distributions, two of which have one additional parameter. For 1973 Dutch earnings

(Kloek and van Dijk, 1978), the distribution does also not do well, notably in the

middle income range, and considerable improvements are possible utilizing three-

and four-parameter models such as the Champernowne and log-t distributions.

Suruga (1982) considered Japanese incomes for 1963“1971. Here the gamma

distribution is outperformed by the Singh“Maddala, Fisk, and beta distributions but does

somewhat better than the lognormal and one-parameter Pareto (II) distribution. In an

extension of Suruga™s work, Atoda, Suruga, and Tachibanaki (1988) considered grouped

data from the Japanese Income Redistribution Survey for 1975, strati¬ed by occupation.

Although among the distributions they employed the Singh“Maddala appears to be the

most appropriate for the majority of strata, it is remarkable that when a model is selected

via information criteria (such as the AIC), the gamma distribution is often preferred over

the more ¬‚exible generalized gamma distribution. In a later study employing individual

data from the same source the gamma is again often indistinguishable from the

generalized gamma distribution (Tachibanaki, Suruga, and Atoda, 1997).

Dagum (1983) ¬t a gamma distribution to 1978 U.S. family incomes. The (four-

and three-parameter) Dagum type III and type I as well as the Singh “ Maddala

distribution perform considerably better; however, the gamma distribution outper-

forms the lognormal by wide margins.

Ransom and Cramer (1983) considered a measurement error model, viewing

observed income as the sum of a systematic component and an independent N(0, s 2 )

error term. For competing models with systematic components following Pareto or

lognormal distributions, they found that the gamma variant is rejected by chi-square

goodness-of-¬t tests for U.S. family incomes for 1960 and 1969.

McDonald (1984) estimated the gamma model for 1970, 1975, and 1980

U.S. family incomes. The distribution is outperformed by three- and four-parameter

families such as the (generalized) beta, generalized gamma, and Singh “ Maddala

distributions, but is superior to all other two-parameter models, notably the lognormal

and Weibull distributions (an exception being the 1980 data for which the Weibull

does somewhat better). Also, the improvements achieved by employing the

generalized gamma are quite small for the 1970 and 1975 data.

Bhattacharjee and Krishnaji (1985) found that the landholdings in 17 Indian

states for 1961 “ 1962 appear to be more adequately approximated by a gamma

distribution with a decreasing density ( p , 1) than by a lognormal distribution. A

log-gamma distribution provides a comparable ¬t.

Bordley and McDonald (1993) employed the gamma distribution for the

estimation of the income elasticity in an aggregate demand model for automotive

data. It turns out that the income-elasticity estimates provided by the gamma

are fairly similar to those associated with distributions providing good approxi-

mations to U.S. income distribution, such as the generalized beta distribution of the

second kind (GB2; see Chapter 6).

Angle (1993b, 1996) found that personal income data from the 1980“ 1987 U.S.

Current Population Survey as well as from the Luxembourg Income Study (for eight

168 GAMMA-TYPE SIZE DISTRIBUTIONS

countries in the 1980s) strati¬ed by levels of education are well approximated by

gamma distributions, with shape parameters that moreover increase with educational

attainment. Somewhat surprisingly, it turns out that the aggregate distribution can

also be ¬tted by a gamma distribution, although ¬nite mixtures of gamma

distributions are known in general not to follow this distribution.

Victoria-Feser and Ronchetti (1994) ¬t a gamma distribution to incomes of

households on income support using the 1979 UK Family Expenditure Survey. They

employed the MLE as well as an optimal B-robust estimator (OBRE) (see above) and

concluded that the latter provides a better ¬t because it gives more importance to the

majority of data.

In the study of Brachmann, Stich, and Trede (1996) utilizing 1984 “1993 German

household incomes, the gamma distribution emerges as the best two-parameter

model. However, only the more ¬‚exible GB2 and Singh “ Maddala distributions

seem to be appropriate for these data.

