The ¬rst equation (5.22) yields an expression for the scale parameter in terms of

P

the shape parameters, b ; b(^ , p) ¼ [ n Xia =(n^ )]1=^ . Upon substituting this

^ ^a ^ pa

^

i¼1

expression into the second equation (5.23), we get

(" ! Á#)À1

ÀPn

X log Xi

n ^

Xia log Xi

À Ài¼1 n Á

p ; p(^ ) ¼ a P

^ ^a ^ : (5:25)

^

a

n i¼1 Xi

i¼1

^

Substituting this into the remaining equation (5.24) gives us an equation in a

!

aX X

^n n

^

Xia þ log(n^ ) ¼ 0,

^

f(^ ) :¼ Àc( p) þ

a log Xi À log p (5:26)

n i¼1 i¼1

157

5.1 GENERALIZED GAMMA DISTRIBUTION

^

where p is determined by (5.25). When trying to solve the preceding equations, for

example, Hager and Bain (1970) reported persistent divergence using an unstabilized

Newton algorithm.

Wingo (1987b) argued that these problems stem from an inappropriate use of zero-

¬nding algorithms striving for fast local convergence that often diverge upon application

to highly nonlinear problems such as the present one. Also, (5.26) is de¬ned only for

a . 0; hence, any iterative numerical procedure must assure that only positive iterates

^

are obtained. Wingo (1987a) showed that a ¼ 0 is a double root of (5.26), which

explains to some extent the numerical problems reported in the earlier literature.

Wingo recommended (1987b) a derivative-free numerical root isolation method

developed by Jones, Waller, and Feldman (1978) that assures globally optimal

maximum likelihood estimators for the parameters. The procedure is able to locate

all of the zeros of the likelihood equations or indicate that none exist. The method

determines all of the real zeros of f(^ )=^ 2 ¼ 0 on a suf¬ciently large interval [f(^ )

aa a

being de¬ned in (5.26)], if any, and obtains estimates of p and b from (5.25) and

(5.22). For the resulting sets of parameter estimates, the likelihoods are compared

and the parameter set with the largest likelihood is selected. Wingo applied this

method to three data sets containing zero, one, and two solutions of f(^ )=^ 2 ¼ 0,

aa

respectively. [In Wingo (1987a) it is also conjectured, based on extensive

computational tests, that f(^ ) ¼ 0 never has more than two positive zeros.]

a

The Fisher information on u ¼ (a, b, p)` is given by

0 1

c( p)

1 1 þ pc( p)

0

B a2 {1 þ c( p)[2 þ c( p)] þ pc ( p)} À À

aC

b

B C

2

B 1C

1 þ pc( p) ap

I (u) ¼ B C,

À (5:27)

B bC

b2

b

B C

@ A

c( p) 1

c0 ( p)

À

b

a

p¬¬¬

a^^

from which the asymptotic covariance matrix of n(^ , b, p)` can be obtained by

inversion.

In our context, Kloek and van Dijk (1978) reported that for their data the

asymptotic correlation matrix of generalized gamma parameter estimates is nearly

^ ^

singular, with highly correlated estimates a and p. This once again underlines the

problems associated with parameter estimation in connection with this distribution.

5.1.7 Empirical Results

Incomes and Wealth

Amoroso (1924 “1925) ¬t a four-parameter generalized gamma distribution to

Prussian incomes of 1912. Some 50 years later Bartels (1977) applied the three-

parameter version to 1969 ¬scal incomes for three regions in the Netherlands and

found that it does better than the gamma and Weibull special cases, but not as good

as the log-logistic and Champernowne distributions.

158 GAMMA-TYPE SIZE DISTRIBUTIONS

Analyzing 1973 Dutch earnings, Kloek and van Dijk (1978) determined the

generalized gamma distribution to be superior to the gamma and lognormal models

but also reported on numerical problems. They preferred other three-parameter

models such as the Champernowne and log-t distributions.

McDonald (1984) estimated the generalized gamma distribution for 1970,

1975, and 1980 U.S. family incomes. It outperforms eight other distributions and

is inferior to only the GB2 and Singh “ Maddala distributions (see the following

chapter).

Atoda, Suruga, and Tachibanaki (1988) considered grouped data from the

Japanese Income Redistribution Survey for 1975, strati¬ed by occupation. Among

the distributions they employed, the Singh “ Maddala appears to be the most

appropriate for the majority of strata. It is noteworthy that when a model is selected

via information criteria such as the AIC, the generalized gamma is always inferior to

one of its special cases, the Weibull and gamma distributions. In a later study

employing individual data from the same source, the generalized gamma was

sometimes the best distribution in terms of likelihood but only marginally better than

its Weibull and gamma special cases (Tachibanaki, Suruga, and Atoda, 1997).

In a comprehensive study employing 15 income distribution models of the beta

and gamma type, Bordley, McDonald, and Mantrala (1996) ¬t the generalized

gamma distribution to U.S. family incomes for 1970, 1975, 1980, 1985, and 1990. It

is outperformed by the (G)B2, (G)B1, Dagum, and Singh “ Maddala distributions”

see the following chapter”but does signi¬cantly better than all two-parameter

models considered.

In an application using 1984 “1993 German household incomes, the generalized

gamma distribution is revealed as inappropriate model for these data (Brachmann,

Stich, and Trede, 1996). Speci¬cally, it does not provide an improvement over the

two-parameter gamma distribution. The data seem to require a more ¬‚exible model

such as the GB2 and Singh “ Maddala distribution.

Actuarial Losses

In the actuarial literature, Cummins et al. (1990) considered 16 loss distributions

when modeling the Cummins and Freifelder (1978) ¬re loss data. They found that

an inverse generalized gamma provides an excellent ¬t but contains too many

parameters. It emerges that one- or two-parameter special cases such as the inverse

gamma, inverse Weibull, and inverse exponential distributions are already

suf¬ciently ¬‚exible. For the Cummins and Freifelder (1978) severity data the

Singh “Maddala distribution is preferable.

5.1.8 Related Distributions

A “quadratic elasticity” distribution with p.d.f.