gamma distribution; hence, (6.83) is a direct generalization of the univariate mixture

representation (6.65). The Takahasi “ Burr distribution possesses Singh “ Maddala

marginals as well as conditionals.

As is well known from the theory of copulas (e.g., Nelsen, 1998), a multivariate

survival function can be decomposed in the form

F (x1 , . . . , xk ) ¼ G[F 1 (x1 ), . . . , F k (xk )],

where the F i (xi ), i ¼ 1, . . . , xk , are the marginal survival functions and G is the

copula (a c.d.f. on [0, 1]k that captures the dependence structure of F). The

dependence structure within the Takahasi “Burr family has been studied by Cook

and Johnson (1981); the copula is

( )Àq

X

k

uiÀ1=q À (k À 1)

G(u1 , . . . , uk ) ¼ 0 , ui , 1, i ¼ 1, . . . , k,

, (6:84)

i¼1

which is often called the Clayton (1978) copula. However, considering data on

annual incomes for successive years, Kleiber and Trede (2003) found that this model

212 BETA-TYPE SIZE DISTRIBUTIONS

does not provide a good ¬t. When combining an elliptical copula (associated with,

e.g., a multivariate normal distribution) with Singh “Maddala marginal distributions,

they obtained rather encouraging preliminary results.

6.3 DAGUM DISTRIBUTIONS

Although introduced as an income distribution only one year after the Singh “

Maddala model, the Dagum distribution is less widely known. Presumably, this is

due to the fact that Dagum™s work was published in the French journal Economie

´

Appliquee, whereas the Singh “Maddala paper appeared in the more widely read

Econometrica. However, in recent years there are indications that the Dagum

distribution is, in fact, a more appropriate choice in many applications.

6.3.1 De¬nition and Motivation

The Dagum distribution is a GB2 distribution with the shape parameter q ¼ 1;

hence, its density is

apxapÀ1

, x . 0,

f (x) ¼ ap (6:85)

b [1 þ (x=b)a ]pþ1

where a, b, p . 0.

Like the Singh “ Maddala distribution considered in the previous section, the

Dagum distribution was rediscovered many times in various ¬elds of science.

Apparently, it occurred for the ¬rst time in Burr (1942) as the third example of

solutions to his differential equation de¬ning the Burr system of distributions. Thus,

it is known as the Burr III distribution. As mentioned above, the Dagum distribution

is closely related to the Singh “ Maddala distribution, speci¬cally

1 1

) $ SM a, , p :

X $ D(a, b, p) ( (6:86)

X b

This relation permits us to translate several results pertaining to the Singh “ Maddala

family to corresponding results for the Dagum distributions.

The Singh “ Maddala is the Burr XII, or simply the Burr distribution, so it is

not surprising that the Dagum distribution is also called the inverse Burr

distribution, notably in the actuarial literature (e.g., Klugman, Panjer, and

Willmot, 1998). Like the Singh “ Maddala, the Dagum distribution can be

considered a generalized log-logistic distribution. The special case where a ¼ p is

sometimes called the inverse paralogistic distribution (Klugman, Panjer, and

Willmot, 1998). Prior to its use as an income distribution, the Dagum family was

proposed as a model for precipitation amounts in the meteorological literature

(Mielke, 1973), where it is called the (three-parameter) kappa distribution.

213

6.3 DAGUM DISTRIBUTIONS

Mielke and Johnson (1974) nested it within the GB2 and called it the beta-K

distribution. In a parallel development”aware of Mielke (1973) but presumably

unaware of Dagum (1977)”Fattorini and Lemmi (1979) proposed the

distribution as an income distribution. [See also Lemmi (1987).] Nonetheless,

the distribution is usually called the Dagum distribution in the income

distribution literature, and we shall follow this convention below.

Dagum (1977) derived his model from the empirical observation that the income

elasticity of the c.d.f. of income is a decreasing and bounded function of F. Starting

from the differential equation,

dlog F(x)

¼ ap{1 À [F(x)]1=p },

h(x, F) ¼ x ! 0, (6:87)

dlog x

subject to p . 0 and ap . 0, one obtains the density (6.85).

Fattorini and Lemmi (1979) independently arrived at the Dagum distribution as

the equilibrium distribution of a continuous-time stochastic process under certain

assumptions on its in¬nitesimal mean and variance (see also Dagum and Lemmi,

1989).

6.3.2 Moments and Other Basic Properties

Like the c.d.f. of the Singh “ Maddala distribution discussed in the previous section,

the c.d.f. of the Dagum distribution is available in closed form, namely,

h x Àa iÀp

, x . 0:

F(x) ¼ 1 þ (6:88)

b

This is also true of the quantile function

F À1 (u) ¼ b[uÀ1=p À 1]À1=a , for 0 , u , 1: (6:89)

As was the case with the Singh “Maddala distribution discussed in the previous

section, the Dagum family was considered in several equivalent parameterizations.

Mielke (1973) and later Fattorini and Lemmi (1979) used (a, b, u) ¼ (1=p, bp1=a , ap),

whereas Dagum (1977) employed (b, d, l) ¼ ( p, a, ba ).

From (6.12), the kth moment exists for Àap , k , a; it equals

bk B( p þ k=a, 1 À k=a) bk G( p þ k=a)G(1 À k=a)

k

:

E(X ) ¼ (6:90)

¼

G( p)

B( p, 1)

[In view of (6.86), this result can alternatively be obtained upon replacing q with p

and a with Àa in (6.46).]

214 BETA-TYPE SIZE DISTRIBUTIONS

Speci¬cally,

bG( p þ 1=a)G(1 À 1=a)

E(X ) ¼ (6:91)

G( p)

and

b2 {G( p)G( p þ 2=a)G(1 À 2=a) À G2 ( p þ 1=a)G2 (1 À 1=a)}

:

var(X ) ¼ (6:92)