1 22

x pÀ1 eÀbxÀg x , x . 0,

f (x) ¼ (5:28)

d( p, b, g)

159

5.1 GENERALIZED GAMMA DISTRIBUTION

was considered by Bordley, McDonald, and Mantrala (1996). Here the normalizing

constant is given by

b

2 2

d( p, b, g) ¼ (2b2 )Àp=2 G( p)eb =(8g ) DÀp p¬¬¬ , (5:29)

2g

where DÀp is a parabolic cylinder function. (An alternative but more lengthy

expression occurs in terms of the con¬‚uent hypergeometric function 1 F1 .)

The quadratic elasticity distribution was originally motivated by Bordley and

McDonald (1993) in connection with the estimation of income elasticity in an

aggregate demand model for speci¬c car lines. The gamma distribution is associated

with a linear income share elasticity [see (5.8)], whereas the data considered by

Bordley and McDonald showed signs of being slightly quadratic. The income share

elasticity of the new distribution is

h(x, f ) ¼ ( p À 1) À bx À 2g 2 x2 : (5:30)

Clearly, the gamma distribution is the special case where g ¼ 0 and a generalized

gamma with a ¼ 2 is obtained for b ¼ 0. The moments of (5.28) are given by

d( p þ k, b, g)

E(X k ) ¼ , (5:31)

d( p, b, g)

where d is de¬ned in (5.29).

This model was ¬tted to U.S. family incomes for 1970, 1975, 1980, 1985, and

1990 by Bordley, McDonald, and Mantrala (1996); the performance was found to be

intermediate between gamma and generalized gamma but inferior to beta-type

distributions with a comparable number of parameters.

A further generalization along these lines is a member of the so-called

generalized exponential family used by Bakker and Creedy (1997, 2000) and

Creedy, Lye, and Martin (1997). Its p.d.f. is given by

È É

f (x) ¼ exp u1 log x þ u2 x þ u3 x2 þ u4 x3 À h , x ! 0: (5:32)

Here exp(h), is the normalizing constant. A generalized gamma distribution with

a ¼ 2 [using the notation of (5.2)] is obtained for u2 ¼ 0 ¼ u4 and the quadratic

elasticity distribution (5.28) is the special case where u4 ¼ 0. An important feature

of this model is that it can accommodate multimodality.

Creedy, Lye, and Martin (1997) estimated this generalized gamma-type

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

(March Supplement), for which it does about as well as a generalized lognormal

distribution and considerably better than the standard gamma and lognormal

distributions. When ¬t to New Zealand wages and salaries for 1991, classi¬ed by age

groups and sex, the distribution is superior to generalized lognormal, lognormal, and

gamma ¬ttings in nine of ten cases for males and in six of ten cases for females, in

terms of chi-square criteria (Bakker and Creedy, 1997, 1998). In a further application

160 GAMMA-TYPE SIZE DISTRIBUTIONS

of this model to male individual incomes (before tax, but including transfer

payments) from the New Zealand Household Expenditure Surveys for each year over

the period 1985 “1994, Bakker and Creedy (2000) found the coef¬cient on x3 to be

statistically insigni¬cant and therefore con¬ned themselves to the special case where

u4 ¼ 0, which is the quadratic elasticity model (5.28). Their data exhibit bimodality

that they attribute to the inclusion of transfer payments in their measure of income.

Following earlier work by, for example, Metcalf (1969) (mentioned in the preceding

chapter) they investigated how macroeconomic variables in¬‚uence the parameters of

distribution over time. It emerges that the rate of unemployment is the primary

in¬‚uence on the shape of the distribution.

5.2 GAMMA DISTRIBUTION

The gamma (more precisely, the Pearson type III) distribution is certainly among the

¬ve most popular distributions in applied statistical work when unimodal and

positive data are available. In economic and engineering applications it has two

rivals”lognormal and Weibull. It is hard to state categorically which one is the

frontrunner. Lancaster (1966) asserted that both Laplace (in the 1836 third edition of

´

his Theorie analytique des probabilites) and Bienayme in 1838 (in his Memoires de

´ ´ ´

l™Academie de Sciences de l™Institute de France) obtained the gamma distribution.

However, these references pertain to normal sampling theory and therefore

essentially to the history of x 2 distributions. For a gamma distribution with a general

(i.e., not limited to half-integers) shape parameter, an early reference predating

Pearson™s (1895) seminal work on asymmetric curves may be attributed to De Forest

(1882 “1883), as pointed out by Stigler (1978).

Here we shall brie¬‚y sketch the basic properties of the gamma distribution and

concentrate on aspects more closely related to size and income distributions.

5.2.1 De¬nition, History, and Basic Properties

The pioneering work marking the initial use of the gamma distribution as an income

distribution is due to the French statistician Lucien March, who in 1898 ¬t

the gamma distribution to various French, German, and U.S. earnings distributions.

March was inspired by Pearson™s work on asymmetric curves. As mentioned above,

some 25 years later Amoroso in 1924 introduced a generalized gamma distribution,

and another 50 years later the standard gamma distribution resurfaced as a size

distribution almost simultaneously but independently in the cybernetics (Peterson

and von Foerster, 1971) and econometrics (Salem and Mount, 1974) literatures.

The p.d.f. of the gamma distribution is

1

x pÀ1 eÀx=b , x . 0,

f (x) ¼ (5:33)

p

b G( p)

where p, b . 0, with p being a shape and b a scale parameter.

161

5.2 GAMMA DISTRIBUTION

As mentioned in the preceding section, the gamma distribution includes the

exponential (for p ¼ 1) and chi-square distributions (for p ¼ n=2, n an integer) as

special cases.

From (5.12), the moments of the gamma distribution are given by

G( p þ k)

E(X k ) ¼ bk : (5:34)

G( p)

Hence, the mean is

E(X ) ¼ bp, (5:35)

and the variance equals

var(X ) ¼ b2 p: (5:36)

The coef¬cient of variation is therefore of the simple form

1

CV ¼ p¬¬¬ : (5:37)

p

The shape factors are

p¬¬¬¬¬ 2

b1 ¼ p¬¬¬ (5:38)

p

and

6

b2 ¼ 3 þ : (5:39)

p

Hence, in the gamma case we have the relation