Closely connected conditions are in terms of two stronger concepts of stochastic

ordering, the convex and star-shaped orderings. For two distributions Fi , i ¼ 1, 2,

supported on [0, 1) or a subinterval thereof, distribution F1 is said to be convex

(star-shaped) ordered with respect to a distribution F2 , denoted as

À1 À1

F1 !conv F2 (F1 !Ã F2 ), if F1 F2 is convex [F1 F2 (x)=x is nonincreasing] on the

support of F2 . It can be shown that the convex ordering implies the star-shaped

ordering, which in turn implies the Lorenz ordering (Chandra and Singpurwalla,

1981; Taillie, 1981).

These criteria are useful when the quantile function is available in a simple closed

form, as is the case with, among others, the Weibull distribution; see Chapter 5.

Several further methods for verifying Lorenz dominance were discussed by Arnold

(1987) or Kleiber (2000a).

Various suggestions have been made as to how to proceed when two Lorenz

curves intersect. In international comparisons of income distributions, this is

particularly common for countries on different economic levels, for example,

industrialized and developing countries. This suggests that the problem can be

resolved by scaling up the Lorenz curves by the ¬rst moment, leading to the

so-called generalized Lorenz curve (Shorrocks, 1983; Kakwani, 1984)

°u

GL(u) ¼ E(X ) Á L(u) ¼ F À1 (t) dt, 0 , u , 1: (2:12)

0

In contrast to the classical Lorenz curve, the generalized Lorenz curve is no longer

scale-free and so it completely determines any distribution with a ¬nite mean (Thistle,

1989a). The associated ordering concept is the generalized Lorenz ordering, denoted

as X1 !GL X2 . In economic parlance, the generalized Lorenz ordering is a welfare

order, since it takes not only distributional aspects into account (as does the Lorenz

ordering) but also size-related aspects such as the ¬rst moment. In statistical terms it is

simply second-order stochastic dominance (SSD), since (e.g., Thistle, 1989b)

°x °x

X1 !GL X2 ( F1 (t) dt ! F2 (t) dt, for all x ! 0:

)

0 0

26 GENERAL PRINCIPLES

Shorrocks and others have provided many empirical examples for which generalized

Lorenz dominance applies; hence, this extension appears to be of considerable

practical importance.

Another variation on this theme is the absolute Lorenz ordering introduced by

Moyes (1987); it replaces scale invariance with location invariance and is de¬ned in

terms of the absolute Lorenz curve

°u

{F À1 (t) À E(X )} dt, 0 , u , 1:

AL(u) ¼ E(X ) Á {L(u) À u} ¼ (2:13)

0

However, these proposals clearly do not exhaust the possibilities. See Alzaid

(1990) for additional Lorenz-type orderings de¬ned via weighting functions that

emphasize certain parts of the Lorenz curves.

2.1.2 Parametric Families of Lorenz Curves

In view of the importance of the Lorenz curve in statistical and economic analyses of

income inequality, it should not come as a surprise that a plethora of parametric

models for approximating empirical Lorenz curves has been suggested. Since the

Lorenz curve characterizes a distribution up to scale, it is indeed quite natural to start

directly from the Lorenz curve (or the quantile function), especially since many

statistical of¬ces report distributional data in the form of quintiles or deciles,

occasionally even in the form of percentiles. In these cases the shape of the income

distribution is only indirectly available and perhaps not even required if an

assessment of inequality associated with the distribution is all that is desired. In

short, does one ¬t a distribution function to the data and obtain the implied Lorenz

curve (and Gini coef¬cient), or does one ¬t a Lorenz curve and obtain the implied

distribution function (and Gini coef¬cient)?

The pioneering effort of Kakwani and Podder (1973) triggered a veritable

avalanche of papers concerned with the direct modeling of the Lorenz curve, of

which we shall only present a brief account. Since any function that passes through

(0, 0) and (1, 1) and that is monotonically increasing and convex in between is a

bona ¬de Lorenz curve, the possibilities are virtually endless. Kakwani and Podder

(1973, 1976) suggested two forms. Their 1973 form is

L(u) ¼ ud eÀh(1Àu) , 0 , u , 1, (2:14)

where h . 0 and 1 , d , 2, whereas the more widely known second form

(Kakwani and Podder, 1976) has a geometric motivation. Introducing a new

coordinate system de¬ned in terms of

1 1

h ¼ p¬¬¬ (u þ v) and p ¼ p¬¬¬ (u À v), (2:15)

2 2

27

2.1 SOME CONCEPTS FROM ECONOMICS

where 0 , u , 1 and v ¼ L(u), this form is given by

p¬¬¬

h ¼ ap ( 2 À p)b :

a

(2:16)

Here a ! 0, 0 a 1 and 0 , b 1. This model amounts to expressing a point

[F(x), F(1) (x)] on the Lorenz curve in the form (p, h), where h is the length of the

ordinate from [F(x), F(1) (x)] on the egalitarian line and p is the distance of the

ordinate from the origin along the egalitarian line. [As pointed out by Dagum (1986),

the new coordinate system (2.15) was initially introduced by Gini (1932) in the

Italian literature.]

Other geometrically motivated speci¬cations include several models based on

conic sections: Ogwang and Rao (1996) suggested the use of a circle™s arc, Arnold

˜

(1986) employed a hyperbolic model, whereas Villasenor and Arnold (1989) used a

segment of an ellipse. Although the resulting ¬t is sometimes excellent, all these

models have the drawbacks that their parameters must satisfy certain constraints

which are not easily implemented in the estimation process and also that the

expressions for the Gini coef¬cients are somewhat formidable (an exception is the

Ogwang “ Rao speci¬cation).

A further well-known functional form is the one proposed by Rasche et al. (1980)

who suggested

L(u) ¼ [1 À (1 À u)a ]1=b , 0 , u , 1, (2:17)

where 0 , a, b 1. This is a direct generalization of the Lorenz curve of the Pareto

distribution (2.7) obtained for b ¼ 1 and a , 1. For a ¼ b the curve is self-

symmetric (in the sense of Section 2.1.1), as pointed out by Anstis (1978).

In order to overcome the drawback of many previously considered functional

forms, namely, a lack of ¬t over the entire range of income, several authors have

proposed generalizations or combinations of the previously considered functions.

Quite recently, Sarabia, Castillo, and Slottje (1999) have suggested a family of

parametric Lorenz curves that synthetizes and uni¬es some of the previously

considered functions. They point out that for any Lorenz curve L0 the following

curves are also Lorenz curves that generalize the initial model L0 :

L1 (u) ¼ ua L0 (u), 0 , u , 1, where either a ! 1 or 0 a , 1 and L000 (u) ! 0.

. 0

L2 (u) ¼ {L0 (u)}g , 0 , u , 1, where g ! 1.

.

L3 (u) ¼ ua {L0 (u)}g , 0 , u , 1, and a, g ! 1.

.

An advantage of this approach is that Lorenz ordering results are easily obtained,

in particular

L1 (u; a1 ) !L L1 (u; a2 ), if and only if a1 ! a2 . 0.

.

L2 (u; g1 ) !L L2 (u; g2 ), if and only if g1 ! g2 . 0.

.

A combination of the preceding two cases yields results for L3.

.

28 GENERAL PRINCIPLES

For the particular choice L0 (u) ¼ 1 À (1 À u)k , k 1, the Lorenz curve of the

Pareto distribution (2.7), Sarabia, Castillo, and Slottje obtained a class of parametric