It is of some at least theoretical interest that income distributions can be

characterized in terms of these generalized Gini coef¬cients, which is equivalent”in

31

2.1 SOME CONCEPTS FROM ECONOMICS

view of (2.22)”to a characterization in terms of the ¬rst moments of the order

statistics. As we shall see in the following chapters, most parametric models for

the size distribution of incomes possess heavy (polynomial) tails, so only a few

of the moments exist. However, these distributions are determined by the sequence

of the associated generalized Gini coef¬cients, provided the mean is ¬nite (Kleiber

and Kotz, 2002).

Another classical index is the Pietra coef¬cient or relative mean deviation,

de¬ned as

EjX À E(X )j

:

P¼ (2:25)

2E(X )

It has the interesting geometrical property of being equal to the maximum distance

between the Lorenz curve and the equality line.

Being a natural scale-free index, the variance of logarithms

VL(X ) ¼ var( log X ) (2:26)

has also attracted some attention, especially in applied work, apparently because of

its simple interpretation in connection with the popular lognormal distribution.

However, Foster and Ok (1999) showed that it can grossly violate the Lorenz

ordering, thus casting some doubt on its usefulness. [Earlier although less extreme

results in this regard were obtained by Creedy (1977).]

Among the many further inequality coef¬cients proposed over the last 100 years,

we shall con¬ne ourselves to two one-parameter families that are widely used in

applied work. These are the Atkinson (1970) measures

&° 1 '1=(1Àe)

1

x1Àe dF(x)

Ae ¼ 1 À , (2:27)

E(X ) 0

where e . 0 is a sensitivity parameter giving more and more weight to the small

incomes as it increases, and the so-called generalized entropy measures (Cowell and

Kuga, 1981)

° 1 "& #

'

xu

1

GEu ¼ À1 dF(x), (2:28)

u(u À 1) 0 E(X )

where u [ Rn{0, 1}. Again, u is a sensitivity parameter emphasizing the upper tail

for u . 0 and the lower tail for u , 0. It is worth noting that for u ¼ 2 we

essentially obtain the squared coef¬cient of variation; hence, this widely used

characteristic of a distribution is of special signi¬cance in connection with size

phenomena. For u ¼ 0, 1 the generalized entropy coef¬cients are de¬ned via a

limiting argument, yielding

& '

°1

x x

T1 :¼ GE1 ¼ dF(x) (2:29)

log

0 E(X ) E(X )

32 GENERAL PRINCIPLES

and

& '

°1

E(X )

T2 :¼ GE0 ¼ dF(x): (2:30)

log

x

0

The latter two measures are known as the Theil coef¬cients”after Theil (1967)

who derived them from information-theoretic considerations”T1 being often

referred to as the Theil coef¬cient and T2 as Theil™s second measure or the mean

logarithmic deviation.

A drawback of both families of coef¬cients is that they are simple functions of the

moments of the distributions; hence, they will only be meaningful for a limited range

of the sensitivity parameters e and u if the underlying distribution possesses only a

few ¬nite moments (Kleiber, 1997). Unfortunately, this is precisely the type of

distributions we shall encounter below.

2.1.4 Sampling Theory of Lorenz Curves and Inequality Measures

In view of the long history of inequality measurement, it is rather surprising that only

comparatively recently the asymptotics of time-honored tools such as the Lorenz

curve and associated inequality measures have been investigated. A possible

explanation is that in earlier literature income data were often believed to come from

censuses. Nowadays it is however generally acknowledged that most data are, in fact,

obtained from surveys (although not necessarily from simple random samples). In

addition, the applications of Lorenz curves extend to other areas such as actuarial

science where samples may be much smaller. This creates the need for an adequate

theory of sampling variation.

As mentioned above, it would by now be easy to write a 500-page monograph

dealing exclusively with inequality measurement, parametric and nonparametric,

classical and computer-intensive. Our following brief account presents the core

results in the sampling theory of Lorenz curves and some popular inequality

measures when raw microdata are available, and only in the case of complete data.

[We do not even discuss the celebrated Gastwirth (1972) bounds on the Gini

coef¬cient and its associated sampling theory.]

Lorenz Curves

The pointwise strong consistency of the empirical Lorenz curve was proved by Gail

and Gastwirth (1978) and Sendler (1979) under the assumption of a ¬nite mean and

uniqueness of the quantile under consideration. The question arises if, and under

what conditions, the entire empirical Lorenz curve can be considered a good

estimator of its theoretical counterpart. This requires the use of more advanced

probabilistic tools from the theory of weak convergence of stochastic processes. The

¬rst results in this area were due to Goldie (1977) who showed that (1) the empirical

Lorenz curve Ln converges almost surely uniformly to the population Lorenz curve

a:s:

sup jLn (u) À L(u)j À 0

!

u[[0,1]

33

2.1 SOME CONCEPTS FROM ECONOMICS

(a Glivenko“Cantelli-type result), and (2) a functional central limit theorem holds.

For the latter one must consider the normed difference between the empirical and

theoretical Lorenz curves”the empirical Lorenz process

p¬¬¬

n{Ln À L};

under appropriate regularity conditions it converges weakly (in the space C[0, 1] of

continuous functions on [0, 1]) to a Gaussian process that is related to (without being

identical to) the familiar Brownian bridge process.

More recent results along these lines include works on the rate of convergence of

the empirical Lorenz process (a function-space law of the iterated logarithm) due to

Rao and Zhao (1995), and subsequently re¬ned by Csorgo and Zitikis (1996, 1997).

¨ ´´

These results allow for the construction of asymptotic con¬dence bands for the

entire Lorenz curves; see Csorgo Gastwirth, and Zitikis (1998). [Con¬dence

¨ ´´,

intervals for single points on the Lorenz curves”a much easier problem”were

obtained by Sendler (1979).]

However, although the empirical process approach considering the entire Lorenz

curve as the random element of interest is perhaps the most appropriate setup from a

theoretical point of view, in applied work it is often suf¬cient to consider the Lorenz

curve at a ¬nite set of points (the deciles, say). A line of research dealing with

inference based on a vector of Lorenz curve ordinates was initiated by Beach and

Davidson (1983). Consider the k points 0 , u1 , u2 , Á Á Á , uk , 1, with

corresponding quantiles F À1 (ui ), i ¼ 1, . . . , k. (Note that under the general

assumption of this chapter”namely, that F be supported on an interval”these

quantiles are unique.) Writing the conditional mean of incomes less than or equal to

F À1 (ui ) as gi :¼ E[X jX F À1 (ui )], i ¼ 1, . . . , k, and m ¼ E(X ) we can express the

corresponding Lorenz curve ordinates in the form

° F À1 (ui ) ° F À1 (ui )

g