Àa !u

x

, 0 , x0 x,

F(x) ¼ 1 À (3:103)

x0

where a, u . 0. The classical Pareto distribution is obtained for u ¼ 1. The p.d.f. is

Àa !uÀ1

x

a ÀaÀ1

f (x) ¼ uax0 x , 0 , x0 x,

1À (3:104)

x0

and the quantile function is given by

F À1 (u) ¼ x0 (1 À u1=u )À1=a , 0 , u , 1: (3:105)

95

3.8 STOPPA DISTRIBUTIONS

Stoppa™s generalized Pareto distribution can be derived from a differential

equation for the elasticity h(x, F) of the distribution function. If one supposes that

(1) the resulting income density is either unimodal or decreasing, (2) the support of

the distribution is [x0 , 1), for some x0 . 0, (3) h(x, F) is a decreasing function of

F(x), with limx!x0 h(x, F) ¼ 1 and limx!1 h(x, F) ¼ 0, the differential equation

!

1 À [F(x)]1=u

F 0 (x)

h(x, F) ¼ Á x ¼ au a, u . 0,

, (3:106)

[F(x)]1=u

F(x)

leads to an income distribution with the p.d.f. (3.104).

For integer values n of u, the distribution can be viewed as the distribution of Xn:n

from a Pareto parent distribution; cf. (3.35). Thus, the distribution itself is closed

under maximization, namely,

X $ Stoppa(x0 , a, u) ¼ Xn:n $ Stoppa(x0 , a, nu): (3:107)

)

In contrast, the Pareto distribution is closed under minimization; cf. (3.34).

Compared to the classical Pareto distribution, the Stoppa distribution is more

¬‚exible since it has an additional shape parameter u that allows for unimodal (for

u . 1) and zeromodal (for u 1) densities. The mode is at

1=a

1 þ ua

u . 1,

xmode ¼ x0 , (3:108)

1þa

and at x0 otherwise. As u increases, the mode shifts to the right. Figure 3.2 illustrates

the effect of the new parameter u.

The kth moment exists for k , a and equals

k

E(X k ) ¼ uxk B 1 À , u : (3:109)

0

a

The Lorenz curve is of the form

Bz (u, 1 À 1=a)

, 0 , u , 1,

L(u) ¼ (3:110)

B(u, 1 À 1=a)

where z ¼ u1=u and Bz denotes the incomplete beta function, and the Gini coef¬cient

is given by

2B(2u, 1 À 1=a)

G¼ À 1: (3:111)

B(u, 1 À 1=a)

It follows that for Xi $ Stoppa(x0 , ai , u), i ¼ 1, 2, with 1 , a1 a2 , we have

X1 !L X2 , and for Xi $ Stoppa(x0 , a, ui ), i ¼ 1, 2, with u1 u2 , we have X1 L X2 .

Analogous implications hold true for the Zenga ordering (Polisicchio, 1990).

Consequently, the Gini coef¬cient is an increasing function of u, for ¬xed a, and a

96 PARETO DISTRIBUTIONS

Stoppa densities: x0 ¼ 1, a ¼ 1:5, and u ¼ 1(1)5 (from top left).

Figure 2

decreasing function of a, for ¬xed u. Also, for ¬xed u and a ! 1 (a ! 1) the

Lorenz curve tends to the Lorenz curve associated with maximal (minimal)

concentration, whereas for ¬xed a and u ! 0 (u ! 1) the Lorenz curve tends to

the Lorenz curve associated with minimal (maximal) concentration (Domma, 1994).

Zenga™s inequality measure j2 is given by (Stoppa, 1990c)

& !'

1 1 1

j2 ¼ 1 À exp c(u þ 1) À c u À þ 1 þ c 1 À Àg , (3:112)

a a a

where g is Euler™s constant.

The parameters of (3.103) can be estimated in several ways. Stoppa (1990b,c)

considered nonlinear least-squares estimation in the Pareto diagram as well as ML

estimation. The corresponding estimators are not available in closed form and must

be derived numerically. In the ML case the parameter x0, de¬ning the endpoint of the

support of F, poses the usual problem arising in connection with threshold

parameters, the likelihood being unbounded in the x0 direction. Stoppa suggested

using modi¬ed ML estimators as discussed by Cohen and Whitten (1988). Starting

values for ML estimation may be obtained from, for example, regression estimators

as proposed by Stoppa (1995). He pointed out that the c.d.f. can be rewritten as

log{[1 À F 1=u (x)]} ¼ alog x0 À alog x,

97

3.9 CONIC DISTRIBUTION

yielding for the elasticity

1

logh(F, x) ¼ log(uaxa ) À alog x À log F(x):

0

u

Thus, log(uaxa ), a and 1=u can be estimated by least squares, say, and estimates for

0

the original parameters are then obtained by solving the de¬ning equations of the

new parameters.

For known x0 the Fisher information matrix is given by

I (u, a)

0 1

n[c(2) À c(u þ 2)]

n

B u2 C

a(u þ 1)

B C

B C (3:113)

u

n

B C

{2logx0 [c(2) À c(u þ 4)]C,

¼B Á þ

B C

a2 a(u þ 1)(u þ 2)

B C

@ A

10