1

{p[(c ( p) À c (1) À 1)2 þ c0 ( p) þ c0 (1)]

I11 ¼ 2 (2 þ p)

a

þ 2[c ( p) À c (1)]}, (6:111)

p À 1 À p[c ( p) À c (1)]

I21 ¼ I12 ¼ , (6:112)

b(2 þ p)

a2 p

I22 , (6:113)

¼2

b (2 þ p)

a

I23 ¼ I32 ¼ , (6:114)

b(1 þ p)

c (2) À c ( p)

I31 ¼ I13 ¼ , (6:115)

a(1 þ p)

1

:

I33 ¼ (6:116)

p2

We note that there are at least two earlier derivations of the Fisher information in

the statistical literature: a detailed one using Dagum™s parameterization due to

Latorre (1988) and a second one due to Zelterman (1987). As mentioned above, the

latter article considers the distribution of log X , a generalized logistic distribution,

using the parameterization (u, s, a) ¼ ( log b, 1=a, p). Latorre (1988) also provided

asymptotic standard errors for the Gini and Zenga coef¬cients derived from MLEs

for the Dagum model.

However, an inspection of the scores (6.107“6.109) reveals that supx k@L=@uk ¼ 1;

thus, the scores function is unbounded in the Dagum case. This implies that the MLE is

rather sensitive to isolated observations located suf¬ciently far away from the majority of

the data. There appears therefore to be some interest in more robust procedures. For a

robust approach to the estimation of the Dagum model parameters using an optimal B-

robust estimator (OBRE), see Victoria-Feser (1995). (The basic ideas underlying this

estimator are outlined in Section 3.6 in connection with the Pareto distribution.)

6.3.6 Extensions

Dagum (1977, 1980a) introduced two further variants of his distribution; thus, we

shall refer to the previously discussed standard version as the Dagum type I

distribution in what follows. The Dagum type II distribution has the c.d.f.

h x Àa iÀp

F(x) ¼ a þ (1 À a) 1 þ , x ! 0, (6:117)

b

220 BETA-TYPE SIZE DISTRIBUTIONS

where a, b, p . 0 and a [ (0, 1). Clearly, this is a mixture of a point mass at the

origin with a Dagum (type I) distribution over the positive hal¬‚ine. Thus, the kth

moment exists for 0 , k , a.

The type II distribution was proposed as a model for income distributions with

null and negative incomes, but more particularly to ¬t wealth distributions, which

frequently present a large number of economic units with null gross assets and with

null and negative net assets.

There is also a Dagum type III distribution, de¬ned via

h xÀa iÀp

F(x) ¼ a þ (1 À a) 1 þ , (6:118)

b

where again a, b, p . 0 but a , 0. Consequently, the support of this variant is

[x0 , 1), x0 . 0, where x0 ¼ {b[(1 À 1=a)1=p À 1]}À1=a is determined implicitly from

the constraint F(x) ! 0. Clearly, for the Dagum (III) distribution the kth moment

exists for k , a.

Both the Dagum type II and type III can be derived from the differential equation

( !1=p )

d[log F(x) À a] F(x) À a

h(x, F) ¼ ¼ ap 1 À , x ! 0,

1Àa

dlog x

subject to p . 0 and ap . 0. This is a generalization of the differential equation

(6.87) considered above.

Investigating the relation between the functional and personal distribution of

income, Dagum (1999) obtained the following bivariate c.d.f. specifying the joint

distribution of human capital and wealth:

F(x1 , x2 ) ¼ (1 þ b1 xÀa1 þ b2 xÀa2 þ b3 xÀa1 xÀa2 )Àp , xi . 0, i ¼ 1, 2: (6:119)

1 2 1 2

If b3 ¼ b1 b2 ,

F(x1 , x2 ) ¼ (1 þ b1 xÀa1 )Àp (1 þ b2 xÀa2 )Àp :

1 2

Hence, the marginals are independent. As far as we are aware, there are no empirical

applications of this multivariate Dagum distribution.

However, there is a recent application of a bivariate Dagum distribution in the

actuarial literature. In the section dealing with the Singh “Maddala distribution, we

presented a bivariate income distribution de¬ned in terms of a copula. This is also

the approach of Klugman and Parsa (1999), who combined Frank™s (1979) copula

& '

(eÀau1 À 1)(eÀau2 À 1)

1

G(u1 , u2 ) ¼ À log 1 þ ,

eÀa À 1

a

0 , ui , 1, i ¼ 1, 2, a = 1, (6:120)

221

6.3 DAGUM DISTRIBUTIONS

with two Dagum marginal distributions Fi , i ¼ 1, 2, and applied the resulting model

F(x1 , x2 ) ¼ G[F1 (x1 ), F2 (x2 )],

to the joint distribution of loss and allocated loss adjustment expense on a single

^

claim. They obtained a ¼ 3:07, indicating moderate positive dependence.

6.3.7 Empirical Results

Although the Dagum distribution was virtually unknown in major English language

economics and econometrics journals until well into the 1990s, there were several

early applications to income and wealth data, most of which appeared in French,

Italian, and Latin American publications.

Incomes and Wealth

Dagum (1977, 1980a) applied his type II distribution to U.S. family incomes of 1960

and 1969, for which the model outperforms the Singh“Maddala, gamma, and

lognormal functions, and in his Encyclopedia of Statistical Sciences entry of 1983 he ¬t

Dagum types I and III as well as Singh“Maddala, gamma, and lognormal distributions

to 1978 U.S. family incomes. Here the Dagum types III and I rank ¬rst and second; both

outperform the (two-parameter) lognormal and gamma distributions by wide margins.

Espinguet and Terraza (1983) employed the Dagum type II distribution when

modeling French wages, strati¬ed by occupation for 1970“ 1978. The distribution is

superior to a Box“ Cox-transformed logistic, the Singh “ Maddala and three-parameter

lognormal and Weibull distributions, as well as a four-parameter beta type I model.

Dagum and Lemmi (1989) ¬t the Dagum type I“ III distributions to Italian

income data from the Banca d™Italia sample surveys of 1977, 1980, and 1984, for

which the ¬t is in general quite satisfactory. The data were disaggregated by sex,

region, and source of income.

Majumder and Chakravarty (1990) considered the Dagum type I distribution when

modeling U.S. income data for 1960, 1969, 1980, and 1983. The distribution

improves upon all other two- and three-parameter models. In a reassessment of

Majumder and Chakravarty™s ¬ndings, McDonald and Mantrala (1993, 1995) ¬t the

Dagum type I distribution to 1970, 1980, and 1990 U.S. family incomes using two

different ¬tting methods and alternative groupings of the data; for the 1970 and 1990

data the distribution performs almost as well as the four-parameter GB2 distribution,

con¬rming Majumder and Chakravarty™s conclusions.

McDonald and Xu (1995) studied 1985 U.S. family incomes. Out of 11

distributions considered, the Dagum type I ranks third in terms of likelihood and

several other criteria, being outperformed only by the four-parameter GB2 and a

¬ve-parameter generalized beta distribution [see (6.142) below].