f (x) ¼ (5:2)

ap

b G( p)

Here b ¼ b1=a is a scale and a, p are shape parameters. This was introduced by

Amoroso (1924 “ 1925) as the family of generalized gamma distributions. Amoroso

originally considered a four-parameter variant de¬ned by X À m, m [ IR, but we

shall con¬ne ourselves to the three-parameter version (5.2). We use the notation

X $ GG(a, b, p). It is sometimes convenient to allow for a , 0 in (5.2); one then

simply replaces a by jaj in the numerator. For clarity, we shall always refer to a

generalized gamma distribution as the distribution with the p.d.f. (5.2) and a . 0;

the variant with a , 0 will be called an inverse generalized gamma distribution and

denoted as InvGG(a, b, p).

There are a number of alternative parameterizations: Amoroso used (s, g, p) ;

(1=a, 1=b, p), Stacy (1962) employed (a, b, d) ; (a, b, ap), whereas Taguchi

(1980) suggested (a, b, h) ; (ap, b, 1=p). We shall use (5.2), which was

also employed by McDonald (1984) and Johnson, Kotz, and Balakrishnan (1994).

The generalized gamma distribution is a fairly ¬‚exible family of distributions; it

includes many distributions supported on the positive hal¬‚ine as special or limiting

cases:

The gamma distribution is obtained for a ¼ 1; hence, if X $ GG(a, b, p), then

.

X 1=a $ Ga(b, p). In particular, the chi-squared distribution with n degrees of

freedom is obtained for a ¼ 1 and p ¼ n=2.

The inverse gamma (or Vinci) distribution is obtained for a ¼ À1.

.

p ¼ 1, a . 0 yields the Weibull distribution.

.

p ¼ 1, a , 0 yields the inverse Weibull distribution (to be called the log-

.

Gompertz distribution in this chapter).

a ¼ p ¼ 1 (a ¼ À1, p ¼ 1) yields the (inverse) exponential distribution.

.

a ¼ 2, p ¼ 1=2 yields the half-normal distribution. This caused Cammillieri

.

(1972) to refer to the generalized gamma distribution as the generalized

seminormal distribution. More generally, all positive even powers and all

149

5.1 GENERALIZED GAMMA DISTRIBUTION

positive powers of the modulus of a normal random variable (with mean zero)

follow a generalized gamma distribution.

As a ! 0, p ! 1, b ! 1 but a2 ! 1=s 2 and bp1=a ! m, the distribution

.

tends to a lognormal LN(m, s 2 ).

For a ! 0, p ! 1, with ap ! r, r . 0, the distribution tends to a power

.

function distribution PF(r, b). Since the power function distribution is the inverse

Pareto distribution [see (3.38)], one directly determines that for a ! 0, p ! À1,

with ap ! Àr, the Pareto distribution Par(b, r) is also a limiting case.

Figure 5.1 summarizes the interrelations between the distributions that are

included in the present chapter.

We should note that the preceding list comprises several of the most popular

lifetime distributions. The generalized gamma distribution is also useful for

discriminating among these models.

5.1.2 The Generalized Gamma Distribution as an Income

Distribution

Esteban (1986) proposed characterizing income distributions in terms of their

income share elasticity. If j(x, x þ h) denotes the share of total income earned by

individuals with incomes in the interval [x, x þ h], we can write j(x, x þ h) ¼

Figure 1 Gamma-type distributions and their interrelations: generalized gamma distribution (GG), inverse

generalized gamma distribution (InvGG), gamma distribution (Ga), Weibull distribution (Wei), inverse

Weibull (¼log-Gompertz) distribution (InvWei), inverse gamma distribution (InvGa), exponential

distribution (Exp), and inverse exponential distribution (InvExp).

150 GAMMA-TYPE SIZE DISTRIBUTIONS

F(1) (x þ h) À F(1) (x) and de¬ne the elasticity h(x, f ) of this quantity

d log j(x, x þ h)

h(x, f ) ¼ lim (5:3)

d log x

h!0

Ð xþh

d log x tf (t) dt

(5:4)

¼ lim

E(X )d log x

h!0

x(x þ h)f (x þ h) À x2 f (x)

(5:5)

¼ lim

E(X )j(x, x þ h)

h!0

xf 0 (x)

:

¼1þ (5:6)

f (x)

It indicates the rate of change for the ¬rst-moment distribution (the distribution of

income shares) at each income level. Since f is a density, the income share elasticity

characterizes an income distribution. It is therefore possible to characterize the

generalized gamma distribution in terms of h(x, f ): Suppose that, for a distribution

supported on (0, 1),

limx!1 h(x, f ) ¼ Àa, for some a . 0 (a type of weak Pareto law).

.

There is at least one interior mode, that is, f 0 (m) ¼ 0 for some m [ (0, 1).

.

h(x, f ) exhibits a constant rate of decline, that is, either h0 (x, f ) ¼ 0 or

.

d log h0 (x, f )

¼ À(1 þ 1)

d log x

for some 1 . À1.

Upon integration, the third assumption can be rephrased as requiring either

d

h(x, f ) ¼ Àa þ , if 1 ¼ 0,

log x

or

d

h(x, f ) ¼ Àa þ , if 1 = 0,

j1jx1

where a and d are constants of integration.

If we combine the ¬rst and third assumption, it follows that both a and 1 must be

positive. If there is to be a unique maximum m, we must further have h(m, f ) ¼ 1

and therefore

!1=1

d

:

m¼

(1 þ a)1

151

5.1 GENERALIZED GAMMA DISTRIBUTION

Hence, d . 0. A reparameterization yields the desired elasticity

x À1