where m [ IR; s, n . 0: As in the case of the t distribution, no closed-form

expression of the c.d.f. is available. From the properties of the t distribution, the

variance of logarithms is given by

ns 2

VL(X ) ¼ , (4:66)

nÀ2

provided n . 2: Apparently it has not been appreciated in the econometrics literature

how heavy the tails of this distribution are. Kleiber (2000b) pointed out that the log -t

distribution does not have a single ¬nite moment, that is, E(X k ) ¼ 1 for

all k [ IRn{0}: However, the Lorenz curve and most of the standard inequality

measures only exist when the mean is ¬nite. Speci¬cally, for (4.65) the variance of

logarithms is the only inequality measure among the common ones that exists,

provided n is suf¬ciently large. (In the case where n ¼ 1; i.e., the log-Cauchy case,

even the variance of logarithms is in¬nite.)

Hogg and Klugman (1983) presented the following interesting mixture

representation for the log t distribution: Suppose X has a lognormal distribution,

parameterized in the form

r¬¬¬¬¬¬ !

Àu(log x À m)2

1u

f (x j u) ¼ , x . 0,

exp

u 2p 2

144 LOGNORMAL DISTRIBUTIONS

that is, log X has a normal distribution with mean m and variance 1=u; and u $

Ga(n, l): Then

° 1 ( r¬¬¬¬¬¬ )

2! !

ln unÀ1 eÀlu

1u (log x À m)

f (x) ¼ du

exp Àu

0 u 2p G(n)

2

ln G(n þ 1=2)

¼ p¬¬¬¬¬¬ :

2pG(n)x[l þ (log x À m)2 ]nþ1=2

Thus, the log -t distribution may be considered a shape mixture of lognormal variates

with inverse gamma weights.

Kloek and van Dijk (1977) ¬t a three-parameter log -t distribution to Australian

family disposable incomes for the period 1966 “1968, disaggregated by age,

occupation, education, and family size. Although for about one half of the samples

they considered, “one may doubt whether it is worthwhile to introduce the extra

parameter [namely, n]” (p. 447), for other cases the ¬t is considerably better. Using

Cox tests (Cox, 1961), they found that the lognormal distribution is rejected when

compared with the log -t; but not vice versa. Overall, they concluded that the log -t

distribution appears to be a useful improvement over the lognormal.

Cummins et al. (1990) applied the log -t distribution to aggregate ¬re losses,

a data set that seems to be better modeled by simpler distributions such as the inverse

exponential or inverse gamma distribution.

Recently, Azzalini and Kotz (2002) ¬t a log-skewed-t distribution to U.S. family

income data for 1970(5)1990 with rather encouraging but preliminary results.

Other generalized lognormal distributions”not to be confused with the

distribution discussed in Section 4.10 above”were considered by Bakker and

Creedy (1997, 1998) and Creedy, Lye, and Martin (1997). Their distributions

arise as the stationary distribution of a certain stochastic model and possesses

the p.d.f.

f (x) ¼ exp{u1 (log x)3 þ u2 (log x)2 þ u3 log x þ u4 x À h}, 0 , x , 1, (4:67)

where exp(h) is the normalizing constant. Clearly, the two-parameter lognormal

distribution is obtained for u1 ¼ u4 ¼ 0: (Observe that the gamma distribution is

also a special case, arising for u1 ¼ u2 ¼ 0.)

Creedy, Lye, and Martin (1997) estimated this generalized lognormal distribution

for individual earnings from the 1987 U.S. Current Population Survey (March

Supplement), for which the model does about as well as a generalized gamma

distribution and much better than the standard gamma and lognormal distributions.

When applied to New Zealand wages and salaries for 1991, classi¬ed by age groups

and sex, the distribution performs again consistently better than the two-parameter

lognormal and about as well as the gamma distribution, in terms of chi-square

145

4.12 RELATED DISTRIBUTIONS

criteria. However, in nine out of ten cases for males and in six out of ten cases for

females it is outperformed by a generalized gamma distribution with the same

number of parameters (Bakker and Creedy, 1997, 1998).

Saving (1965) used a SB -type distribution (Johnson, 1949; see also Section 2.4) as

a model for ¬rm sizes that is a four-parameter lognormal-type distribution on a

bounded domain.

Two additional distributions closely related to the lognormal, the Benini and

Benktander type I distributions, will be discussed in Chapter 7.

CHAPTER FIVE

Gamma-type Size Distributions

For our purposes, gamma-type distributions comprise all distributions that are

members of the generalized gamma family introduced by Luigi Amoroso in the 1920s,

including the classical gamma and Weibull distributions and simple transformations

of them, such as inverted or exponentiated forms.

The literature dealing with these models is rather substantial, notably in

engineering and more recently in medical applications, and detailed accounts of the

gamma and Weibull distributions are, for instance, available in Chapters 17 and 21

of Johnson, Kotz, and Balakrishnan (1994). There is a book-length treatment of

estimation for the gamma distribution, written by Bowman and Shenton (1988).

Our exposition therefore focuses on the “size aspects” and log-gamma and log-

Gompertz distributions, two distributions whose applications (as of today) appear to

be mainly in connection with size phenomena.

5.1 GENERALIZED GAMMA DISTRIBUTION

In the American literature the generalized gamma distribution is most often referred

to as the Stacy (1962) distribution although by now it is acknowledged that

Amoroso™s (1924 “1925) paper in Annali di Matematica was probably the ¬rst work

in which the generalized gamma distribution appeared. To this, we add the less

known fact that D™Addario in the 1930s dealt with this generalization of the gamma

distribution. In a report by A. C. Cohen (1969) entitled A Generalization of the

Weibull Distribution, this distribution was rediscovered again! It is not out of the

question that a more thorough search would locate a source for this distribution

during the years 1940 “ 1960.

147

148 GAMMA-TYPE SIZE DISTRIBUTIONS

5.1.1 De¬nition and Interrelations

The classical gamma distribution is de¬ned by

1

z pÀ1 eÀz=b , z . 0,

f (z) ¼ (5:1)

b p G( p)

where p, b . 0. If a power transformation X a , a . 0, of some random variable X

follows this distribution, the p.d.f. of X is

a a