f (x) ¼ uþ1 exp u m þ x . 0,

x 2

where u . 0, m [ IR, s . 0, and

log x À m À us 2

2

L(x; m þ us , s) ¼ F x . 0,

,

s

denotes the c.d.f. of the two-parameter lognormal distribution.

The kth moment of the Pareto-lognormal distribution exists for k , u and equals

u 2

ekmþ(km) =2 :

E(X k ) ¼

uÀk

Similarly to the Pareto distribution, the Pareto-lognormal family is closed with

respect to the formation of moment distributions.

106 PARETO DISTRIBUTIONS

Colombi showed that the distribution is unimodal and provided an implicit

expression for the mode. He also discussed suf¬cient conditions for Lorenz ordering

and derived the Gini coef¬cient, which is of the form

s 2us À s

2

eus(usÀs) F À p¬¬¬

G ¼ 1 À 2F À p¬¬¬ þ ,

2u À 1

2 2

an expression that is seen to be decreasing in s.

Moreover, he ¬t the distribution to Italian family incomes for 1984 and 1986.

For the 1984 data the model is outperformed by both the Dagum type I and

Singh “Maddala distribution in terms of likelihood and minimum chi-square, but for

the 1986 data the situation is reversed.

We should like to add that several further distributions included in this book,

notably those of Chapter 6, may also be considered “generalized Pareto

distributions,” in that they possess a polynomially decreasing upper tail but are

more ¬‚exible than the original Pareto model in the lower-income range.

CHAPTER FOUR

Lognormal Distributions

Naturally we cannot present all that is known about the lognormal distribution in a

chapter of moderate length within the scope of this book. For example, in

quantitative economics and ¬nance, the lognormal distribution is ubiquitous and it

arises, among other things, in connection with geometric Brownian motion, the

standard model for the price dynamics of securities in mathematical ¬nance.

At least two books have been devoted to the lognormal distribution: Aitchison and

Brown (1957) presented the early contributions with an emphasis on economic

applications, whereas the compendium edited by Crow and Shimizu (1988) contains

wider coverage. The lognormal distribution is also systematically covered in a 50-page

chapter in Johnson, Kotz, and Balakrishnan (1994). Below we shall concentrate on the

“size” aspects of the distribution and emphasize topics that were either omitted or only

brie¬‚y covered in these three sources. These include a generalized lognormal

distribution for which the literature available is mainly in Italian.

4.1 DEFINITION

The p.d.f. of the lognormal distribution is given by

& '

1 1 2

f (x) ¼ p¬¬¬¬¬¬ exp À 2 (log x À m) , x . 0: (4:1)

2s

x 2ps

Thus, the distribution arises as the distribution of X ¼ exp Y ; where Y $ N(m, s 2 ):

Hence, the c.d.f. is given by

log x À m

F(x) ¼ F , x . 0, (4:2)

s

where F denotes the c.d.f. of the standard normal distribution.

107

108 LOGNORMAL DISTRIBUTIONS

There is also a three-parameter (shifted) lognormal distribution, to be discussed in

Section 4.7 below.

4.2 HISTORY AND GENESIS

As far as economic size distributions are concerned, the pioneering study marking

initial use of the lognormal distribution was Gibrat™s thesis of 1931. (See Appendix

A for a short biography of Robert Gibrat.) Gibrat asserted that the income of an

individual (or the size of a ¬rm) may be considered the joint effect of a large number

of mutually independent causes that have worked during a long period of time. At a

certain time t it is assumed that the change in some variate X is a random proportion

of a function g(XtÀ1 ) of the value XtÀ1 already attained. Thus, the underlying law of

motion is

Xt À XtÀ1 ¼ et g(XtÀ1 ),

where the et ™s are mutually independent and also independent of XtÀ1 : In the special

case where g(X ) ; X ; the so-called law of proportionate effect results, meaning that

the change in the variate at any step of the process is a random proportion of the

previous value of the variate. Thus,

Xt À XtÀ1 ¼ et XtÀ1 ,

which may be rewritten as

Xt À XtÀ1

¼ et ,

XtÀ1

whereby through summation over t

X Xt À XtÀ1 X

n n

et :

¼

XtÀ1

t¼1 t¼1

Assuming that the effect at each step is small, we get

° Xn

X Xt À XtÀ1

n

dX

¼ log Xn À log X0 ,

%

XtÀ1 X

X0

t¼1

which yields

X

n

et :

log Xn ¼ log X0 þ