2 2 2

2 2

v ¼p (x À m) dF(x) þ (1 À p) (x À m) dF(x) À :

4

m 0

Although the Gini and Pietra coef¬cients are directly related to the Lorenz curve

and can therefore be treated using results on the asymptotics of linear functions of

order statistics, this is not true for the Atkinson and generalized entropy measures.

However, these quantities are simple functions of the moments of the size

38 GENERAL PRINCIPLES

distribution, so it is natural to estimate them by the method of moments. For

complete random samples the large sample properties of the resulting estimators

1=e

m(e)

^

An,e ¼ 1 À (2:40)

m(1)

and

& '

1 m(u)

c

GEn,u À1 , (2:41)

¼

u(u À 1) m(1)u

where m(e) denotes the sample moment of order e, were derived by ¬¬Thistle (1990)

p¬

and Kakwani (1990). Itp¬¬¬also convenient to denote the variance of nm(e) by s 2 (e)

is p¬¬¬

and the covariance of nm(e) and nm(e 0 ) by g (e, e 0 ).

Provided the required moments exist, a straightforward application of the delta

method yields

^

p¬¬¬ An,e À Ae d

n p¬¬¬¬¬¬¬¬¬¬ À N (0, 1), (2:42)

!

vA (e)

where

!2 ( )

!

em(e) 2 2

1 À Ae 2em(e)

s 2 (e) À

vA (e) ¼ g (e, 1) þ s, (2:43)

em(e) m m

with m(e) :¼ E(X e ) and s 2 (e) :¼ m(2e) À m(e)2 Thistle also showed that the

^

estimator Ae is strongly consistent and that (2.43) can be consistently estimated by

replacing the population moments with their sample counterparts, thus allowing for

the construction of asymptotic con¬dence intervals and tests based on (2.42).

An analogous result is available for the generalized entropy measures

c

p¬¬¬ GEn,u À GEu d

n p¬¬¬¬¬¬¬¬¬¬¬¬¬ À N (0, 1), (2:44)

!

vGE (u)

where

È22 É

1

m s (u) À 2umm(u)g(u, 1) þ u 2 m2 (u)s 2 ,

vGE (u) ¼ (2:45)

u 2 (u À 1)2 m 2(uþ1)

Again, (2.45) can be estimated consistently so that asymptotic tests and con¬dence

intervals based on (2.44) are feasible.

Alternatively, con¬dence intervals for inequality measures may be obtained via

the bootstrap method; see Mills and Zandvakili (1997) for a bootstrap approach in

39

2.1 SOME CONCEPTS FROM ECONOMICS

the case of the Theil and classical Gini coef¬cients and Xu (2000) for the

generalized Gini case.

2.1.5 Multivariate Lorenz Curves

With the increasing use of multivariate data, multivariate Lorenz curves are no doubt

the wave of the future. So far only a limited number of results are available on this

challenging concept. The following exposition draws heavily on Koshevoy and

Mosler (1996).

Taguchi (1972a,b) suggested de¬ning a bivariate Lorenz curve”or rather

surface”as the set of points [s, t, LT (s, t)] [ R3 , where

þ

°

xa (x) dF(x),

s¼

R2

þ

°

xa (x)˜ 1 dF(x),

t¼ x

R2

þ

°

LT (s, t) ¼ xa (x)˜ 2 dF(x)

x

R2

þ

and

&

1, x a,

xa (x) ¼

0, otherwise,

with x ¼ (x1 , x2 ), a ¼ (a1 , a2 ), 0 aj 1, and xi ¼ xi =E(Xi ), i ¼ 1, 2. [x a is

˜

de¬ned in the componentwise sense.]

This may be called the Lorenz“Taguchi surface. A problem with Taguchi™s proposal

is that it does not treat the coordinate random variables in a symmetric fashion.

Arnold (1983, 1987) introduced the following de¬nition which is somewhat

easier to handle. The Lorenz “Arnold surface LA is the graph of the function

Ð j 1 Ð j2

x1 x2 dF(x1 , x2 )

A

L (u1 , u2 ) ¼ Ð 0 Ð 0 (2:46)

11

0 0 x1 x2 dF(x1 , x2 )

where

° ji

dF (i) (xi ),

ui ¼ 0 ui 1, i ¼ 1, 2,

0

F (i) , i ¼ 1, 2, being the marginals of F and ji [ Rþ . An appealing feature of this

approach is that if F is a product distribution function, F(x1 , x2 ) ¼ F (1) (x1 ) Á F (2) (x2 ),

then LA (u1 , u2 ) is just the product of the marginal Lorenz curves. Hence for two

40 GENERAL PRINCIPLES

product distribution functions F and G, LA (u1 , u2 ) LA (u1 , u2 ) if and only if the