1

a [ (0, 2],

, (7:26)

1 þ s a jtja

245

7.3 CHAMPERNOWNE DISTRIBUTION

but does not have a simple expression for the density and c.d.f. Speci¬cally, it

appears in a mixture representation of the Linnik distribution discussed by

Kozubowski (1998).

These interesting theoretical properties in the spirit of modern developments in

the theory of statistical distributions could lead to a better understanding of the

hidden structure of the Champernowne distribution and perhaps provide an avenue

for further generalizations and discoveries. Needless to say, the original motivation

and model leading to this distribution are quite removed from its genesis related to

the stable and Linnik laws. See Kleiber (2003b) for further details and applications

of the above interrelations.

In the case where jlj , 1, the moments E(X k ) of (7.17) exist only for

Àa , k , a. They are [Champernowne (1952), for k ¼ 1, and Fisk (1961a), for a

general k]

p sin ku=a

E(X k ) ¼ xk : (7:27)

0

u sin kp=a

The coef¬cient of variation is therefore given by

s¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬

u u p

À 1:

CV ¼ (7:28)

cot cot

p a a

Harrison (1974) presented the Atkinson measures of inequality for the

distribution (7.16) that are given by

p e=(1Àe) & sin[(1 À e)u=a] '1=(1Àe) sin(p=a)

:

Ae ¼ 1 À (7:29)

u sin[(1 À e)p=a] sin(u=a)

Here e is the inequality aversion parameter. In the case where e ¼ 1, the last formula

is not applicable. Instead, one obtains the simple expression

x0 sin(p=a)

A1 ¼ 1 À ¼1À , (7:30)

E(X ) sin(u=a)

an inequality measure that was previously proposed by Champernowne (1952).

Harrison (1974) showed graphically that the measure Ae is, for a ¬xed e,

monotonically decreasing in u, for 0 , u , p, approaching complete equality of

incomes for u ! p, that is, for a point mass concentrated at x0 .

All three forms of the Champernowne distribution discussed above possess the

property of symmetry for the distribution of the income power. There is, however, an

246 MISCELLANEOUS SIZE DISTRIBUTIONS

asymmetric (on a log-scale) four-parameter generalization of the case where jlj , 1,

given by the c.d.f. (Champernowne, 1952)

8 & '!

sin u

1

>

>1 À

> (s À 1)u þ 2 arctan , for 0 x x0 ,

< cos u þ (x=x0 )sa

(1 þ s)u

F(x) ¼ & '

> sin u

2s

>

>1 À

: for x0 , x,

,

arctan

cos u þ (x=x0 )a

(1 þ s)u

(7:31)

where the new parameter s may be viewed as a skewness parameter, since when

s . 1, the curve exhibits positive skewness, and for s , 1, the curve is negatively

skewed. The case where s ¼ 1 yields the distribution (7.18). The p.d.f. of (7.31) is

given by

2sasin u(x=x0 )c

f (x) ¼ , (7:32)

x(1 þ s)u[1 þ 2cos u(x=x0 )c þ (x=x0 )ac ]

where c ¼ sa for x x0 and c ¼ a for x . x0. [We note that (7.31) is sometimes

referred to as “Champernowne™s ¬ve-parameter formula” (see, e.g., Campano,

1987), although (7.31) de¬nes a four-parameter distribution. This is presumably due

to Champernowne™s unusual notation (7.12), where n is a function of the remaining

three parameters y0 , a, and l.]

As far as the estimation is concerned, only methods for grouped data have been

discussed in the literature. Rather than employing the computationally complex maxi-

mum likelihood method”note the precomputer era of his work!”Champernowne

(1952) attributed special importance to methods which yield “solutions that agree

with the observed distribution in such economically important matters as the total

number of incomes, the average income and the position and slope of the Pareto

line for high incomes” (pp. 597 “598). Most of the methods he considered start

with the total number of incomes and some form of average income (mean, median,

mode), proceed with an estimation of Pareto™s alpha by a regression technique, and

¬nally obtain u via interpolation.

He ¬t the three-parameter distribution to 1929 U.S. family incomes, 1938

Japanese incomes (previously also considered by Hayakawa, 1951), 1930

Norwegian incomes, and 1938 “1939 UK incomes, and the four-parameter model

to 1918 (previously considered by Davis, 1941a,b) and 1947 U.S. incomes. The ¬t of

the 1918 data is comparable to that provided by Davis™s model (see Section 7.2).

In addition, it emerges that for 1933 Bohemian incomes the two-parameter

Champernowne (i.e., Fisk) distribution provides as good a ¬t as several three-

parameter models.

Harrison (1974) suggested a minimum-distance estimator, determining

parameters simultaneously using an iterative generalized least-squares approach.

Analyzing British data for the years 1954 “ 1955, 1959“ 1960 and 1969 “1970, he

247

7.4 BENKTANDER DISTRIBUTIONS

contrasted it with Champernowne™s methods and found his estimator to be more

reliable.

When ¬tted to the distribution of 1969 ¬scal incomes in three regions of the

Netherlands, it outperforms gamma, generalized gamma Weibull, and log-logistic

distributions (Bartels, 1977). A Box“ Cox-transformed Champernowne distribution

does even better on these data.

Kloek and van Dijk (1977) analyzed the distribution of Australian family incomes

during 1966 “1968. Using minimum x2 as well as maximum likelihood estimates,

they noted that, compared to the lognormal, gamma, log Pearson type IV, and log-t

distributions, the Champernowne function “. . .¬ts rather well, but . . .the parameters

cannot easily be interpreted” (p. 446). Employing Cox tests (Cox, 1961), they found

that “. . . neither the log-t [when] compared with the Champernowne nor the log

Pearson IV compared with the Champernowne can be rejected, and vice versa.” In

other words, the distributions are statistically on the same footing.

Kloek and van Dijk (1978) utilized the four-parameter Champernowne

distribution when analyzing Dutch earnings data for 1973. Estimating parameters

by the minimum x2 method, they found that the distribution outperforms the two-

parameter lognormal and two-parameter gamma distributions and is comparable to

the three-parameter generalized gamma and the three-parameter log-t distributions.

Fattorini and Lemmi (1979) ¬t the three-parameter Champernowne distribution to

Italian, U.S., and Swedish data for the years 1967“ 1976, 1954 “ 1957, 1960 “1962,

and 1965 “1966, respectively. They found that it almost always outperforms the

lognormal distribution, but their kappa 3 (i.e., Dagum type I) distribution is even

better, in terms of the sum of squared errors and the Kolmogorov distance.

Interestingly enough, their estimates of l turn out to be numerically greater than 1 for