Hence, the coef¬cient of variation is

s¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬

G( p)G( p þ 2=a)G(1 À 2=a)

CV ¼ À1 (6:93)

G2 ( p þ 1=a)G2 (1 À 1=a)

and the shape factors are

p¬¬¬¬¬ G2 ( p)l3 À 3G( p)l2 l1 þ 2l3

1

b1 ¼ (6:94)

2 3=2

[G( p)l2 À l1 ]

and

G3 ( p)l4 À 4G2 ( p)l3 l1 þ 6G( p)l2 l2 À 3l4

1 1

b2 ¼ , (6:95)

22

[G( p)l2 À l1 ]

where we have set li ¼ G(1 À i=a)G( p þ i=a), i ¼ 1, 2, 3, 4. These expressions are

not easily interpreted. p¬¬¬¬¬

From a moment-ratio diagram”a graphical display of ( b1 , b2 )”of the Dagum

and the closely related Singh “Maddala distributions (Rodriguez, 1983; Tadikamalla,

1980) it may be inferred that both distributions allow for various degrees of positive

skewness and leptokurtosis, and even for a considerable degree of negative skewness,

although this feature does not seem to be of particular interest in our context.

Tadikamalla (1980, p. 342) observed “that although the Burr III [¼Dagum] distri-

bution covers all of the region . . . as covered by the Burr XII [¼Singh “ Maddala]

distribution and more, much attention has not been paid to this distribution.” Kleiber

(1996) noted that the same has happened in the econometrics literature.

The mode of this distribution is at

1=a

ap À 1

, if ap . 1,

xmode ¼b (6:96)

aþ1

215

6.3 DAGUM DISTRIBUTIONS

and at zero otherwise. This built-in ¬‚exibility is an attractive feature in that the

model can approximate income distributions, which are usually unimodal, and

wealth distributions, which are zeromodal.

From (6.89) the median is (Dagum, 1977)

xm ¼ b[21=p À 1]À1=a :

Moments of the order statistics can be obtained in an analogous manner to the

Singh “Maddala case. In view of the relation (6.86) presented above and the

corresponding result (6.53) for Singh “Maddala minima, we have the closure property

X $ D(a, b, p) ¼ Xn:n $ D(a, b, np) (6:97)

)

and thus, using (6.90), we obtain

bk G(np þ 1=a)G(1 À 1=a)

:

E(Xn:n ) ¼ (6:98)

G(np)

As in the Singh “ Maddala case, the moments of other order statistics can be

obtained using recurrence relations. This will be necessary for the computation of

generalized Gini coef¬cients where expectations of sample minima are required.

Domma (1997) provided some further distributional properties of the sample median

and the sample range.

To the best of our knowledge, the hazard rate and mean excess function of the

Dagum distribution have not been investigated in the statistical literature.

Nonetheless, from the general properties of regularly varying functions (see

Chapter 2) we can infer that the hazard rate is decreasing for large x, speci¬cally

r(x) [ RV1 (À1), and similarly that the mean excess function is increasing,

e(x) [ RV1 (1).

6.3.3 Representations and Characterizations

There are comparatively few explicit characterizations of the Dagum (Burr III)

distribution in the statistical literature. However, in view of the close relationship

with the Singh “ Maddala distribution, all characterizations presented for that

distribution translate easily into characterizations of the Dagum distribution.

For example, El-Saidi, Singh, and Bartolucci (1990) showed that the functional

equation

a ! a ! a !

b b b

. xy ¼ P 1 þ .x ÁP 1þ .y ,

P 1þ

X X X

where a, b . 0 and x, y . 1, characterizes the Dagum (which they called a genera-

lized log-logistic) distribution among all continuous distributions supported on

[0, 1). This follows directly from the corresponding characterization of the

216 BETA-TYPE SIZE DISTRIBUTIONS

Singh “Maddala distribution considered above via the relation (6.86). Ghitany

(1996) observed that this characterization can be considered a restatement of the

well-known characterization of the exponential distribution in terms of its lack of

memory property, as described in connection with the Singh “ Maddala distribution.

Utilizing the mixture representation of the GB2 distribution (6.21), the Dagum

distribution can be considered a compound generalized gamma distribution whose

scale parameter follows an inverse Weibull distribution

^

GG(a, u, p) InvWei(a, b) ¼ D(a, b, p): (6:99)

u

6.3.4 Lorenz Curve and Inequality Measures

Since the quantile function of the Dagum distribution is available in closed form, its

(normalized) integral, the Lorenz curve, is also of a comparatively simple form,

namely (Dagum, 1977),

1 1

L(u) ¼ Iz pþ ,1À , 0 u 1, (6:100)

a a

where z ¼ u1=p .

A subclass of the Dagum distributions, de¬ned by

h x Àa iÀ1þ1=a

x . 0,

F(x) ¼ 1 þ , (6:101)

b

where a . 1, exhibits symmetric Lorenz curves (in the sense of Chapter 2).

Interestingly, this was noted by Champernowne (1956, p. 182) long before the

distribution was proposed as an income distribution. However, Champernowne did

not develop the model further.

From (6.24) and (6.25) the necessary and suf¬cient conditions for Lorenz domi-

nance are

a2 :

a1 p1 a2 p2 and a1 (6:102)

This was derived by Kleiber (1996) from the corresponding result for the Singh “

Maddala distribution using (6.86); for a different approach see Kleiber (1999b,

2000a). [Dancelli (1986) had shown somewhat earlier that (income) inequality is

decreasing to zero for both a ! 1 and p ! 1 and increasing to 1 for a ! 1 and

p ! 0, respectively, keeping the other parameter ¬xed.]

Klonner (2000) presented necessary as well as suf¬cient conditions for ¬rst-order

stochastic dominance within the Dagum family. The conditions a1 ! a2 ,

a1 p1 a2 p2 , and b1 ! b2 are suf¬cient for X2 !FSD X1, whereas the conditions

a1 ! a2 and a1 p1 a2 p2 are necessary. (See the corresponding conclusions for the

Singh “Maddala distribution for an interpretation of these results.)

217

6.3 DAGUM DISTRIBUTIONS