{[u, Z(u)] j u [ (0, 1)}

is the Zenga concentration curve. Note that F(1) F implies F À1 F(1) , so that the

À1

Zenga curve belongs to the unit square. It is also evident from (2.52) that the curve is

scale-free.

It is instructive to compare the Zenga and Lorenz curves: For the Lorenz curve,

the amount of concentration associated with the u Á 100% poorest of the population

is described by the difference u À L(u). Rewriting the normalized form [u À L(u)]=u

in the form

F(1) [F À1 (u)]

u À L(u)

¼1À ,

F[F À1 (u)]

u

we see that concentration measurement according to Lorenz amounts to comparing

the c.d.f. F and the c.d.f. of the ¬rst-moment distribution F(1) at the same abscissa

F À1 (u). An alternative approach consists of comparing the abscissas at which F and

F(1) take the same value u. This is the idea behind the Zenga curve. Zenga refers to

Z(u) as a point measure of inequality.

If the Zenga curve is an alternative to the Lorenz curve, the question of what

corresponding summary measures look like arises. There are several possibilities to

aggregate the information contained in the point measure (2.52) into a single

coef¬cient. Zenga (1984, 1985) proposed two measures; the ¬rst suggested taking

the arithmetic mean of the Z(u), u [ [0, 1],

°1 °1

F À1 (u)

j¼ Z(u) du ¼ 1 À du, (2:53)

À1

0 F(1) (u)

0

43

2.2 HAZARD RATES, MEAN EXCESS FUNCTIONS, AND TAILWEIGHT

whereas the second utilizes the geometric mean of the ratios F À1 (u)=F(1) (u) and is

À1

therefore given by

(° " # )

1 À1

F (u)

j2 ¼ 1 À exp du : (2:54)

log À1

F(1) (u)

0

j can be rewritten in the form

°1

1

F À1 [L(u)] du:

j¼1À

E(X ) 0

Compared to the Lorenz curve, the Zenga curve is somewhat more dif¬cult to

interpret; it is neither necessarily continuous nor is it convex (or concave). For a

classical Pareto distribution with c.d.f. F(x) ¼ 1 À (x=x0 )Àa , 0 , x0 x, and a . 1,

we obtain F À1 (u) ¼ x0 (1 À u)À1=a and F(1) (u) ¼ x0 (1 À u)À1=(aÀ1) and therefore

À1

Z(u) ¼ 1 À (1 À u)1=[a(aÀ1)] , 0 , u , 1:

Clearly, the Zenga curve of the Pareto distribution is an increasing function on

[0, 1], approaching the u axis with increasing a. Recall from (2.7) that in connection

with the Lorenz ordering, an increase in a is associated with a decrease in inequality.

It is therefore natural to call a distribution F2 less concentrated than another

distribution F1 if its Zenga curve is nowhere above the Zenga curve associated with

F1 and thus to de¬ne a new ordering via

X1 !Z X2 :( Z1 (u) ! Z2 (u) for all u [ (0, 1): (2:55)

)

In general, the Lorenz and Zenga orderings are unrelated; it is however interesting

that the Zenga measure j satis¬es the Lorenz ordering (Berti and Rigo, 1995).

Further research in connection with Zenga and Lorenz orderings may be worthwhile.

2.2 HAZARD RATES, MEAN EXCESS FUNCTIONS,

AND TAILWEIGHT

Researchers in the actuarial sciences have addressed the problem of distinguishing

among various skewed probability distributions given sparse observations at the

right tail. Speci¬cally, it has been suggested to employ what Benktander (1963)

called the mortality of claims

f (x)

r(x) ¼ , x ! 0, (2:56)

1 À F(x)

44 GENERAL PRINCIPLES

and what Benktander and Segerdahl (1960) called the average excess claim

Ð1

(t À x) dF(t)

x

Ð1

e(x) ¼ E(X À x j X . x) ¼ , x ! 0, (2:57)

dF(t)

x

for distinguishing among potential models. See Benktander (1962, 1963) and

Benktander and Segerdahl (1960) for some early work in the actuarial literature.

The mortality of claims and the average excess claim are more widely known

under different names. The former is usually called the “hazard rate” or “failure rate”

(in reliability theory) and also the “force of mortality” (in life insurance) or

“intensity function” (in extreme value theory); the latter is also known as the “mean

residual life function” (notably in biometrics and engineering statistics) or the “mean

excess function” (in actuarial applications). We shall use the terms hazard rate and

mean excess function in the sequel.

The hazard rate gives the rate at which the risk of large claims is decreasing when

x grows. Shpilberg (1977) argued that this function has a direct connection with the

physical progress of ¬re in ¬re insurance: Most ¬res are extinguished quickly after

they start and the amount of any related claims remains slight. However, if early

extinction fails, then the chance of rapidly stopping the ¬re decreases, which in

the case of large risk units results in large claims and consequently long tails of the

distribution. In Benktander™s (1963) view, the lower the claims™ rate of mortality, the

skewer and more dangerous is the claim distribution.

Integration by parts shows that the mean excess function can alternatively be

expressed in the form

°1

1

e(x) ¼ F (t) dt, x0 x; (2:58)

F (x) x

where F ¼ 1 À F: Conversely, in the continuous case the c.d.f. can also be recovered

from the mean excess function via

& °x '

e(x0 ) 1

x ! x0 :

F(x) ¼ 1 À dt , (2:59)

exp À

e(x) x0 e(t)

Often (but not always), we shall encounter the case where x0 ¼ 0. It is a direct