À log a

log-Gompertz À log F (0, 0) 0

x,1

a

Fisk 1ÀF (0, 0) 0

1 À (1 À F)b

x,1

a

Singh “ Maddala (0, þ) 0

F(1 À F)À1

1 À F 1=b x,1

a

Dagum (I) (0, þ) 0

F À a 1=b

x,1

a

Dagum (II) 1À (þ, þ) 0

1Àa

F À a 1=b

x,1

a 0 , x0

Dagum (III) 1À (À, þ)

1Àa

and q an income distribution is obtained. Table 2.6 provides a selection of models

that can be deduced from Dagum™s system.

Among those models, the Dagum (II) distribution is mainly used as a model of

wealth distribution.

Since the Fisk distribution is also known as the log-logistic distribution (see

Chapter 6) and the Burr III and Burr XII distributions are generalizations of

this distribution, Dagum (1983) referred to his system as the generalized

logistic-Burr system. Needless to say, this collection of distributions can be enlarged

further by introducing location and scale parameters or using transformation

functions.

CHAPTER THREE

Pareto Distributions

The Pareto distribution is the prototypical size distribution. In view of the

unprecedented information explosion on this distribution during the last two

decades, it would be very easy to write a four-volume compendium devoted to this

magical model rehashing the wealth of material available in the periodical and

monographic literature. This distribution”attributed to Vilfredo Pareto (1895)”in

analogy with the Lorenz curve is the pillar of statistical income distributions.

However, due to space limitations we can in this volume only provide a brief but

hopefully succinct account, and we shall concentrate on economic and actuarial

applications. For the literature up to the early 1980s, we refer the interested reader to

the excellent text by Arnold (1983). We shall emphasize contributions from the last 20

years. These include, among others, the unbiased estimation of various Pareto

characteristics. We shall also discuss numerous recent generalizations of the Pareto

distribution in which Stoppa™s contributions play a prominent role (not suf¬ciently

well represented in the English language literature).

3.1 DEFINITION

The classical Pareto distribution is de¬ned in terms of its c.d.f.

Àa

x

, x ! x0 . 0,

F(x) ¼ 1 À (3:1)

x0

where a . 0 is a shape parameter (also measuring the heaviness of the right tail) and

x0 is a scale. The density is

axa0

, x ! x0 . 0,

f (x) ¼ (3:2)

xaþ1

59

60 PARETO DISTRIBUTIONS

and the quantile function equals

F À1 (u) ¼ x0 (1 À u)À1=a , 0 , u , 1: (3:3)

We shall use the standard notation X $ Par(x0 , a).

In his pioneering contributions at the end of the nineteenth century, Pareto (1895,

1896, 1897a) suggested three variants of his distribution. The ¬rst variant is the classical

Pareto distribution as de¬ned in (3.1). Pareto™s second model possesses the c.d.f.

x À m Àa

x ! m,

F(x) ¼ 1 À 1 þ , (3:4)

x0

and is occasionally called the three-parameter Pareto distribution. The special case

where m ¼ 0,

x Àa

F(x) ¼ 1 À 1 þ , x ! 0, (3:5)

x0

where x0 , a . 0, is often referred to as the Pareto type II distribution. This distribution

was rediscovered by Lomax (1954) some 50 years later in a different context. In our

classi¬cation, the Pareto type II distribution falls under “beta-type distributions”;

therefore, it will not be discussed in the present chapter but rather in Chapter 6 below,

although in more general form. It can be considered a special case of the Singh“

Maddala distribution (case a ¼ 1, in the notation of Chapter 6); there is also a simple

relation with the Pareto type I model, namely,

X $ Par(II)(x0 , a) ( X þ x0 $ Par(x0 , a): (3:6)

)

It is worth emphasizing that the term “Pareto distribution” is used in connection with

both the Pareto type I and Pareto type II versions. Rytgaard (1990) asserted that “Pareto

distribution” usually means Pareto type I in the European and Pareto type II in the

American literature, but we have not been able to verify this pattern from the references

available to us.

It should also be noted that the Pareto type II distribution belongs to the second

period of Stoppa™s (1993) classi¬cation involving two parameters (see Section 2.3).

The third distribution proposed by Pareto”the Pareto type III distribution

(Arnold, 1983, uses a different terminology!)”has the c.d.f.

CeÀbx

, x ! m,

F(x) ¼ 1 À (3:7)

(x À m)a

where m [ IR, b, a . 0, and C is a function of the three parameters. It arises from

the introduction of a linear term bx in the doubly logarithmic representation

log {1 À F(x)} ¼ log C À a log (x À m) À bx:

61

3.2 HISTORY AND GENESIS

The exponential term in (3.7) assures the ¬niteness of all moments. For income data

^

the values of b are usually very small”Pareto (1896) obtained a value of b ¼

0:0000274 for data on the Grand Duchy of Oldenburg in 1890”so the Pareto type III

distribution does not seem to be attractive in the sense of income distribution theory

and applications. On the other hand, Creedy (1977) pointed out that “ . . .the values of

b and [m] are not invariant with respect to the units of measurement, and b is likely to

be very small since the units of x are large and are elsewhere transformed by taking