same Fisher information, although using a different parameterization from which the

equivalence is not easily recognized.) Prentice also discussed a reparameterization

that is useful for discriminating between special and limiting cases of the GB2.

6.1.6 Empirical Results

Incomes and Wealth

Vartia and Vartia (1980) ¬t a four-parameter shifted B2 distribution (under the

name of scaled and shifted F distribution) to the 1967 distribution of taxed income

in Finland. Using ML and method-of-moments estimators, they found that their

model ¬ts systematically better than the two- and three-parameter lognormal

distributions.

McDonald (1984) estimated the GB2 and B2 distributions for 1970, 1975, and

1980 U.S. family incomes. The GB2 outperforms 10 other distributions (mainly of

the beta and gamma type), whereas the B2 ¬t is comparable to the ¬t obtained using

gamma or generalized gamma distributions but is inferior to the Singh “ Maddala

distribution for each one of the data sets considered.

Butler and McDonald (1989) gave GB2 parameter estimates for U.S. Caucasian

family incomes from 1948 “1980, using maximum likelihood estimators applied to

grouped data. They found shape parameters a [ (2:8, 8:4), roughly decreasing over

196 BETA-TYPE SIZE DISTRIBUTIONS

time, p [ (0:17, 0:6), and q [ (0:4, 1:7), with larger values of p and q pertaining to

the post-1966 data. This shows that the tail indexes ap, aq are roughly constant over

time, while there are considerable changes in the center of the distribution.

Majumder and Chakravarty (1990) considered the GB2 distribution when

modeling U.S. income data for 1960, 1969, 1980, and 1983. The distribution ranks

second when compared to their own distribution. However, there are some

contradictory results in their paper, and in a reassessment of their ¬ndings

emphasizing that the Majumder “Chakravarty distribution is merely a reparame-

trization of the GB2 distribution, McDonald and Mantrala (1993, 1995) ¬t this

model to 1970, 1980, and 1990 U.S. family incomes using two different methods of

estimation and alternative groupings of the data. The distribution provides the best ¬t

among all the models considered. Majumder and Chakravarty™s paper provides us

with an important lesson in ¬tting income data and signals the dangers and pitfalls

when data are not properly scrutinized.

In a study of the aggregate demand for speci¬c car lines, Bordley and McDonald

(1993) employed the GB2 distribution for the direct estimation of income elasticity

from population income distribution. Their results are consistent with those of

traditional econometric studies of automotive demand.

McDonald and Xu (1995) studied 1985 U.S. family incomes. Out of

11 distributions considered”mainly of the gamma and beta type”the GB2 ranks

¬rst (being observationally equivalent to the more general GB distribution, which is

brie¬‚y mentioned in Section 6.5 below) in terms of likelihood.

In a very comprehensive already mentioned study employing 15 income

distribution models (again of the beta and gamma type), Bordley, McDonald, and

Mantrala (1996) ¬t the GB2 distribution to U.S. family incomes for 1970, 1975,

1980, 1985, and 1990. For all but the 1970 data (where the ¬ve-parameter GB

distribution”see Section 6.5 below”provides a slight improvement) the GB2

outperforms every other distribution, with the improvements relative to the nested

models being statistically signi¬cant. Bordley, McDonald, and Mantrala concluded

that the GB2 distribution is the best-¬tting four-parameter distribution for these data.

Brachmann, Stich, and Trede (1996) ¬t the GB2 (and many of its subfamilies and

limiting cases) to individual household incomes from the German Socio-Economic

Panel (SOEP) for the years 1984 “1993. A comparison with nonparametric density

estimates shows that only the GB2 and, to a lesser extent, the Singh “ Maddala

distribution to be discussed in the following section are satisfactory. However, the

means and Gini coef¬cients appear to be slightly underestimated.

