Zenga ordering among the Dagum distributions was studied by Polisicchio

(1990). Similar to the Singh “ Maddala case, it turns out that a1 a2 implies

X1 !Z X2 , for a ¬xed p, and analogously that p1 p2 implies X1 !Z X2 , for a ¬xed

a. Under these conditions, we know from (6.68) that the distributions are also

Lorenz-ordered, speci¬cally X1 !L X2. However, a complete characterization of the

Zenga order within the family of Dagum distributions seems to be currently

unavailable.

The Gini coef¬cient is (Dagum, 1977)

G( p)G(2p þ 1=a)

G¼ À 1: (6:103)

G(2p)G( p þ 1=a)

The generalized Gini coef¬cients can be obtained as follows: Combining the

well-known recurrence relation (Arnold and Balakrishnan, 1989, p. 7)

X

n

n jÀ1

jÀi

E(Xk:n ) ¼ (À1) E(Xj:j )

j iÀ1

j¼1

and the expression for the expectations of Dagum maxima (6.98) yields

Xn

G( p) n G( jp þ 1=a)

(À1) jÀ1 :

Gn ¼ (6:104)

j

G( p þ 1=a) j¼1 G( jp)

The Zenga index j2 is (Latorre, 1988)

& '

E(X log X )

j2 ¼ 1 À exp E( log X ) À

E(X )

& !'

1 1 1

c ( p) þ c 1 À Àc pþ À c (1) :

¼ 1 À exp

a a a

6.3.5 Estimation

Dagum (1977) discussed ¬ve methods for estimating the model parameters and

recommended a nonlinear least-squares method minimizing

Xn h x Àa iÀp o

n

i

Fn (xi ) À 1 þ , (6:105)

b

i¼1

a minimum distance technique based on the c.d.f. A further regression-type

estimator utilizing the elasticity (6.87) was considered by Stoppa (1995).

218 BETA-TYPE SIZE DISTRIBUTIONS

The log-likelihood for a complete random sample of size n is

X

n

log L ¼ nlog a þ nlog p þ (ap À 1) log xi À naplog b À ( p þ 1)

i¼1

Xh x a i

n

i

log 1 þ , (6:106)

‚

b

i¼1

yielding the likelihood equations

X xi X log(xi =b)

n n

n

þp ¼ ( p þ 1) , (6:107)

log

1 þ (b=xi )a

a b

i¼1 i¼1

X

n

1

np ¼ ( p þ 1) , (6:108)

1 þ (b=xi )a

i¼1

X xi X h x a i

n n

n i

:

þa log 1 þ (6:109)

log ¼

p b b

i¼1 i¼1

However, likelihood estimation in this family is not without problems:

Considering the distribution of log X , a generalized logistic distribution, Zelterman

(1987) showed that there is a path in the parameter space along which the likelihood

becomes unbounded. This implies that the global maximizer of the likelihood does

not de¬ne a consistent estimator of the parameters. Fortunately, there nonetheless

exists a sequence of local maxima that yields consistent estimators (Abberger and

Heiler, 2000).

Apparently unaware of these problems, Domanski and Jedrzejczak (1998)

´

provided a simulation study for the performance of MLEs for samples of size n ¼

1,000(1,000)10,000. It emerges that estimates of the shape parameters a, p can be

considered as unbiased for samples of sizes 2,000 “3,000, and as approximately

normally distributed and ef¬cient for n ! 7,000. Reliable estimation of the scale

parameter seems to require even larger samples. Estimators appear to be unbiased

for n ! 4,000, but even for n ¼ 10,000 there are considerable departures from

normality.

Analogously to the Singh “Maddala distribution, we can obtain the Fisher

information matrix

0 1

" #

I11 I12 I13

2

@ log L @ I21 I23 A,

I (u) ¼ ÀE I22 (6:110)

¼:

@ui @uj i,j

I31 I32 I33

219

6.3 DAGUM DISTRIBUTIONS

where u ¼ (a, b, p)` , from the information matrix of the GB2 distribution (6.30).

This yields