scale and a, p, q are shape parameters. The new parameter c [ [0, 1] permits a

smooth transition between the special cases c ¼ 0, the GB1 distribution, and c ¼ 1,

the GB2 distribution. The moments of this generalized distribution exist for all k if

c , 1 or for k , aq if c ¼ 1; they are

bk B( p þ k=a, q) kk k

k

;pþqþ ;c :

E(X ) ¼ 2 F1 p þ ,

B( p, q) aa a

However, when ¬tting this model to 1985 family incomes, McDonald and Xu found

that the GB2 subfamily is selected (in terms of likelihood and several other criteria).

Thus, it appears that the ¬ve-parameter GB distribution does not provide additional

¬‚exibility, at least in our context. The authors of this book remain skeptical about the

usefulness and meaning of ¬ve-parameter distributions and consider the distribution

described by (6.142) to be just a curious theoretical generalization.

233

6.5 (GENERALIZED) BETA DISTRIBUTION OF THE FIRST KIND

A further application of beta type I income distribution models is given in Pham-

Gia and Turkkan (1997), who derived the density of income X ¼ X1 þ X2 , where X1

is true income and X2 an independent (additive) reporting error under the assumption

that Xi , i ¼ 1, 2, follow B1( pi , qi ) distributions with general support [0, ui ]. The

density of X can be expressed in terms of an Appell function (a bivariate

hypergeometric function).

6.5.2 Empirical Applications

In an in¬‚uential paper, Thurow (1970) applied the B1 distribution to U.S. Census

Bureau constant dollar (1959) income distributions for households (families and

unrelated individuals) for every year from 1949 “ 1966, strati¬ed by race. [The

estimated parameters are not given in Thurow (1970) but in McDonald (1984).]

He also studied the impact of various macroeconomic factors on the parameters

of the distribution via regression techniques. In particular, his results raise

questions as to whether economic growth is associated with a more egalitarian

distribution and also suggest that in¬‚ation may lead to a more equal distribution

for whites. However, McDonald (1984) expressed some doubts concerning T

Thurow™s results. He pointed out that none of the estimated densities is -shaped

(as is to be expected) and that the implied Gini coef¬cients differ from census

estimates by about 30%, concluding that theses differences highlight estimation

problems.

McDonald and Ransom (1979a) employed the B1 distribution for approximating

U.S. family incomes for 1960 and 1969 through 1975. When utilizing three different

estimators, it turns out that the distribution is preferable to the gamma and lognormal

distributions but inferior to the Singh “ Maddala, which has the same number of

parameters.

McDonald (1984) estimated both the B1 and GB1 distribution for 1970, 1975,

and 1980 U.S. family incomes. The performance of the GB1 is comparable to the

generalized gamma and B2 distribution”both of which have the same number of

parameters”but inferior to the GB2 or Singh “ Maddala distributions.

The B1 distribution was also ¬tted to Japanese income data, in grouped form,

from the 1975 Income Redistribution Survey by Atoda, Suruga, and Tachibanaki

(1988). Four occupational classes as well as primary and redistributed incomes were

considered, and ¬ve different estimation techniques applied. The distribution

provides a considerably better ¬t than the (generalized) gamma and lognormal

distributions, but the Singh “ Maddala is superior.

Bordley, McDonald, and Mantrala (1996) ¬t the (G)B1 and GB distributions to

U.S. family incomes for 1970, 1975, 1980, 1985, and 1990. The GB distribution is

observationally equivalent to the GB2 distribution for four of the data sets; for the

1970 data it provides a slight improvement. The GB1 comes in ¬fth out of 15 beta-

and gamma-type distributions, being observationally equivalent to the generalized

gamma distribution for all these years. However, the three-parameter Dagum type I

and Singh “ Maddala distributions perform considerably better.

234 BETA-TYPE SIZE DISTRIBUTIONS

Brachmann, Stich, and Trede (1996) estimated both the B1 and GB1 distributions

for household income data from the German Socio-Economic Panel (SOEP) for

1984 “1993. They noted that the ML estimation of the GB1 proved to be rather

dif¬cult since the gradient of the log likelihood in the parameter b was rather small.

Also, both models tend to underestimate the mean for these data.

All these studies appear to suggest that there are several distributions supported

on an unbounded domain which provide a considerably better ¬t than a GB1 or B1

distribution.

CHAPTER SEVEN

Miscellaneous Size Distributions

In this chapter we shall study a number of size distributions that may not be in the

mainstream of current research, but are de¬nitely of historical interest as well as

containing potential applications. We have tried to unify the results scattered in the

literature, sometimes in the most unexpected sources.

7.1 BENINI DISTRIBUTION

As was discussed in detail in Chapter 3 (the reader may wish to consult Section 3.2),

Vilfredo Pareto (1896, 1897)”the father of the statistical-probabilistic theory of

´

income distributions”announced in his classical works La courbe de la repartition de

´

la richesse and Cours d™economie politique the remarkable discovery that the survival

function of an income distribution is approximately linear in a double-logarithmic

plot. He provided empirical veri¬cations of his law for a multitude of data, showing

that the relationship is valid for any (geographical) location, any time period (for

which the data are available), and any economic level of a country or region. An alert

and energetic Italian statistician and demographer, Rodolfo Benini (a short biography

of Benini is presented in Appendix A) was able to con¬rm almost immediately in 1897

that the Pareto law indeed holds for incomes as well as various other economic

variables. While analyzing additional different data sources, Benini subsequently

discovered in 1905 “ 1906 that for the distribution of legacies a quadratic (rather than

linear) function

log F (x) ¼ a0 À a1 log x À a2 ( log x)2 (7:1)

provides a better ¬t. This leads to a distribution with the c.d.f.

F(x) ¼ 1 À exp{Àa( log x À log x0 ) À b( log x À log x0 )2 }, (7:2)

235

236 MISCELLANEOUS SIZE DISTRIBUTIONS

where x ! x0 . 0 and a, b . 0. For parsimony, Benini (1905) considered only the

case where a ¼ 0, that is,

F(x) ¼ 1 À exp{Àb( log x À log x0 )2 } (7:3)

Àb( log xÀlog x0 )

x

x ! x0 . 0:

¼1À ,

x0

Subsequently, this idea was used by several other Italian economists and

statisticians, including Bresciani Turroni (1914) and Mortara (1917, 1949). They

however introduced higher-order terms. Independently, a well-known Austrian

statistician Winkler (1950) some 45 years later also suggested that, in the Pareto

diagram, a higher-order polynomial in log x may provide an even better ¬t to empirical

income distributions than the Pareto distribution and he ¬t a quadratic”that is, the

original Benini distribution (7.1)”to the U.S. income distribution of 1919.

Also independently, but somewhat later, in the actuarial literature, the distribution

with c.d.f. (7.1) was proposed as a model for the size-of-loss distribution. In Head

(1968) it appears as a nameless distribution resulting in a better ¬t than the Pareto

distribution for several empirical ¬re loss-severity distributions. Ramachandran

(1969) also found this model to be preferable to the Pareto distribution when dealing

with UK ¬re losses. DuMouchel and Olshen (1975) called it an approximate

lognormal distribution, whereas Shpilberg (1977) referred to it as the quasi-

lognormal distribution. We shall refer to all the variants (i.e., a ¼ 0 and a = 0) of