a

^^

from which an estimate of the asymptotic covariance matrix of (u, a) may be obtained

^^

by numerical inversion of I °u ; a Þ. Properties of estimators of functions of a and/or u

can be derived by means of the delta method. It should be noted that most of the

functions arising in connection with inequality measurement are scale-free and are

therefore functions of only a and u. In particular, Stoppa (1990c) derived the

asymptotic distributions of the ML estimates of the Gini index (3.111) and of Zenga™s

inequality measure (3.112).

Stoppa (1990b) proposed a second generalized Pareto distribution”which we

shall call the Stoppa type II distribution”with the c.d.f.

& !Àa 'u

xÀc

x . c,

F(x) ¼ 1 À , (3:114)

x0

which is seen to be a two-parameter Stoppa type I distribution amended with location

and scale parameters c and x0 . It can be derived along similar lines as the type I model.

The procedure leading to the Stoppa distribution (power transformation of the c.d.f.)

was recently applied to more general distributions by Zandonatti (2001); see Chapter 6

for generalizations of the Pareto (II) and Singh“Maddala distributions, among others.

3.9 CONIC DISTRIBUTION

In spite of being an unpublished discussion paper, Houthakker (1992) has received

substantial attention among the admittedly relatively narrow circle of researchers and

users in the area of statistical income distributions.

98 PARETO DISTRIBUTIONS

Resurrecting the geometrical approach to income distributions that takes the

Pareto diagram as the starting point, Houthakker introduced a ¬‚exible family of

generalized Pareto distributions de¬ned in terms of conic sections. A conic section

in the Pareto diagram is given by

c0 U 2 þ 2c1 UV þ c2 V 2 þ 2c3 U þ 2c4 V þ c5 ¼ 0, (3:115)

where U ¼ log{F (x)}, V ¼ log x, and c0 , c1 , . . . , c5 are parameters. In order to

retain the weak Pareto law, a conic section with a linear asymptote is required.

Consequently, circles and ellipses have to be excluded and the only admissible conic

section in our context is the hyperbola. Since exp U must de¬ne a survival function,

further constraints have to be imposed. It is not dif¬cult but somewhat tedious to

derive the resulting c.d.f. and p.d.f. We present the basic properties of the conic

distributions using a reparameterization suggested by Kleiber (1994). He utilizes the

earlier work of Barndorff-Nielsen (1977, 1978) on so-called hyperbolic distributions

that leads to a more transparent functional form.

The c.d.f. of a general conic distribution is given by

& '

q¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬

F(x) ¼ 1 À exp Àh 1 þ ( log x À l)2 þ j ( log x À l) þ m , x ! x0 . 0,

(3:116)

p¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬

h, m ¼ h 1 þ ( log x0 À l)2 À j ( log x0 À l). It is also

where h . 0, À1 , j

required that

h( log x0 À l)

p¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬ À j ! 0: (3:117)

1 þ ( log x0 À l)2

The density is therefore

'" #

&q¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬

h( log x À l)

f (x) ¼ xÀjÀ1 exp Àh 1 þ ( log x À l)2 À jl þ m Á p¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬ À j :

1 þ ( log x À l)2

(3:118)

Houthakker discussed two subclasses of the conic distributions in detail. These

are the conic-linear distributions de¬ned via the condition h À j ¼ 0 and the conic-

quadratic models de¬ned by f (x0 ) ¼ 0. The latter condition is equivalent to the l.h.s.

of (3.117) being equal to zero. Consequently, for non-“quadratic” conic

distributions, we always have f (x0 ) . 0.

It is not dif¬cult to see that the c.d.f. is asymptotically of the form xÀ(hÀj) ; hence,

the kth moment of this distribution exists only for k , h À j, and h À j plays the

role of Pareto™s a. For the moments we obtain a rather formidable expression

involving a complete as well as an incomplete modi¬ed Bessel function of the third

kind (also known as a MacDonald function), which will not be given here. The

99

3.10 A ˜˜LOG-ADJUSTED™™ PARETO DISTRIBUTION

probably easiest way to the Gini coef¬cient is via the representation in terms of

moments of order statistics (2.22); the resulting expression also involves incomplete

Bessel functions.

Kleiber (1994) pointed out that a simple regression estimator may be used to obtain

parameter estimates for the conic distribution. In a doubly logarithmic representation

of (3.116), only the parameter l enters in a nonlinear fashion, so a quick method is to

estimate m, h, j via linear regression while performing a grid search over l. However,

p¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬

it is evident from (3.116) that the regressors 1 þ ( log x À l)2 and log x À l are

highly collinear, suggesting that the model is essentially overparameterized. Indeed,

Brachmann, Stich, and Trede (1996) reported on numerical problems when trying to

¬t linear and quadratic conic distributions to German household incomes.

3.10 A “LOG-ADJUSTED” PARETO DISTRIBUTION

Ziebach (2000) proposed a generalization of the Pareto distribution by introducing a

logarithmic adjustment term that allows for a more ¬‚exible shape. Like Houthakker

(1992), he started from the Pareto diagram, specifying a c.d.f. with the representation

log{1 À F(x)} ¼ Àa log x À b log(log x) þ g:

Clearly, the new term does not affect the asymptotic linearity in the doubly logarithmic

representation, so that the resulting distribution obeys the weak Pareto law. The

condition F(x0 ) ¼ 0, for some x0 . 0, yields eg ¼ xa ( log x0 )b , leading to the c.d.f.

0

x a log x b

0 0

, 1 , x0

F(x) ¼ 1 À x, (3:119)

x log x

where either a . 0 and b ! Àalog x0 or a ¼ 0 and b . 0. The density is given by a

more complex expression

xa (log x0 )b (alog x þ b)

f (x) ¼ 0 aþ1 1 , x0

, x, (3:120)

x (log x)bþ1

which is decreasing for b ! 0 and unimodal for

&q¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬ '!

1

( log x0 þ 1)2 þ 4alog x0 À ( log x0 þ 1) :

b [ Àalog x0 , Àalog x0 þ

2

Only if b ¼ Àalog x0 , do we have f (x0 ) ¼ 0; otherwise, f (x0 ) . 0. Figure 3.3

illustrates the effect of the new parameter b.

The moments of the distribution are

axk

À kb(a À k)bÀ1 xa ( log x0 )b G[Àb; (a À k)log x0 ],

k 0

E(X ) ¼ (3:121)

0

aÀk

100 PARETO DISTRIBUTIONS