X1 þ X2

follows a beta type I distribution with parameters p and q. In view of (6.18), it is

therefore possible to de¬ne the B2 distribution via

X $ W À1 À 1,

191

6.1 (GENERALIZED) BETA DISTRIBUTION OF THE SECOND KIND

as was done by, for example, Feller (1971, p. 50), who notes that the resulting

distribution is sometimes named after Pareto. This caused Arnold (1983) to refer to

the GB2 distribution, which can evidently be expressed as

X $ b[W À1 À 1]1=a (6:20)

as the Feller “ Pareto distribution.

The representation (6.16) can also be considered a mixture representation,

namely, as a scale mixture of generalized gamma distributions with inverse

generalized gamma weights (Venter, 1983; McDonald and Butler, 1987). Recall that

the generalized gamma distribution has the density

a a

xapÀ1 eÀ(x=u) :

f (x) ¼ ap

u G( p)

Now, if the parameter u $ InvGG(a, b, q), then, in an obvious notation,

^

GG(a, u, p) InvGG(a, b, q) ¼ GB2(a, b, p, q): (6:21)

u

Evidently, this is just a restatement of (6.16). For the beta distribution of the second

kind, that is, the case where a ¼ 1, the above representation simpli¬es to

^

Ga(u, p) InvGa(b, q) ¼ B2(b, p, q), (6:22)

u

where InvGa denotes an inverse gamma (or Vinci) distribution.

Thus, the GB2 distribution and its subfamilies have a theoretical justi¬cation as a

representation of incomes arising from a heterogeneous population of income

receivers, or, in actuarial terminology, as a representation of claims arising from a

heterogeneous population of exposures. This argument was used by Parker (1997)

when utilizing a B2 distribution (6.22) as the distribution of self-employment

income; he argued that individuals are heterogeneous with regard to entrepreneurial

characteristics.

Table 6.1 presents the mixture representations for all the distributions discussed in

the present chapter (McDonald and Butler, 1987).

6.1.4 Lorenz Curves and Inequality Measures

The Lorenz curve of the GB2 distribution, which exists whenever aq . 1, cannot

be obtained directly from the Gastwirth representation since the quantile function

is not available in closed form. However, the Lorenz curve is available in terms

of the ¬rst-moment distribution in the following manner: Butler and McDonald

(1989) observed that the normalized incomplete moments, that is, the c.d.f.™s

of the higher-order moment distributions, can be expressed as the c.d.f.™s of

192 BETA-TYPE SIZE DISTRIBUTIONS

Table 6.1 Mixture Representations for Beta-type Size Distributions

Distribution Structural Distribution Mixing Distribution

GG(x; a, u, p)

GB2(x; a, b, p, q) InvGG(u; a, b, q)

Ga(x; u, p)

B2(x; b, p, q) InvGa(u; b, q)

Wei(x; a, u)

SM(x; a, b, q) InvGG(u; a, b, q)

GG(x; a, u, p)

Dagum(x; a, b, p) InvWei(u; a, b)

Wei(x; a, u)

Fisk(x; a, b) InvWei(u; a, b)

Exp(x; u)

Lomax(x; b, q) InvGa(u; b, p)

Source: McDonald and Butler (1987).

GB2 distributions with different sets of parameters. Namely, for X $ GB2(a, b,

p, q) and

Ðx

t k f (t) dt

0

0 , x , 1,

F(k) (x) ¼ ,

E(X k )

we have

k k

F(k) (x) ¼ F x; a, b, p þ , q À , 0 , x , 1: (6:23)

a a

This closure property is of practical importance in that computer programs used

to evaluate the distribution function of the GB2 distribution can also be used to

evaluate the higher-order moment distributions by merely changing the values of the

parameters. In particular, theoretical Lorenz curves of GB2 distributions can be

obtained by plotting F(1) (x) against F(x), 0 , x , 1. This closure property is

special to the GB2 and B2 distributions; it does not extend to any of the subfamilies

discussed in the following sections.

Using the ratio representation (6.16), Kleiber (1999a) showed that for

Xi $ GB2(ai , bi , pi , qi ), i ¼ 1, 2,

a1 a2 , a1 p1 a2 p2 and a1 q1 a2 q2 (6:24)

together imply X1 !L X2. This generalizes earlier results obtained by Wil¬‚ing

(1996b). However, the condition is not necessary. The necessary conditions for

Lorenz dominance are

a2 q2 :

a1 p1 a2 p2 and a1 q1 (6:25)

This result was ¬rst obtained by Wil¬‚ing (1996b); for an alternative approach see

Kleiber (2000a). It is worth noting that, although a full characterization of Lorenz

193

6.1 (GENERALIZED) BETA DISTRIBUTION OF THE SECOND KIND

order within the GB2 family is currently unavailable, the conditions (6.24) and

(6.25) are strong enough to yield complete characterizations for all the subfamilies

considered in the following sections.

Sarabia, Castillo, and Slottje (2002) obtained some Lorenz ordering results for

a˜˜

nonnested models. In particular, if X $ GG(˜ , b, p), Y $ GB2(a, b, p, q), and

˜ ˜˜

aq . 1, a ! a, ap ! a, then Y !L X .

As noted in the previous section, the GB2 distribution has heavy tails; hence, only

a few of the moments exist. This implies that inequality measures such as the

generalized entropy measures only exist for sensitivity parameters within a certain

range (Kleiber, 1997) that is often rather narrow in practice.

The above relation (6.23) also provides a simple way to obtain the Pietra index of

inequality, namely,

P ¼ F(m) À F(1) (m),

where m is the mean of X (Butler and McDonald, 1989).

The Gini coef¬cient of the GB2 is available in McDonald (1984) as a lengthy

expression involving the generalized hypergeometric function 3 F2

& !

2B(2p þ 1=a, 2q À 1=a) 1 1

G¼ 3 F2 1, p þ q, 2p þ ; p þ 1, 2( p þ q); 1

pB( p, q)B( p þ 1=a, q À 1=a) p a

!'