(the classical Gini coef¬cient), we obtain

XÈ

n É

^ n,2 ¼ 1 À 1 (n À i þ 1)2 À (n À i)2 Xi:n

G

X n n2 i¼1

1 XX n n

jXi À Xj j,

¼

X n n2 i¼1 j¼1

36 GENERAL PRINCIPLES

the numerator of which is recognized as a variant of a measure of spread commonly

referred to as Gini™s mean difference; see, for example, David (1968).

From (2.37) we moreover see that the weight function (1 À t)n is smooth and

bounded (for n . 1) on [0, 1]. Using limit theory for such functions of order

statistics (see, e.g., Stigler, 1974), Zitikis and Gastwirth obtained for n . 1 under the

sole assumption that E(X 2 ) is ¬nite

p¬¬¬ d

^ 2

n(Gn,n À Gn ) À N (0, sF,n ),

!

where

1È É

sF (n, n) þ 2(Gn À 1)sF (1, n) þ (Gn À 1)2 sF (1, 1)

2

sF,n ¼ (2:38)

m2

with, for a, b [ {1, n},

°1 °1

{F(s ^ t) À F(s)F(t)}{1 À F(s)}aÀ1 {1 À F(t)}bÀ1 ds dt:

sF (a, b) ¼

0 0

Alternative expressions for the asymptotic variance of the classical Gini index

(n ¼ 2) may be found in Goldie (1977) and Sendler (1979).

To apply this result for the construction of approximate con¬dence intervals, a

2

consistent estimate of (2.38) is required. A simple nonparametric estimate of sF,n is

obtained by replacing F in (2.38) by its empirical counterpart Fn . This yields the

estimator

1È É

sn (n, n) þ 2(Gn,n À 1) sn (1, n) þ (Gn,n À 1)2 sn (1, 1)

s2 ¼

n,n

2

X

with, for a, b [ {1, n},

XX

nÀ1 nÀ1

f(n) (a, b)(Xiþ1:n À Xi:n )(Xjþ1:n À Xj:n ),

sn (a, b) ¼ ij

i¼1 j¼1

where

& '

i aÀ1 j bÀ1

ij ij

f(n) (a, b) ¼ ab ^ :

1À 1À

À

ij

nn nn n n

Under the previously stated assumptions, Zitikis and Gastwirth showed that this

estimator is strongly consistent, implying that an approximate 100(1 À a)%

con¬dence interval for Gn is given by

p¬¬¬

^

Gn,n + za=2 sn,n = n,

37

2.1 SOME CONCEPTS FROM ECONOMICS

where za=2 denotes the a=2 fractile of the standard normal distribution. For the case

where instead of complete data only a vector of Lorenz curve ordinates is available,

Barrett and Pendakar (1995) provided the asymptotic results for the generalized Gini

measures following the Beach and Davidson (1983) approach.

A similar derivation yields the asymptotic distribution of the Pietra coef¬cient,

although the presence of absolute values provides some additional complications. A

natural estimator of (2.25) is

Pn

jXi À X j D

^ i¼1

¼: :

Pn ¼ (2:39)

2nX X

Writing

X X X

n

jXi À X j ¼ (X À X i ) þ (Xi À X ),

i¼1 Xi .X

Xi X

we see that 2nD may be expressed in the form

!

X X

n

jXi À X j ¼ 2 N X À Xi ,

i¼1 Xi X

where N denotes the random number of observations less than X .

Gastwirth (1974) showed that if E(X 2 ) , 1 and the underlying density is

continuous in the neighborhood of m ¼ E(X ), the numerator and denominator of

(2.39) are asymptotically jointly bivariate normally distributed. Hence, an

application of the delta method yields

p¬¬¬ d

^ 2

n(Pn À P) À N (0, sP );

!

2

where the asymptotic variance sP has the somewhat formidable representation

!

°m

v 2 d2 s 2 d

2

À 3 ps 2 À 2

sP (x À m) dF(x) ,

¼ 2þ

4m2 m

m 0

with p ¼ F(m), d ¼ EjX À E(X )j and

°1 °m