this particular case, we have the following result: If E(log X ) and E(log2 X ) are

prescribed, the lognormal distribution is the maximum entropy distribution on [0, 1)

(Kapur, 1989, p. 68). (The result may, of course, also be stated as prescribing the

geometric mean and variance of logarithms.)

From (4.3) we know that products of independent lognormal random variables

are also lognormally distributed. This can be extended to a characterization

parallelling Cramer™s (1936) celebrated characterization of the normal distribution: If

´

Xi ; i ¼ 1, 2; are independent and positive, then their product X1 X2 is lognormally

distributed if and only if each Xi follows a lognormal distribution.

4.5 LORENZ CURVE AND INEQUALITY MEASURES

Unfortunately, the lognormal Lorenz curve cannot be expressed in a simple closed

form; it is given implicitly by

L(u) ¼ F[FÀ1 (u) À s 2 ], 0 , u , 1: (4:16)

It follows directly from the monotonicity of F that the Lorenz order is linear within

the family of two-parameter lognormal distributions, speci¬cally

)2 2

X1 ! L X2 ( s1 ! s2 : (4:17)

This basic result can be derived in various other ways: First, as was noted

above, the parameter exp(m) is a scale parameter and hence plays no role in

116 LOGNORMAL DISTRIBUTIONS

connection to the Lorenz ordering. Thus, for Xi $ LN(0, si2 ), i ¼ 1, 2; we have

d

X1 ¼ f(X2 ); where f(x) ¼ xs1 =s2 ; cf. (4.13). Therefore, F1 F2 ¼ f(F2 F2 ) ¼ f:

À1 À1

This function is clearly convex iff s1 ! s2 : Hence, X1 is more spread out than X2

in the sense of the convex (transform) order that is known to imply the Lorenz

ordering (see Chapter 2). Second, a slightly different argument due to Fellman

(1976) uses the fact that the function f is star-shaped, that is, that f(x)=x is

increasing on [0, 1); which implies that X1 is more spread out than X2 in the sense

of the star-shaped ordering, an ordering that is also known to imply the Lorenz

ordering (see Chapter 2). (In fact, the star-shaped ordering is intermediate between

the convex and Lorenz orderings.) For a third approach, see Arnold et al. (1987),

who showed that the result (4.17) follows also from the strong unimodality of the

normal distribution.

The lognormal Lorenz curve has an interesting geometric property, namely, it is

symmetric about the alternate diagonal of the unit square; see Figure 4.3. This can be

veri¬ed analytically using Kendall™s condition (2.8). At the point of intersection with

Figure 3 Lognormal Lorenz curves: s ¼ 0:5(0:5)3 (from left to right).

117

4.5 LORENZ CURVE AND INEQUALITY MEASURES

the alternate diagonal, that is, the point [w(s=2), w(Às=2)]; the slope of the Lorenz

curve is unity (e.g., Moothathu, 1981).

The Lorenz curve may alternatively be represented in terms of the ¬rst-

moment distribution. The c.d.f.™s of the higher-order moment distributions of

lognormal distributions can be expressed as the c.d.f.™s of lognormal distributions

with a different set of parameters (see, e.g., Aitchison and Brown, 1957).

Speci¬cally, for X $ LN(m, s 2 ) we have

F(k);m,s 2 (x) ¼ F(0) (x; m þ ks 2 , s 2 ), (4:18)

where F(0) stands for the c.d.f. of the lognormal distribution.

This closure property is most useful in connection with size phenomena. In

particular, the lognormal Lorenz curve can now be expressed as

{[u, L(u)]ju [ [0, 1]} ¼ {[F(0) (x), F(1) (x)]jx [ [0, 1)}:

It can also be exploited for the derivation of various inequality measures. The

Pietra and Gini coef¬cients are remarkably simple; the former is given by

s

P ¼ 2F À1 (4:19)

2

and the Gini coef¬cient equals

s

G ¼ 2F p¬¬¬ À 1: (4:20)

2

The Theil measure is

!

X X 1

¼ s 2,

T1 ¼ E (4:21)

log

E(X ) E(X ) 2

which coincides with the expression for Theil™s second measure T2 : The variance of

logarithms is, of course,

VL(X ) ¼ var(log X ) ¼ s 2 : (4:22)

It is interesting that the preceding three coef¬cients are closely related in the case of

the lognormal distribution, namely,

1

T1 ¼ T2 ¼ VL(X )

2

[Theil (1967), see also Maasoumi and Theil (1979)]. It should also be noted that

(4.22) is increasing in s; in agreement with (4.17); thus, in the lognormal case the

118 LOGNORMAL DISTRIBUTIONS

variance of logarithms satis¬es the Lorenz ordering (Hart, 1975). (As mentioned in

Chapter 2, this is not true in general.)

Two modi¬cations of the ¬rst Theil coef¬cient with the median and the mode

replacing the mean were provided by Shimizu and Crow (1988, p. 11). They are

! 2

s

X X 2

¼ s exp

Tmed (X ) ¼ E log

xmed xmed 2

and

! 2

X X 3s

2

:

Tmode (X ) ¼ E ¼ 2s exp

log

xmode xmode 2

The Zenga curve is of the form (Zenga, 1984)

2

Z(u) ¼ 1 À eÀs , 0 u 1, (4:23)

and hence constant. Note that it is an increasing function of the shape parameter s ;

hence, inequality is increasing in s according to both Lorenz and Zenga curves

(Polisicchio, 1990). Zenga™s ¬rst index is given by (Zenga, 1984)

j ¼ 1 À exp(Às 2 ), (4:24)

which is also the expression for his second measure j2 (Pollastri, 1987a). Thus,

like the two Theil coef¬cients, the two Zenga coef¬cients coincide in the lognormal

case.