where cos u ¼ l and Àp , u , p. (Actually, one must con¬ne oneself to 0 , u , p

because otherwise the model may not be identi¬able.) The latter expression can be

rewritten as

asin u(x=x0 )aÀ1

f (x) ¼

ux0 [1 þ 2cos u(x=x0 )a þ (x=x0 )2a ]

asin u(x=x0 )aÀ1

,

¼

ux0{[cos u þ (x=x0 )a ]2 þ sin2 u}

which integrates to the c.d.f.

!

sin u

1

x . 0:

F(x) ¼ 1 À arctan , (7:18)

cos u þ (x=x0 )a

u

Parameters a and x0 play the same role as in the case where l ¼ 1. Unfortunately,

the new parameter u evades simple interpretation; Champernowne (1952) noted that

it may be regarded as a parameter for adjusting the kurtosis of the distribution of log

income. Harrison (1974) observed that, for u ! p, the distribution approaches a

point mass concentrated at x0 . For u ! 0, the distribution becomes the one with

l ¼ 1, that is, the Fisk distribution. The role of the parameter u is illustrated in

Figure 7.2.

Figure 7.2 Champernowne densities: a ¼ 1:5, x0 ¼ 1, u ¼ d Á p, and d ¼ 0:1(0:1)0:9 (from left to right).

243

7.3 CHAMPERNOWNE DISTRIBUTION

A stochastic model leading to the Champernowne distribution as the equilibrium

distribution was brie¬‚y discussed by Ord (1975).

Equation (7.18) implies that the quantile function is available in closed form; it is

& '1=a

sin u

F À1 (u) ¼ x0 À cos u , 0 , u , 1: (7:19)

tan[u(1 À u)]

As always, this is an attractive feature for simulation purposes.

The mode of this form of the Champernowne distribution occurs at

(p¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬ )1=a

2 À sin2 u À cos u

a

:

xmode ¼ x0 (7:20)

aþ1

Finally, for l . 1 the density becomes

asinh h

, x . 0,

f (x) ¼ (7:21)

a

hx[(x=x0 ) þ 2cosh h þ (x=x0 )Àa ]

and the corresponding c.d.f. is

a

x þ eh xa

1 0

F(x) ¼ 1 À , (7:22)

log a Àh xa

2h x þe 0

where cosh h ¼ l. The mode of this form of the Champernowne distribution

occurs at

(p¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬ )1=a

a2 À sinh2 h À cosh h

:

xmode ¼ x0 (7:23)

aþ1

In view of the earlier remarks, all three forms of the Champernowne distribu-

tion may be regarded as “generalized log-logistic distributions.” However,

Champernowne (1952) reported that the majority of income distributions which

he graduated throughout his investigations give values of l numerically less than

1. Most authors (e.g., Harrison, 1974; Kloek and van Dijk, 1977, 1978; Campano,

1987) who have used the Champernowne distribution have con¬ned themselves to

the case jlj , 1. We shall follow their convention below. (The case l ¼ 1, of course,

is dealt with in Chapter 6.)

During the course of preparing the section on the Champernowne distribution, the

authors”to their delight”were able to discover in the widely scattered literature a

number of somewhat unexpected sources in which the distribution (7.16) is

discussed, albeit under different names. (We are not sure whether Champernowne”

who continued to develop and modify his original distributions for some 35 years”

was aware of these new forms and descriptions of his model.)

244 MISCELLANEOUS SIZE DISTRIBUTIONS

This fact (similar to the situation with the Benini distribution; see Section 7.1) is

another indication of the wide diversity of sources for derivations and uses of

statistical distributions, especially in the last 50 years. [The Champernowne

distribution is possibly not the mainstream distribution utilized in standard income

studies; it was revisited after some 30 years in a paper by Campano (1987), who

found it to be appropriate to more recent income data.]

Our observations pertain to the appearance of the distribution in the applied and

theoretical probabilistic literature and to its strong relation to stable distributions of

various kinds, which are of increasing importance in ¬nancial applications.

1. The three-parameter Champernowne distribution (7.18) with a ¼ 1 is nothing

but a truncated Cauchy distribution, that is, the distribution with the density

1

À 1 , x , 1,

f (x) ¼ , (7:24)

ps[1 þ {(x À q)=s}2 ]

truncated from below at zero [for the Champernowne distribution (7.18),

q ¼ Àcos u and s ¼ sin u]. This fact was overlooked even in the

comprehensive two-volume compendium Continuous Univariate Distri-

butions by Johnson, Kotz, and Balakrishnan (1994, 1995), who discussed

these distributions in different volumes without connecting them.

2. More unexpectedly, another special case of the Champernowne distribution,

namely, the case where a [ (0, 2] and u ¼ pa=2, is discussed in Section 3.3

of the classical book on stable distributions by Zolotarev (1986). There it

occurs in connection with the distribution of functions (ratios) of independent

strictly stable variables with the same characteristic exponent a. Zolotarev also

provided an alternative integral representation of this special case of the

Champernowne c.d.f.

°1

1 þ jx þ yja dy

1 1

:

Fa (x) ¼ þ (7:25)

log

1 þ jx À yja y

2 ap2 0

He further pointed out that Fa (Á) is a distribution function even for a . 2 and

moreover, for any integer n ! 1, F2n (Á) is a mixture of Cauchy distributions

with a linearly transformed argument.

3. The Champernowne distribution is also associated with the so-called Linnik

(1953) distribution (a generalization of the Laplace distribution) that possesses

a simple characteristic function