In the Italian literature the representation (2.3) in terms of the quantile function

was used as early as 1915 by Pietra who obviously was not aware of Lorenz™s

contribution. It has also been popularized by Piesch (1967, 1971) in the German

literature.

In the era preceding Gastwirth™s (1971) in¬‚uential article (reviving the interest in

Lorenz curves in the English statistical literature), the Lorenz curve was commonly

de¬ned in terms of the ¬rst-moment distribution. The moment distributions are

de¬ned by

°x

1

t k f (t) dt, x ! 0, k ¼ 0, 1, 2, . . . ,

F(k) (x) ¼ (2:5)

E(X k ) 0

provided E(X k ) , 1. Hence, they are merely normalized partial moments. Like the

higher-order moments themselves, the higher-order moment distributions are

dif¬cult to interpret; however, the c.d.f. F(1) (x) of the ¬rst-moment distribution

simply gives the share of the variable X accruing to the population below x. In the

context of income distributions, Champernowne (1974) refers to F(0) , that is, the

underlying c.d.f. F, as the people curve and to F(1) as the income curve.

It is thus not dif¬cult to see that the Lorenz curve can alternatively be expressed as

{(u, L(u))} ¼ {(u, v)ju ¼ F(x), v ¼ F(1) (x); x ! 0}: (2:6)

Although the representation (2.3) is often more convenient for theoretical

considerations, the “moment distribution form” (2.6) also has its moments,

especially for parametric families that do not admit a quantile function expressed

in terms of elementary functions. In the following chapters, we shall therefore use

whatever form is more tractable in a given context. It is also worth noting that several

of the distributions considered in this book are closed with respect to the formation

of moment distributions, that is, F(k) is of the same type as F but with a different set

of parameters (Butler and McDonald, 1989). Examples include the Pareto and

lognormal distributions and the generalized beta distribution of the second kind

discussed in Chapter 6.

It follows directly from (2.3) that the Lorenz curve has the following properties:

L is continuous on [0, 1], with L(0) ¼ 0 and L(1) ¼ 1.

.

L is increasing.

.

L is convex.

.

Conversely, any function possessing these properties is the Lorenz curve of a certain

statistical distribution (Thompson, 1976).

Since any distribution is characterized by its quantile function, it is clear from

(2.3) that the Lorenz curve characterizes a distribution in L up to a scale parameter

(e.g., Iritani and Kuga, 1983). It is also worth noting that the Lorenz curve itself may

be considered a c.d.f. on the unit interval. This implies, among other things, that this

23

2.1 SOME CONCEPTS FROM ECONOMICS

“Lorenz curve distribution””having bounded support”can be characterized in

terms of its moments, and moreover that these “Lorenz curve moments” characterize

the underlying income distribution up to a scale, even if this distribution is of the

Pareto type and only a few of the moments exist (Aaberge, 2000).

By construction, the quantile function associated with the “Lorenz curve

distribution” is also a c.d.f. It is sometimes referred to as the Goldie curve, after

Goldie (1977) who studied its asymptotic properties.

Although the Lorenz curve has been used mainly as a convenient graphical tool

for representing distributions of income or wealth, it can be used in all contexts

where “size” plays a role. As recently as 1992, Aebi, Embrechts, and Mikosch have

used Lorenz curves under the name of large claim index in actuarial sciences. Also,

the Lorenz curve is intimately related to several concepts from engineering statistics

such as the so-called total-time-on-test transform (TTT) (Chandra and Singpurwalla,

1981; Klefsjo, 1984; Heilmann, 1985; Pham and Turkkan, 1994). It continues to ¬nd

¨

new applications in many branches of statistics; recently, Zenga (1996) introduced a

new concept of kurtosis based on the Lorenz curve (see also Polisicchio and Zenga,

1996).

As an example of Lorenz curves, consider the classical Pareto distribution (see

Chapter 3) with c.d.f. F(x) ¼ 1 À (x=x0 )Àa , x ! x0 . 0, and quantile function

F À1 (u) ¼ x0 (1 À u)À1=a , 0 , u , 1. The mean E(X ) ¼ a x0 =(a À 1) exists if and

only if a . 1. This yields

L(u) ¼ 1 À (1 À u)1À1=a , 0 , u , 1, (2:7)

provided a . 1. We see that Lorenz curves from Pareto distributions with a different

a never intersect. Empirical Lorenz curves occasionally do intersect, so Pareto

distributions may not be useful in these situations. Figure 2.2 depicts the Lorenz

curves of two Pareto distributions, with a ¼ 1:5 and a ¼ 2:5.

It is natural to study the geometric aspects of Lorenz curves, for example, their

symmetry (or lack thereof ) with respect to the alternate diagonal

{(x, 1 À x)jx [ [0, 1]}, the line perpendicular to the line of equal distribution. A

general condition for self-symmetry was given by Kendall (1956) in the form of a

functional equation for the density

!3=2 !

E(X ) x

f (x) ¼ g log , (2:8)

x E(X )

where g( y) is an even function of y.

Clearly, the Lorenz curve of the Pareto distribution (2.7) does not possess this

symmetry property. The best known example of a distribution with self-symmetric

Lorenz curves is the lognormal; see Figure 4.3 in Chapter 4. See also

Champernowne (1956), Taguchi (1968), and especially Piesch (1975) for further

details on the geometry of Lorenz curves.

24 GENERAL PRINCIPLES

Figure 2.2 Lorenz curves of two Pareto distributions: a ¼ 1:5 (solid) and a ¼ 2:5 (dashed).

Arnold et al. (1987) observed that every distribution F which is strongly

unimodal and symmetric about 0 leads to a self-symmetric Lorenz curve

representable as Lt (u) ¼ F[F À1 (u) À t], u [ (0, 1), t ! 0. The prime example is

the normal distribution that leads to a lognormal Lorenz curve.

Figure 2.2 also prompts us to compare two distributions, in a global sense, by

comparing their corresponding Lorenz curves. If two Lorenz curves do not intersect,

it may perhaps be appropriate to call the distribution with the lower curve “more

unequal” or “more variable,” and indeed a stochastic ordering based on this notion,

the Lorenz partial ordering, was found to be a useful tool in many applications. For

X1 , X2 [ L, the Lorenz ordering is de¬ned as

for all u [ [0, 1]:

X1 !L X2 :( F1 !L F2 :( LX1 (u) LX2 (u), (2:9)

) )

Here X1 is said to be larger than X2 (or more unequal) in the Lorenz sense. From

(2.3) it is evident that the Lorenz ordering is scale-free; hence,

a, b . 0:

X1 !L X2 ( a Á X1 !L b Á X2 , (2:10)

)

25

2.1 SOME CONCEPTS FROM ECONOMICS

Economists usually prefer to denote the situation where LX1 LX2 as X2 !L X1 ,

because F2 is, in a certain sense, associated with a higher level of economic welfare

(Atkinson, 1970). We shall use the notation (2.9) that appears to be the common one

in the statistical literature, employed by Arnold (1987) or Shaked and Shanthikumar

(1994), among others.

Among the methods for verifying Lorenz ordering relationships, we mention that

d

if X2 ¼ g(X1 ), then (Fellman, 1976)

g(x)

is nonincreasing on (0, 1) ¼ X1 !L g(X1 ): (2:11)

)

x

Under the additional assumption that g be increasing on [0, 1), the condition is also

necessary (Arnold, 1987). This result is useful, among other things, in connection