, Grimmett and Stirzaker (2001), Gut
(2009) and the more advanced Gut (2005).

1.2.2 Statistics
We assume that the reader has met point estimation and properties of estimators (for example, the idea of an unbiased estimator), confidence intervals

and hypothesis tests (for example, t tests, χ2 tests, Kolmogorov–Smirnov test).
We further assume a working knowledge of maximum likelihood estimators
and their large sample properties. Familiarity with plots, such as histograms
and quantile (or Q–Q) plots, is assumed, in addition to familiarity with the
empirical distribution function. Useful references are DeGroot and Schervish
(2002) and Casella and Berger (1990). The introduction to §2.4 contains an
overview of some ideas and methods in statistics. At various points in the book
we use more advanced statistical ideas – whenever we do this, references to
appropriate texts are given.

1.2.3 Simulation
We take as prerequisite some knowledge of simulation of observations from
a given distribution using a pseudo-random number generator and various
techniques, such as the inverse transform (or inversion or probability integral transform) method. For more details and background, see, for example,
chapter 11 in DeGroot and Schervish (2002) and chapter 6 in Morgan (2000).

1.2.4 The statistical software package R
The simulations, statistical analyses and numerical approximations in this book
are carried out using the statistical software package R. We assume familiarity