by Majumder and Chakravarty (1990) is simply a reparameterization of the GB2

distribution. This observation escaped researchers for at least ¬ve years in spite of

the fact that the papers appeared in not unrelated journals: one oriented toward

applications and the other of a more theoretical bent.

6.1.1 De¬nition and Interrelations

The c.d.f. of the GB2 distribution may be introduced using an alternative expression

for the incomplete beta function ratio that is obtained upon setting u :¼ t=(1 þ t)

in (6.1)

°z

t pÀ1

1

z . 0:

Iz ( p, q) ¼ pþq dt, (6:3)

B( p, q) 0 (1 þ t)

Introducing additional scale and shape parameters b and a and setting z :¼ (x=b)a , we

get a distribution with c.d.f.

x a

x . 0,

F(x) ¼ Iz ( p, q), where z ¼ , (6:4)

b

with corresponding density

axapÀ1

x . 0:

f (x) ¼ , (6:5)

bap B( p, q)[1 þ (x=b)a ]pþq

Here all four parameters a, b, p, q are positive, b is a scale, and a, p, q are shape

parameters. It is not dif¬cult to see that the GB2 density is regularly varying at in¬nity

with index Àaq À 1 and regularly varying at the origin with index Àap À 1; thus, all

three shape parameters control the tail behavior of the model. Nonetheless, these three

parameters are not on an equal footing: If the distribution of Y ¼ log X , with density

aeap( yÀlog b)

À1 , y , 1,

f ( y) ¼ , (6:6)

B( p, q)[1 þ ea( yÀlog b) ]pþq

is considered, a turns out to be a scale parameter, whereas p and q are still shape

parameters.

185

6.1 (GENERALIZED) BETA DISTRIBUTION OF THE SECOND KIND

We see that the larger the value of a, the thinner the tails of the density (6.5) are,

whereas the relative values of p and q are important in determining the skewness of

the distribution of log X . Figures 6.1 “6.3 illustrate the effect of the three shape

parameters; each graph keeps two parameters constant and varies the remaining one.

(Note that there is considerable variation in the shape of the density for small a and p

in Figures 6.1 and 6.2.)

As an income distribution, (6.5) was proposed by McDonald (1984) and

independently as a model for the size-of-loss distribution in actuarial science by

Venter (1983), who called it a transformed beta distribution. A decade earlier, it was

brie¬‚y discussed by Mielke and Johnson (1974) in a meteorological application as a

generalization of two distributions that are included in the following sections under

the names of the Singh “ Maddala and Dagum distributions, respectively. It may also

be considered a generalized F distribution, and it appears under this name in, for

example, Kalb¬‚eisch and Prentice (1980). The distribution is further referred to as a

Feller “Pareto distribution by Arnold (1983), who introduced an additional location

parameter, and it was rediscovered, in a different parameterization, in the

econometrics literature by Majumder and Chakravarty (1990). McDonald and

Mantrala (1993, 1995) observed that the Majumder “ Chakravarty model is

equivalent to the GB2 distribution. We shall use McDonald™s (1984) notation below.

The case where a ¼ 1, that is, the beta distribution of the second kind (B2) with

p.d.f. is a member of the Pearson system of distributions (see Chapter 2), namely,

xpÀ1

x . 0,

f (x) ¼ , (6:7)

bp B( p, q)[1 þ x=b]pþq

Figure 1 GB2 densities: p ¼ 0:5, q ¼ 2, and a ¼ 1, 2, 3, 4, 6, 8, 12, 16 (from left to right).

186 BETA-TYPE SIZE DISTRIBUTIONS

Figure 2 GB2 densities: a ¼ 5, q ¼ 0:5, and p ¼ 0:17, 0.25, 0.5, 1, 2, 4 (from left to right).

the Pearson VI distribution. As an income distribution, it was proposed some-

what earlier than the GB2, in an application to Finnish data (Vartia and Vartia,

1980).

The GB2 can also be considered a generalized log-logistic distribution: Setting

a ¼ b ¼ p ¼ q ¼ 1 in (6.6), we get

ey

À1 , y , 1,

f ( y) ¼ ,

(1 þ ey )2

which is the density of a standard logistic distribution (see, e.g., Johnson, Kotz, and

Balakrishnan, 1995, Chapter 23). Speci¬cally, the distribution of (6.6) is a skewed

generalized logistic distribution, symmetry being attained only for p ¼ q. McDonald

and Xu (1995) referred to (6.6) as the density of an exponential GB2 distribution.

Recently, Parker (1999a,b) derived GB2 and B2 earnings distributions from

microeconomic principles (a neoclassical model of optimizing ¬rm behavior),

thereby providing some rationale as to why such distributions may be observed. In

his model the shape parameters p and q are functions of the output-labor elasticity

and the elasticity of income returns with respect to human capital, thus permitting

some insight into the potential causes of observed inequality trends.

The GB2 model is most useful for unifying a substantial part of the size

distributions literature. It contains a large number of income and loss distributions as

special or limiting cases: The Singh “ Maddala distribution is obtained for p ¼ 1, the

Dagum distribution for q ¼ 1, the beta distribution of the second kind (B2) for

187

6.1 (GENERALIZED) BETA DISTRIBUTION OF THE SECOND KIND

GB2 densities: a ¼ 5, p ¼ 0:3, and q ¼ 0:25, 0.5, 1, 2, 4 (from right to left).

Figure 3

a ¼ 1, the Fisk (or log-logistic) distribution for p ¼ q ¼ 1, and the Lomax (or Pareto

type II) distribution for a ¼ p ¼ 1. Figure 6.4 illustrates these interrelations. [For

completeness (and symmetry!) we include an inverse form of the Lomax

distribution, although there appears to be hardly any work dealing explicitly with

this distribution.] Apart from the B2 distribution, which is included in the present

section, these models will be discussed in greater detail in the following sections.

Furthermore, the generalized gamma distribution (see Chapter 5) emerges as a

limiting case upon setting b ¼ q1=a b and letting q ! 1. Consequently, the gamma

and Weibull distributions are also limiting cases of the GB2, since both are special

cases of the generalized gamma distribution.

6.1.2 Moments and Other Basic Properties

In (6.4) the GB2 distribution was introduced via the incomplete beta function ratio.

Utilizing the relation with Gauss™s hypergeometric function 2 F1 (e.g., Temme, 1996),

we obtain

zp

Iz ( p, q) ¼ 2 F1 ( p, 1 À q; p þ 1; z), 0 z 1, (6:8)

pB( p, q)

where

X (a1 ) (a2 ) xn