of a(a À 1) and ap À 1. The salient features now follow easily from the preceding

conditions; Table 5.1 summarizes the results.

S T

A closer look reveals that - and -shaped hazard rates are only possible if

neither a nor p equals 1; hence, these cases cannot occur with the gamma or Weibull

distributions.

5.1.4 Lorenz Curve and Inequality Measures

The quantile function of the generalized gamma distribution is not available in

closed form; hence, we must use the representation of the Lorenz curve in terms of

the ¬rst moment distribution. From Butler and McDonald (1989) we know that the

kth moment distribution is

k

F(k) (x; a, b, p) ¼ F x; a, b, p þ , x ! 0, (5:17)

a

and therefore of the same form as the underlying distribution. Utilizing (2.6), we obtain

& ! '

1

j x [ (0, 1) :

{[u, L(u)]} ¼ F(x; a, b, p), F x; a, b, p þ (5:18)

a

Table 5.1 Hazard Rates of Generalized Gamma Distributions

Sign of a(a 2 1) Sign of ap 2 1 Shape of r(x)

2 2 decreasing

T

þ

2 S-shaped

þ 2 -shaped

þ þ increasing

Source: Glaser (1980), McDonald and Richards (1987).

155

5.1 GENERALIZED GAMMA DISTRIBUTION

Regarding the Lorenz ordering, Taillie (1981) asserted (without detailed derivation)

the result

a2 p2 :

X1 ! L X2 ( a1 a2 and a1 p1 (5:19)

)

A detailed proof using a density crossing argument was later provided by Wil¬‚ing

(1996a).

For the Gini coef¬cient McDonald (1984) derived the expression

&

1 1 1 1

G ¼ 2pþ1=a 2 F1 1, 2p þ ; p þ 1;

2 B( p, p þ 1=a) p a 2

'

1 1 11

:

2 F1 1, 2p þ ; p þ 1 þ ; (5:20)

À

p þ 1=a a a2

Special cases of this result were already known to Amoroso (1924 “1925).

5.1.5 A Compound Generalized Gamma Distribution

Starting from (5.2), compound gamma distributions can be constructed by assigning

(joint) distributions to the parameters a, b, p. If the parameter b itself follows a

three-parameter inverse generalized gamma distribution (with the same parameter a

as the structural distribution),

jaj aqÀ1 Àb a

b (a , 0),

f (b) ¼ e

G(q)

the resulting compound distribution has the p.d.f. (Malik, 1967; Ahuja, 1969)

axapÀ1

x . 0,

f (x) ¼ , (5:21)

B( p, q)(1 þ xa ) pþq

which is the density of a power transformation of a random variable following a

Pearson type VI distribution (or beta distribution of the second kind). This family

will be studied in greater detail in the following chapter; for the moment it should be

noted that equation (5.21) provides the link between Chapters 5 and 6.

5.1.6 Estimation

In early work with the generalized gamma distribution, there were signi¬cant

problems in developing inference procedures. The essential dif¬culty is the esti-

mation of the additional (compared to the classical gamma distribution) shape

parameter a. In fact, if a is known, one can apply the transformation X a and use the

156 GAMMA-TYPE SIZE DISTRIBUTIONS

methods appropriate to gamma distributions (as described in, e.g., Bowman and

Shenton, 1988).

Much of the dif¬culty with the model arises because distributions with rather

different sets of parameters can look very much alike. For example, the work of

Johnson and Kotz (1972) showed that for certain values a , 0 (an inverse generalized

gamma distribution in our terminology) two generalized gamma distributions exist for

p¬¬¬¬¬

certain constellations of the shape factors b1 and b2 . Consequently, it will not be

possible to estimate such distributions by moments alone.

Maximum likelihood estimation is also not straightforward. Unfortunately, the

likelihood function is in general not unimodal, nor does it necessarily exhibit a

maximum. There are two main approaches for obtaining MLEs. The ¬rst consists of the

direct maximization of the likelihood; see Lawless (1980) who used a parameterization

of the distribution of Y ¼ log X proposed by Prentice (1974). Here the likelihood is

maximized over a subset of the parameters, with the remaining parameters temporarily

held ¬xed at some initial values. This is followed by a heuristic interpolation scheme that

attempts to re¬ne further the values of the ¬xed parameters. This approach is not very

ef¬cient computationally and guarantees at most a local maximum of the likelihood

function. It also relies to some extent on the judgment of the statistician when

determining appropriate values for the parameters that are held ¬xed.

The second approach, proposed by Hager and Bain (1970), suggests solving a

scalar nonlinear equation derived from the likelihood equations

!a

^

X Xi

n

p ¼ 0, (5:22)

Àn^ þ

^

i¼1 b

! !a !

^

X X Xi

n n

n Xi Xi

^

þp ¼ 0, (5:23)

log log

À

^ ^ ^

^

a b i¼1 b b

i¼1

!

X Xi

n

^

Ànc(^ ) þ a

p ¼ 0: (5:24)

^

i¼1 b