Bordley, McDonald, and Mantrala (1996) ¬t the gamma distribution to U.S.

family incomes for 1970, 1975, 1980, 1985, and 1990. Although it is outperformed

by all three- and four-parameter models, notably the GB2, Dagum, and Singh “

Maddala distributions, it turns out to be the best two-parameter model except for

1980 and 1985, where the Weibull distribution does equally well.

Creedy, Lye, and Martin (1997) estimated the gamma distribution for individual

earnings from the 1987 U.S. Current Population Survey (March Supplement). The

performance is comparable to, although slightly worse than, the one provided by a

generalized gamma and a generalized lognormal distribution. The standard

lognormal distribution does much worse for these data.

From these studies it would seem that the gamma distribution is perhaps the best

two-parameter model for approximating the size distribution of personal income.

Actuarial Losses

In the actuarial literature Ramlau-Hansen (1988) ¬t a gamma distribution to

windstorm losses for a portfolio of single-family houses and dwellings in Denmark

for the period 1977 “1981. However, there are problems with his choice because the

empirical skewness and coef¬cient of variation are very close, whereas for a gamma

distribution the former necessarily equals twice the latter [see (5.40)].

Cummins et al. (1990), who employed 16 loss distributions when modeling

the Cummins and Freifelder (1978) ¬re loss data, found the gamma distribution to be

among the worst distributions they considered. Speci¬cally, the data seem to require

a model with much heavier tails, such as an inverse gamma distribution.

5.3 LOG-GAMMA DISTRIBUTION

If there is a lognormal distribution that enjoys wide applicability, why should we

deprive ourselves of a log-gamma distribution? A logarithmic transform is often a

sensible operation to smooth the data, especially if we con¬ne ourselves to the

interval (1, 1).

169

5.3 LOG-GAMMA DISTRIBUTION

5.3.1 De¬nition and Basic Properties

If Y follows a two-parameter gamma distribution, the random variable X ¼ exp Y is

said to possess a log-gamma distribution. The density of X is therefore

b p ÀbÀ1

{log (x)}pÀ1 , 1

f (x) ¼ x x, (5:56)

G( p)

where p . 0, b . 0. Here both parameters b, p are shape parameters. Many

distributional properties of this model are given in Taguchi, Sakurai, and Nakajima

(1993), who also discuss a bivariate form. [The parameterization (5.56) differs

slightly from the one used in connection with the gamma distribution in that,

compared to (5.33), we use 1=b instead of b. (5.56) appears to be the standard

parameterization of this distribution in the literature.]

Despite its genesis, the log-gamma distribution is perhaps best considered a

generalized Pareto distribution since the classical Pareto type I with a unit scale is

the special case where p ¼ 1. (The parameter b now plays the role of Pareto™s a.)

Although the Pareto type I has a decreasing density, the log-gamma is more ¬‚exible

in that it allows for unimodal densities. Speci¬cally, the density is decreasing for

p 1, whereas for p . 1 an interior mode exists. The parameter b determines the

shape in the upper income range, whereas p governs the lower tail.

In order to enhance the ¬‚exibility of this distribution, it is sometimes desirable to

introduce a scale parameter, yielding

b p bb ÀbÀ1 n xopÀ1

, 0,b

f (x) ¼ x x, (5:57)

log

G( p) b

where p . 0, b . 0, and b . 0 is the scale. A further generalization along the lines

of the generalized gamma distribution is the generalized log-gamma”or rather log-

“generalized gamma””distribution given by the p.d.f. (Taguchi, Sakurai, and

Nakajima, 1993)

abap n xoapÀ1 h x a i

exp À b log 0,b

f (x) ¼ , x, (5:58)

log

xG( p) b b

where a, b, p, b . 0. The logarithm of a random variable with this density follows a

four-parameter generalized gamma distribution. In our presentation of the basic

properties of the log-gamma distribution, we shall con¬ne ourselves to the two-

parameter case (5.56).

The moments exist for k , b, in which case they are given by

p

b

E(X k ) ¼ : (5:59)

bÀk

170 GAMMA-TYPE SIZE DISTRIBUTIONS