What is statistical inference?

In statistical inference experimental or observational data are modelled as the observed values of random variables, to provide a framework from which inductive conclusions may be drawn about the mechanism giving rise to the data.

We wish to analyse observations x = (x1, …, xn) by:

1. Regarding x as the observed value of a random variable X = (X1, …, Xn) having an (unknown) probability distribution, conveniently specified by a probability density, or probability mass function, f (x).

2. Restricting the unknown density to a suitable family or set F. In parametric statistical inference, f (x) is of known analytic form, but involves a finite number of real unknown parameters θ = (θ1, …, θd). We specify the region Θ ⊆ ℝd of possible values of θ, the parameter space. To denote the dependency of f (x) on θ, we write f (x; θ) and refer to this as the model function. Alternatively, the data could be modelled non-parametrically, a non-parametric model simply being one which does not admit a parametric representation. We will be concerned almost entirely in this book with parametric statistical inference.

The objective that we then assume is that of assessing, on the basis of the observed data x, some aspect of θ, which for the purpose of the discussion in this paragraph we take to be the value of a particular component, θi say. In that regard, we identify three main types of inference: point estimation, confidence set estimation and hypothesis testing.

This chapter develops the key ideas in the Bayesian approach to inference. Fundamental ideas are described in Section 3.1. The key conceptual point is the way that the prior distribution on the unknown parameter θ is updated, on observing the realised value of the data x, to the posterior distribution, via Bayes’ law. Inference about θ is then extracted from this posterior. In Section 3.2 we revisit decision theory, to provide a characterisation of the Bayes decision rule in terms of the posterior distribution. The remainder of the chapter discusses various issues of importance in the implementation of Bayesian ideas. Key issues that emerge, in particular in realistic data analytic examples, include the question of choice of prior distribution and computational difficulties in summarising the posterior distribution. Of particular importance, therefore, in practice are ideas of empirical Bayes inference (Section 3.5), Monte Carlo techniques for application of Bayesian inference (Section 3.7) and hierarchical modelling (Section 3.8). Elsewhere in the chapter we provide discussion of Stein's paradox and the notion of shrinkage (Section 3.4). Though not primarily a Bayesian problem, we shall see that the James–Stein estimator may be justified (Section 3.5.1) as an empirical Bayes procedure, and the concept of shrinkage is central to practical application of Bayesian thinking. We also provide here a discussion of predictive inference (Section 3.9) from a Bayesian perspective, as well as a historical description of the development of the Bayesian paradigm (Section 3.6).

Introduction

Although we have previously used the term premium, we have not formally defined it. A premium is the payment that a policyholder makes for complete or partial insurance cover against a risk. In this chapter we describe and discuss some ways in which premiums may be calculated, but we consider premium calculation from a mathematical viewpoint only. In practice, insurers have to take account not only of the characteristics of risks they are insuring, but other factors such as the premiums charged by their competitors.

We denote by ПX the premium that an insurer charges to cover a risk X. When we refer to a risk X, what we mean is that claims from this risk are distributed as the random variable X. The premium ПX is some function of X, and a rule that assigns a numerical value to ПX is referred to as a premium calculation principle. Thus, a premium principle is of the form ПX = φ(X) where φ is some function. In this chapter we start by describing some desirable properties of premium calculation principles. We then list some principles and consider which of the desirable properties they satisfy.

Properties of premium principles

There are many desirable properties for premium calculation principles. The following list is not exhaustive, but it does include most of the basic properties for premium principles.

Introduction

In this chapter we continue our study of the classical risk model. We start with a useful result concerning the probability that ruin occurs without the surplus process first attaining a specified level. This result will be applied in Sections 8.4 and 8.5 and Exercise 9. We then consider the insurer's deficit when ruin occurs and provide a means of finding the distribution of this deficit. We extend this study by considering the insurer's largest deficit before the surplus process recovers to level 0. Following this, we consider the distribution of the insurer's surplus immediately prior to ruin. We then consider the distribution of the time to ruin, and conclude with a discussion of a problem which involves modifying the surplus process through the payment of dividends.

In this chapter we use the same assumptions and notation as in Chapter 7.

A barrier problem

Let us consider the following question: what is the probability that ruin occurs from initial surplus u without the surplus process reaching level b > u prior to ruin? An alternative way of expressing this question is to ask what is the probability that ruin occurs in the presence of an absorbing barrier at b? We denote this probability by ξ(u, b), and let χ(u, b) denote the probability that the surplus process attains the level b from initial surplus u without first falling below zero.

