n n

jxj , 1:

2 F1 (a1 , a2 ; b; x) ¼ , (6:9)

(b)n n!

n¼0

188 BETA-TYPE SIZE DISTRIBUTIONS

Figure 4 Beta-type size distributions and their interrelations: generalized beta distribution of the second

kind (GB2), Dagum distribution (D), beta distribution of the second kind (B2), Singh“Maddala

distribution (SM), inverse Lomax distribution (IL), Fisk (log-logistic) distribution (Fisk), Lomax

distribution (L).

The c.d.f. can also be expressed in the form (McDonald, 1984)

!

[(x=b)a =(1 þ (x=b)a )]p (x=b)a

x . 0:

F(x) ¼ 2 F1 p, 1 À q; p þ 1; ,

1 þ (x=b)a

pB( p, q)

(6:10)

If an expression for the survival function is required, we may use the complementary

relation

I1Àz ( p, q) ¼ 1 À Iz (q, p)

in conjunction with (6.4).

The mode of the GB2 distribution occurs at

1=a

ap À 1

, if ap . 1,

xmode ¼b (6:11)

aq þ 1

and at zero otherwise.

As noted above, the density of the GB2 distribution is regularly varying at in¬nity

with index Àaq À 1 and also regularly varying at the origin with index Àap À 1.

This implies that the moments exist only for Àap , k , aq. They are

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

E(X k ) ¼ : (6:12)

¼

G( p)G(q)

B( p, q)

189

6.1 (GENERALIZED) BETA DISTRIBUTION OF THE SECOND KIND

Note that this last expression is, when considered a function of k, equal to the

moment-generating function mY (Á) of Y ¼ log X , in view of the relation

mY (k) ¼ E(ekY ) ¼ E(X k ). This point of view is useful for the computation of the

moments of log X , which are required for deriving, for example, the Fisher

information matrix of the GB2 distribution and its subfamilies.

It is easy to see that the GB2 is closed under power transformations:

a

r r

r . 0:

X $ GB2(a, b, p, q) ¼ X $ GB2 , b , p, q , (6:13)

)

r

Also, Venter (1983) observed that the GB2 distribution is closed under inversion, in

the sense that

1 1

) $ GB2 a, , q, p :

X $ GB2(a, b, p, q) ( (6:14)

X b

Thus, it is sometimes convenient to allow for a , 0 in (6.5), one then simply

replaces a by jaj in the numerator.

The hazard rate of the GB2 distributions can exhibit a wide variety of

shapes. Considering the special case where p ¼ q ¼ 1, it is clear that it can at

least be monotonically decreasing as well as unimodal. Since a detailed

analysis is rather involved, we refer the interested reader to McDonald and

Richards (1987).

In view of the density being regularly varying and general results for the mean

excess function of such distributions (see Chapter 2), it follows that the mean excess

function is asymptotically linearly increasing

e(x) [ RV1 (1): (6:15)

6.1.3 Characterizations and Representations

It is possible to characterize the B2 distribution in terms of maximum entropy among

all distributions supported on [0, 1): If both E[log X ] and E[log(1 þ X )] are

prescribed, then the maximum entropy p.d.f. is the B2 density. Hence, this distribution

is characterized by the geometric means of X and 1 þ X (Kapur, 1989, p. 66).

An extension of the classical maximum entropy approach was considered by

Leipnik (1990), who used a relative maximum entropy principle that leads to several

income distributions included in the present chapter, such as the GB2, GB1 (see

Section 6.5), and Singh “ Maddala distributions. The idea here is that income

recipients are affected by ordinal as well as cardinal considerations, and therefore

subjectively reduce their incomes by multiplication with a subjective, but not

individualized, reduction factor f[x, 1 À F(x)], which depends on the actual income

x as well as the income status, here measured by the proportion 1 À F(x) of income

receivers earning more than a preassigned income x. This leads to an adjusted

190 BETA-TYPE SIZE DISTRIBUTIONS

income j(x) ¼ xf[x, 1 À F(x)] that is perhaps best interpreted as utility. A relative

income entropy density f is now de¬ned in terms of the p.d.f. f and the marginal

Ð

subjective income @j[x, F(x)]=@x and determined by the maximization of f (x) dx

under constraints on Ej(X ) and Ej2 (X ), the ¬rst two moments of j with respect to F.

The resulting nonlinear differential equation (according to Leipnik of a type that is

not much studied outside of hydrology and astrophysics) is sometimes solvable and

leads, under appropriate speci¬cations of the adjustment function f, to income

distributions of the generalized beta type.

As mentioned in the preceding chapter, Malik (1967) and Ahuja (1969) both

showed that if X1 $ GG(a, 1, p), X2 $ GG(a, 1, q), and X1 and X2 are independent

(note the identical shape parameter a!),

X1

$ GB2(a, 1, p, q): (6:16)

X2

The relation (6.16) is perhaps more familiar in the form

1=a

X1

$ GB2(a, 1, p, q): (6:17)

X2

where now X1 $ Ga(1, p), X2 $ Ga(1, q), and X1 and X2 are independent, which is a

generalization of the well-known relation between the standard gamma and B2

distributions

X1

$ B2(1, p, q): (6:18)

X2

These relations can be exploited to obtain random samples from the GB2

distribution: There is a large number of gamma random number generators (see

Devroye, 1986), from which generalized gamma samples can be obtained by a power

transformation. Simulating independent data from two generalized gamma

distributions with the required shape parameters we arrive at GB2 samples via the

above relation.

All this can be further rephrased, utilizing the familiar relation between the

gamma distribution and the classical beta distribution (Pearson type I).

If X1 $ Ga(1, p), X2 $ Ga(1, q), and X1 and X2 are independent, the random

variable

X1