by its mean excess function (e.g., Kotz and Shanbhag, 1980).

There is a simple relationship between the mean excess function and the hazard

Ð1

rate r(x): Rewriting (2.58) in the form F (x)e(x) ¼ x F (t) dt and differentiating, we

0

see that F (x)e0 (x) þ F (x)e(x) ¼ ÀF (x), which can be rearranged in the form

0

F (x) 1 þ e0 (x)

:

r(x) ¼ À (2:60)

¼

e(x)

F (x)

45

2.2 HAZARD RATES, MEAN EXCESS FUNCTIONS, AND TAILWEIGHT

In the context of income distributions, the mean excess function arises in

connection with the somewhat neglected van der Wijk™s (1939) law. It asserts that the

average income of everybody above a certain level x is proportional to x itself.

Formally,

Ð1

tf (t) dt

Ðx1 ¼ g x, for some g . 0: (2:61)

f (t) dt

x

The l.h.s. is clearly the mean excess or mean residual life function, and since this

function characterizes a distribution (within the class of continuous distributions),

the r.h.s. de¬nes a speci¬c income distribution. In the case of (2.61), this is the

Pareto type I distribution; see Chapter 3 below.

In order to discuss the properties of the functions r(x) and e(x) for large losses”

or, in economic applications, large incomes”in a uni¬ed manner, it is convenient to

introduce a concept from classical analysis.

To set the stage, the classical Pareto distribution possesses the survival function

F (x) ¼ xÀa , x ! 1, for some a . 0: (2:62)

Its slow decrease implies that the moments E(X k ) exist if and only if k , a. This is

typical for many size distributions; hence, it is useful to have a general framework

for describing distributions of the Pareto type.

This framework is provided by the concept of regularly varying functions. A

function g : Rþ ! Rþ is regularly varying at in¬nity with index r [ R,

symbolically, g [ RV1 (r), if

g(tx)

¼ tr (2:63)

lim

x!1 g(x)

for all t . 0. If r ¼ 0, g is called slowly varying at in¬nity. Clearly, xr , xr log x,

xr log log x, r = 0, are all regularly varying at in¬nity with index r. The functions

log x, log log x are slowly varying. In general, a regularly varying function g [

RV1 (r) has a representation of the form g(x) ¼ xr ˜(x), where ˜ is a slowly varying

function to which we refer as the slowly varying part of g. In our context, the function

g of (2.63) is the density, distribution function, or survival function of the size

distribution. We refer to Bingham, Goldie, and Teugels (1987) for an encyclopedic

treatment of regularly varying functions. Here we require only some rather basic

properties of these functions, all of which may be found in Chapter 1 of Bingham,

Goldie, and Teugels:

RV 1. The concept can be extended to regular variation at points x0 other than

in¬nity; one then replaces g(x) by g(x0 À 1=x) in the above de¬nition.

46 GENERAL PRINCIPLES

RV 2. Products and ratios of regularly varying functions are also regularly

varying. Speci¬cally, if f [ RV1 (r) and g [ RV1 (t), then fg [

RV1 (r þ t) and f =g [ RV1 (r À t).

RV 3. Derivatives and integrals of regularly varying functions are also regularly

varying, under some regularity conditions (which we omit). Loosely

speaking, the index of regular variation increases by 1 upon integration, it

decreases by 1 upon differentiation. The precise results are referred to as

Karamata™s theorem and the monotone density theorem, respectively.

We can now describe the behavior of the hazard rate and the mean excess

function of distributions with regularly varying tails. It is clear from (2.62) that the

Pareto distribution is the prototypical size distribution with a regularly varying tail.

The hazard rate of a Pareto distribution is given by

a

r(x) ¼ ,1 x, (2:64)

x

which is in RV1 (À1), and for the mean excess function a straightforward calculation

yields [compare van der Wijk™s law (2.61)]

x

e(x) ¼ , 1 x, (2:65)

aÀ1

which is in RV1 (1).

These properties can be generalized to distributions with regularly varying tails: If

we are given a distribution with F [ RV1 (Àa), a . 0, it follows from property

RV 2 that

r [ RV1 (À1): (2:66)

Similarly, the mean excess function of such a distribution can be shown to possess

the property

e [ RV1 (1): (2:67)

In view of the r.h.s. of (2.58) and property RV 3, the latter result is quite transparent.

The preceding results imply that empirical distributions possessing slowly

decaying hazard rates or approximately linearly increasing mean excess functions

can be modeled by distributions with regularly varying tails. This is indeed a

popular approach in applied actuarial work. The work of Benktander and

Segerdahl (1960) constitutes an early example of the use of mean excess plots.

More recently, Hogg and Klugman (1983, 1984); Beirlant, Teugels, and Vynckier

(1996); and Embrechts, Kluppelberg, and Mikosch (1997) suggested employing

¨

the empirical mean excess function for selecting a preliminary model. In addition,

Benktander and Segerdahl (1960) and Benktander (1970) de¬ned two new loss

47

2.2 HAZARD RATES, MEAN EXCESS FUNCTIONS, AND TAILWEIGHT

distributions in terms of their mean excess function; see Section 7.4 for further

details.

In connection with income distributions, the concept of regular variation is also

useful in several respects. Firstly it helps to clarify the meaning of Pareto™s

coef¬cient a. Clearly, the usefulness of (regression-type) estimates of the Pareto

parameter is questionable in the absence of an underlying exact Pareto distribution.

On the other hand, empirical Pareto plots are often approximately linear for large

incomes and thus a Pareto-type distribution seems to be an appropriate model. How

does one de¬ne “Pareto type”?

In the economic literature, Mandelbrot (1960) referred to the relation

1 À F(x)

¼ 1 (for all x):

xÀa

as the strong Pareto law. This is equivalent to F following an exact Pareto

distribution. If this property is to be retained for large incomes, an appropriate

condition appears to be

1 À F(x)

¼ 1: (2:68)

lim