0

P

where z(n) ¼ 1 jÀn , n . 1, is the Riemann zeta function. If we add scale and

j¼1

location parameters, the Davis density becomes

bn 1

(x À x0 )ÀnÀ1 0 , x0

f (x) ¼ , x, (7:10)

exp{b(x À x0 )À1 } À 1

G(n)z(n)

where x0 , b . 0 and n . 1. Utilizing the well-known series expansion

X Bn

1

x

xn ,

¼

xÀ1

e n!

n¼0

where Bn are the Bernoulli numbers, it is not dif¬cult to see that the parameter n is

related to Pareto™s a as n ¼ a þ 1.

240 MISCELLANEOUS SIZE DISTRIBUTIONS

For x close to x0 , the density is approximately of the form

f (x) ™ C Á (x À x0 )ÀnÀ1 exp{Àb(x À x0 )À1 }, 0 , x0 x,

and therefore resembles the Vinci (inverse gamma) distribution of Section 5.2.

From (7.9) we directly obtain the moments

bk G(n À k)z(n À k)

k

E[(X À x0 ) ] ¼ , (7:11)

G(n)z(n)

provided k þ 1 , n.

It is quite remarkable that the Davis distribution”not easily related to other

continuous univariate distributions”is a member of D™Addario™s (1949) generating

system of income distributions; cf. Section 2.4. [See also Dagum (1990a, 1996).]

Davis (1941a,b) ¬t his model to the distribution of income among personal-

income recipients in the United States in 1918 and obtained a value of n in the

vicinity of 2.7. His method of estimation is a two-stage procedure: After estimating

^ ^

Pareto™s parameter a by least-squares in the Pareto diagram, yielding n ¼ a þ 1, he

essentially determined b from the estimating equation (7.11) for k ¼ 1.

The distribution was later used by Champernowne (1952), who reconsidered

Davis™s data but found the model not to ¬t as well as his own three-parameter

distribution. We shall next study the Champernowne distribution.

7.3 CHAMPERNOWNE DISTRIBUTION

Champernowne (1937, 1952) considered the distribution of log-income Y ¼ log X ,

also termed “income power,” as the starting point and assumed that it has a density

function of the form

n

À 1 , y , 1,

f ( y) ¼ , (7:12)

cosh[a( y À y0 )] þ l

where a, l, y0 , n are positive parameters, n being the normalizing constant and

hence a function of the others. (The reader will hopefully consult the biography of

Champernowne presented in the appendix to learn more about this colorful person.)

The function given by (7.12) de¬nes a symmetrical distribution, with median y0 ,

whose tails are somewhat heavier than those of the normal distribution. It is

included in a large family of generalized logistic distributions due to Perks (1932),

a British actuary. In view of the relation cosh y ¼ (e y þ eÀy )=2, (7.12) may be

rewritten as

n

:

f ( y) ¼

1=2ea( yÀy0 ) þ l þ 1=2eÀa( yÀy0 )

241

7.3 CHAMPERNOWNE DISTRIBUTION

This explains why (7.12) de¬nes a generalization of the logistic distribution. For

y0 ¼ 0, a ¼ 1, n ¼ 1=2, and l ¼ 1 we obtain

ey

1

f ( y) ¼ ,

¼

e y þ 2 þ eÀy (1 þ e y )2

the density of the standard logistic distribution, in view of sech y ¼ (cosh y)À1 also

known as the sech square distribution.

If we set log x0 ¼ y0 , the density function of the income X ¼ exp Y is given by

n

, x . 0:

f (x) ¼ (7:13)

þ l þ 1=2(x=x0 )a ]

Àa

x[1=2(x=x0 )

By construction, x0 is the median value of income. The form of the c.d.f. depends on

the value of l. There are three variants: jlj , 1, l ¼ 1, and l . 1, which are

discussed in Champernowne™s publications at some length.

The simplest case occurs for l ¼ 1, only brie¬‚y mentioned by Champernowne

(1952). However, it was discussed in greater detail by Fisk (1961a,b) and is therefore

often referred to as the Fisk distribution (see Section 6.4). Its density is

a xaÀ1

a

, x . 0:

f (x) ¼ (7:14)

¼

2x{cosh[alog(x=x0 )] þ 1} xa [1 þ (x=x0 )a ]2

0

Here x0 is a scale parameter and a . 0 is a shape parameter. The parameter a is the

celebrated Pareto™s alpha. This special case of the Champernowne distribution is also

a special case of the Dagum type I distribution (for p ¼ 1) and of the Singh “

Maddala distribution (for q ¼ 1). Consequently, all the distributional properties of

this model were presented in Chapter 6. (In the notation of Chapter 6, a equals a and

x0 equals b.) The c.d.f. of (7.14) is

& Àa 'À1

1 x

x . 0:

F(x) ¼ 1 À a ¼ 1þ , (7:15)

1 þ (x=x0 ) x0

For jlj , 1, the Champernowne density can be written as

asin u

, x . 0,

f (x) ¼ (7:16)

2ux{cosh[alog(x=x0 )] þ cos u}

or alternatively,

asin u

f (x) ¼ , (7:17)

þ 2cos u þ (x=x0 )a ]

Àa

ux[(x=x0 )

242 MISCELLANEOUS SIZE DISTRIBUTIONS