x!1

This is Mandelbrot™s weak Pareto law. [Further weak Pareto laws were introduced by

Kakwani (1980b) and Esteban (1986).] Condition (2.68) is closely related to regular

variation, although the two concepts are not equivalent, as claimed by Merkies and

Steyn (1993). The problem is that there are regularly varying functions such as

F (x) / xa log x (for large x) for which the slowly varying part log x is unbounded.

Here F is clearly regularly varying, but (2.68) does not hold. It would thus seem that

Mandelbrot™s weak Pareto law should perhaps be rephrased as requiring that the size

distribution be regularly varying in the upper tail.

A second issue concerning the meaning of a in the absence of an underlying

Pareto distribution is its relation to income inequality. For an exact Pareto

distribution, the situation is clear (see Section 3.5 below): A smaller a is associated

with greater inequality in the sense of the Lorenz ordering and several associated

inequality measures. What can be said if the income distribution is just of the Pareto

type? Authors such as Bowman (1945) argued that a steeper Pareto curve is

associated with a more equal distribution of income, but a formal proof of this fact

has been lacking until recently.

Kleiber (1999b, 2000a) showed how this can be proven using properties of

regularly varying functions within the framework of the Lorenz ordering. Brie¬‚y, the

argument runs as follows.

The property X1 !L X2 is equivalent to (Arnold, 1987)

°x °x

˜ ˜ for all x [ Rþ

F 1 (t) dt ! F 2 (t) dt (2:69)

0 0

48 GENERAL PRINCIPLES

and also to

°1 °1

˜ ˜ for all x [ Rþ :

F 1 (t) dt ! F 2 (t) dt (2:70)

x x

˜

Here F i denotes the c.d.f. of the mean-scaled random variable Xi =E(Xi ), i ¼ 1, 2. If

F i [ RV1 (Àai ), ai . 0, i ¼ 1, 2, it follows using property RV 3 under appropriate

regularity conditions that the integrated upper tails are in RV1 (Àai þ 1). Hence, we

obtain from (2.70)

Ð1

˜

x F 1 (t) dt

for all x [ Rþ , ( a1

g(x) :¼ Ð a2 :

! 1, )

1˜

F (t) dt

2

x

This shows that if size distributions with regularly varying tails are ordered in the

Lorenz sense, then the more unequal distribution necessarily exhibits heavier tails

(namely, a smaller a). An analogous argument yields that if Fi [ RV0 (Àbi ), bi . 0,

i ¼ 1, 2, then X1 !L X2 implies b1 b2 . Hence, there exists a similar condition for

the lower tails.

This argument provides a useful tool for deriving the necessary conditions for

Lorenz dominance in parametric families. Many distributions studied in this book

are regularly varying at in¬nity and/or the origin, and the index of regular variation

can usually directly be determined from the density or c.d.f. Consequently, necessary

conditions for the Lorenz ordering are often available in a simple manner.

As a by-product, it turns out that Pareto™s alpha can be considered an inequality

measure even in the absence of an underlying exact Pareto distribution, provided it is

interpreted as an index of regular variation. See Kleiber (1999b, 2000a) for further

details and implications in the context of income distributions.

The preceding discussion has shown how Pareto tail behavior can be formalized

using the concept of regular variation. Although many of the distributions studied in

detail in the following chapters are of this type, there are some that cannot be

discussed within this framework.

To conclude this section, we therefore introduce a somewhat broader

classi¬cation of size distributions according to tail behavior. The distributions we

shall encounter below comprise three types of models that we may call Pareto-type

distributions, lognormal-type distributions and gamma-type distributions, respec-

tively. A preliminary classi¬cation is given in Table 2.2 (for x ! 1).

Here we have distributions with polynomially decreasing tails (type I), expo-

nentially decreasing tails (type III) as well as an intermediate case (type II). These

three types can be modi¬ed to enhance ¬‚exibility in the left tail.

Type I. The basic form is clearly the Pareto (I) distribution that is zeromodal.

Unimodal generalizations are of the forms (1) f (x) / eÀ1=x xÀa , leading to

49

2.3 SYSTEMS OF DISTRIBUTIONS

Table 2.2 Three Types of Size Distributions

Type I Type II Type III

(Pareto type) (Lognormal type) (Gamma type)

a

f (x) $ xÀa ¼ eÀa log x f (x) $ eÀ( log x) f (x) $ eÀax

distributions of the inverse gamma (or Vinci) type that exhibit a light (non-

Paretian) left tails or (2) f (x) / xp (1 þ x)À(aþp) , leading to distributions of the

beta (II) type that exhibit heavy (Paretian) left tails and may therefore be

considered “double Pareto” distributions.

Type II. For a ¼ 2 this yields a distribution of lognormal type. For a general a

the densities will be unimodal due to the difference in behavior of log x for

0 , x , 1 and x ! 1.

Type III. The prototypical type III distribution is the exponential distribution

with density f (x) ¼ aeÀax that is zeromodal. A more ¬‚exible shape is obtained

upon introducing a polynomial term, leading to densities of type

f (x) / xp eÀax , the prime example being the gamma density. “Weibullized”

versions also fall under type III.

2.3 SYSTEMS OF DISTRIBUTIONS

The most widely known system of statistical distributions is the celebrated Pearson

system, derived by Karl Pearson in the 1890s in connection with his work on

evolution. It contains many of the best known continuous univariate distributions.

Indeed, we shall encounter several members of the Pearson system in the following

chapters, notably Chapters 5 and 6 that comprise models related to the gamma and

beta distributions.

The Pearson densities are de¬ned in terms of the differential equation

(x À a)f (x)

f 0 (x) ¼ , (2:71)

c0 þ c1 x þ c2 x2

where a, c0 , c1 , c2 are constants determining the particular type of solution. (The

equation originally arose from a corresponding difference equation satis¬ed by the

hypergeometric distribution by means of a limiting argument.) The most prominent

solution of (2.71) is the normal p.d.f. that is obtained for c1 ¼ c2 ¼ 0. All solutions

are unimodal; however, the maxima may be located at the ends of the support.

There are three basic types of solutions of (2.71), referred to as types I, VI, and

IV, depending on the type of roots of the quadratic in the denominator (real and

opposite signs, real and same sign, and complex, respectively). Ten further types

arise as special cases; see Table 2.3. (For the Pearson type XII distributions, the

constants g and h are functions of the skewness and kurtosis coef¬cients.)

50 GENERAL PRINCIPLES

Table 2.3 The Pearson Distributions

Type Density Support

(1 þ x)m1 (1 À x)m2 21 x 1