Parker (1997) considered a three-parameter B2 distribution when studying UK

self-employment incomes from 1976 “ 1991. Starting from a dynamic economic

model of self-employment income, he found that incomes above some threshold

g , 0 follow a two-parameter B2 distribution. He estimated parameters by modi¬ed

method-of-moments estimators (MMME), as advocated by Cohen and Whitten

(1988) for models containing a threshold parameter, ¬nding that incomes became

progressively more unequally distributed, a fact he attributed to an increase of

heterogeneity among the self-employed.

197

6.2 SINGH “ MADDALA DISTRIBUTION

Actuarial Losses

In actuarial science, the already mentioned paper by Cummins et al. (1990) used the

GB2 family to model the annual ¬re losses experienced by a major university

(Cummins and Freifelder, 1978). It turns out that, in terms of likelihood, the full

¬‚exibility of the GB2 is not required and a one-parameter limiting case, the inverse

exponential distribution, agrees very well with the four-parameter GB2. The same

authors also considered data on the severity of ¬re losses and ¬t the GB2 to both

grouped and individual observations. Here three-parameter special and limiting cases

of the GB2 such as the Singh “ Maddala and inverse generalized gamma distributions

are selected.

Cummins et al. (1990) provided an important lesson about unnecessary

overparameterization. In summary, the GB2 family is indeed an attractive, ¬‚exible,

elegant, and ingenious family but it involves four parameters. The experience

collected by statistical applications during the last 100 years teaches us that four-

parameter distributions can sometimes be almost omnipotent and do not allow us

to penetrate to the crux of the matter: in our case the mechanism and factors that

determine the size distributions. One (almost) becomes nostalgic about Pareto™s

simple model proposed at the end of the nineteenth century about 90 years before

McDonald™s pioneering effort. Perhaps the leap forward is just too great?

(Evidently, without the disastrous consequences of the Chinese leap forward in the

late 1950s.)

6.1.7 Extensions

Zandonatti (2001) presented a generalization of the GB2 following Feller™s de¬nition

of a GB2 distribution; cf. (6.20). The distribution is de¬ned by (we omit an

additional location parameter)

X ¼ b[W À1=u À 1]1=a ,

where W denotes a standard beta( p, q) variable, and possesses the density

xa iÀup & x a iÀu 'qÀ1

auxaÀ1 h h

:

f (x) ¼ a 1þ 1À 1þ (6:41)

b B( p, q) b b

We may call the distribution given by (6.41) the Zandonatti distribution. Clearly,

for u ¼ 1 we obtain (6.5).

6.2 SINGH “MADDALA DISTRIBUTION

The Singh “ Maddala distribution introduced by Singh and Maddala in 1975 and in a

more polished form in 1976 has received special attention in the literature of income

distributions. Although it was discovered under the above name before the GB2

198 BETA-TYPE SIZE DISTRIBUTIONS

distribution, it would be convenient to treat it as a special case of the GB2. In the

recent econometric literature, it is often compared with the Dagum distribution (see

the following section), perhaps due to the similarity of their c.d.f.™s.

6.2.1 De¬nition and Motivation

The Singh “Maddala distribution is a special case of the GB2 distribution, with

p ¼ 1. Its density is

aqxaÀ1

x . 0,

f (x) ¼ a , (6:42)

b [1 þ (x=b)a ]1þq

where all three parameters a, b, q are positive. Here b is a scale parameter and a, q

are shape parameters; q only affects the right tail, whereas a affects both tails. We

shall use the notation GB2(a, b, 1, q) ; SM(a, b, q).

This distribution was independently rediscovered many times in several loosely

related areas. Consequently, it is known under a variety of names: It seems that it was

¬rst considered by Burr (1942), where it appears as the twelfth example of solutions

of a differential equation de¬ning the Burr system of distributions [see Kleiber,

(2003a) for a recent survey of this family]. It is therefore usually called the Burr XII

distribution, or”being the most popular Burr distribution”simply the Burr

distribution. Kakwani (1980b, p. 24) reported that it was proposed as an income

distribution as early as 1958 in an unpublished paper presented at a meeting of the

Econometric Society by Sargan. It is also known as the Pareto (IV) distribution