distribution with a ¬nite mean, the condition F(1) (x) ¼ F(x) for all x . x0 . 0

[where F(1) denotes the c.d.f. of the ¬rst-moment distribution] characterizes the

Pareto distribution.

Moothathu (1990b) obtained the following characterization in terms of the

independence of certain random variables. For a ¬xed integer k ! 2, consider the

i.i.d. random variables Z1 , . . . , Zk , with P(Z1 . 1) ¼ 1 and let Y ¼ (Y1 , . . . , YkÀ1 )

kÀ1

be a further vector of random variables supported on IRþ , independent of the Z™s.

Set T ¼ 1 þ Y1 þ Á Á Á þ YkÀ1 and de¬ne

n o

T =Y1 T =YkÀ1

T

V ¼ min Z1 , Z2 , . . . , Zk :

Now the random variable Z1 follows a Pareto distribution if and only if V and Y are

independent. (The result follows from a characterization of the Weibull distribution

obtained in the same paper.) This result has potential applications in that it enables

one to treat a problem of testing k-sample homogeneity as a problem of testing

independence.

3.5 LORENZ CURVE AND INEQUALITY MEASURES

The Lorenz curve, which exists whenever a . 1, is given by

L(u) ¼ 1 À (1 À u)1À1=a , 0 , u , 1: (3:51)

As was already mentioned in the preceding chapter, it follows that Pareto Lorenz

curves never intersect and that, for Xi $ Par(x0 , a),

X1 !L X2 ( a1 a2 , (3:52)

)

provided ai . 1, i ¼ 1, 2. There is an interesting alternative but less direct argument

leading to this result. Arnold et al. (1987) observed that every distribution F

corresponding to an unbounded random variable and possessing a strongly unimodal

78 PARETO DISTRIBUTIONS

density generates an ordered family of Lorenz curves via Lt (u) ¼ F(F À1 (u) À t),

t ! 0. The Pareto distribution admits such a representation, the generating

distribution being the (Gumbel-type) extreme value distribution.

The Gini coef¬cient of the Pareto distribution is

1

G¼ , (3:53)

2a À 1

and the generalized Gini coef¬cients are given by (Kleiber and Kotz, 2002)

nÀ1

n ! 2:

Gn ¼ , (3:54)

na À 1

Hence for a ¼ 1:5, the value originally obtained by Pareto for most of his data, we

have G ¼ 0:5.

The Pietra index equals

(a À 1)À1

:

P¼ (3:55)

aa

and the Theil coef¬cient is

a

1

:

T1 ¼ (3:56)

À log

aÀ1 aÀ1

All of these expressions are decreasing with increasing a, showing that the

parameter a, or rather its inverse, may be considered a measure of inequality.

In the Italian literature the Zenga curve and inequality measures derived from it

have also been considered. As was mentioned in the preceding chapter, the Zenga

curve is given by (Zenga, 1984)

Z(u) ¼ 1 À (1 À u)1=[(aÀ1)a] , 0 , u , 1, (3:57)

and the two Zenga coef¬cients are (Zenga, 1984)

°1

1

j¼ Z(u) du ¼ (3:58)

1 þ a(a À 1)

0

and (Pollastri, 1987a)

& '

À1

j2 ¼ 1 À exp : (3:59)

a(a À 1)

This shows that the Zenga curve of the Pareto distribution is an increasing function

on [0, 1], approaching the x axis with increasing a, and that the two coef¬cients are

decreasing as a increases.

79

3.6 ESTIMATION

3.6 ESTIMATION

The estimation of Pareto characteristics is covered in depth in Arnold (1983) and

Johnson, Kotz, and Balakrishnan (1994). Here we shall only include the classical

regression-type estimators, ML estimation, and several recent developments, notably

in connection with UMVU estimation. For the method of moments, quantile and

Bayes estimators, as well as methods based on order statistics, we refer the interested

reader to the above-mentioned works.

3.6.1 Regression Estimators

Since the Pareto distribution was originally discovered as the distribution whose

survival function is linear in a double-logarithmic plot, the Pareto diagram, it is

not surprising that the regression estimators of its parameters have been used from

the very beginning. The least-squares estimators

P P P

Àn n log Xi Á log F (Xi ) þ n log Xi Á n log F (Xi )

i¼1 i¼1 i¼1

^

aLS ¼ (3:60)

À Pn Á2

Pn 2

n i¼1 ( log Xi ) À i¼1 log Xi

^

and x0, de¬ned by

^

^ ^

log F (X ) ¼ aLS Á log x0 À aLS Á log (X )

(where a bar denotes averaging), are still quite popular in applied work; Quandt

(1966b) has shown that they are consistent.

Early writers such as Pareto (1896, 1897a,b), Benini (1897), or Gini (1909a)

employed Cauchy regression (e.g., Linnik, 1961), a method that is seldom used

nowadays. Interestingly, in a fairly recent small Monte Carlo study Pollastri (1990)

found that Cauchy™s method is often slightly better, in terms of MSE, than least-

squares regression. More recently, Hossain and Zimmer (2000) recommended that

the least-squares estimators be generally preferred over the maximum likelihood and

related estimators for estimating x0 , and also for estimating a for small values of the

parameter (a 4).

3.6.2 Maximum Likelihood Estimation

The likelihood for a sample from a Pareto distribution is

Y axa

n

0