Introduction

In this chapter we consider the aggregate claims arising from a general insurance risk over a short period of time, typically one year. We use the term ‘risk’ to describe a collection of similar policies, although the term could also apply to an individual policy. As indicated in Chapter 1, at the start of a period of insurance cover the insurer does not know how many claims will occur, and, if claims do occur, what the amounts of these claims will be. It is therefore necessary to construct a model that takes account of these two sources of variability. In the following, we consider claims arising over a one-year time interval purely for ease of presentation, but any unit of time can be used.

We start by modelling aggregate claims in Section 4.2 as a random variable, S, and derive expressions for the distribution function and moments of S. We then consider the important special case when the distribution of S is compound Poisson, and we give an important result concerning the sum of independent compound Poisson random variables. In Section 4.4 we consider the effect of reinsurance on aggregate claims, both from the point of view of the insurer and the reinsurer.

The remainder of the chapter is devoted to the important practical question of calculating an aggregate claims distribution. In Section 4.5 we introduce certain classes of counting distribution for the number of claims from a risk.

This book is designed for final-year university students taking a first course in insurance risk theory. Like many textbooks, it has its origins in lectures delivered in university courses, in this case at Heriot-Watt University, Edinburgh, and at the University of Melbourne. My intention in writing this book is to provide an introduction to the classical topics in risk theory, especially aggregate claims distributions and ruin theory.

The prerequisite knowledge for this book is probability theory at a level such as that in Grimmett and Welsh (1986). In particular, readers should be familiar with the basic concepts of distribution theory and be comfortable in the use of tools such as generating functions. Much of Chapter 1 reviews distributions and concepts with which the reader should be familiar. A basic knowledge of stochastic processes is helpful, but not essential, for Chapters 6 to 8. Throughout the text, care has been taken to use straightforward mathematical techniques to derive results.

Since the early 1980s, there has been much research in risk theory in computational methods, and recursive schemes in particular. Throughout the text, recursive methods are described and applied, but a full understanding of such methods can only be obtained by applying them. The reader should therefore by prepared to write some (short) computer programs to tackle some of the examples and exercises.

Many of these examples and exercises are drawn from materials I have used in teaching and examining, so the degree of difficulty is not uniform.

Introduction

Ruin theory is concerned with the level of an insurer's surplus for a portfolio of insurance policies. In Chapter 4 we considered the aggregate amount of claims paid out in a single time period. We now consider the evolution of an insurance fund over time, taking account of the times at which claims occur, as well as their amounts. To make our study mathematically tractable, we simplify a real life insurance operation by assuming that the insurer starts with some non-negative amount of money, collects premiums and pays claims as they occur. Our model of an insurance surplus process is thus deemed to have three components: initial surplus (or surplus at time zero), premiums received and claims paid. For the model discussed in this chapter, if the insurer's surplus falls to zero or below, we say that ruin occurs.

The aim of this chapter is to provide an introduction to the ideas of ruin theory, in particular probabilistic arguments. We use a discrete time model to introduce ideas that we apply in the next two chapters where we consider a continuous time model. Indeed, we will meet analogues of results given in this chapter in these next two chapters. We start in Section 6.2 by describing our model, then in Section 6.3 we derive a general equation for the probability of ruin in an infinite time horizon, and consider situations in which it is possible to obtain an explicit solution for this probability.

Introduction

This book is about risk theory, with particular emphasis on the two major topics in the field, namely risk models and ruin theory. Risk theory provides a mathematical basis for the study of general insurance risks, and so it is appropriate to start with a brief description of the nature of general insurance risks. The term general insurance essentially applies to an insurance risk that is not a life insurance or health insurance risk, and so the term covers familiar forms of personal insurance such as motor vehicle insurance, home and contents insurance, and travel insurance.

Let us focus on how a motor vehicle insurance policy typically operates from an insurer's point of view. Under such a policy, the insured party pays an amount of money (the premium) to the insurer at the start of the period of insurance cover, which we assume to be one year. The insured party will make a claim under the insurance policy each time the insured party has an accident during the year that results in damage to the motor vehicle, and hence requires repair costs. There are two sources of uncertainty for the insurer: how many claims will the insured party make, and, if claims are made, what will be the amounts of those claims? Thus, if the insurer were to build a probabilistic model to represent its claims outgo under the policy, the model would require a component that modelled the number of claims and another that modelled the amounts of those claims.