Figure 3 “Log-adjusted” Pareto densities: x0 ¼ 2, a ¼ 1:5, b ¼ À1:03, À1(:2)0 (from bottom left).

(Here b ¼ 1:03 represents the boundary of the admissible parameter space.)

for k , a, and

log x0

E(X k ) ¼ xk þ kxk Á ,

0 0

bÀ1

for k ¼ a and b . 1; otherwise, they do not exist. Equation (3.121) illustrates the

effect of the new parameter b: The ¬rst term is simply the kth moment of a classical

Pareto distribution, the second terms describes the adjustment due to b.

The Lorenz curve is available via the ¬rst-moment distribution and the Gini

coef¬cient via the representation (2.22) in terms of expectations of order statistics,

but the resulting expressions are again somewhat involved.

The “log-Pareto” distribution is quite similar to the log-gamma distribution

& 'bÀ1

ab xa ÀaÀ1 x

0

, 0 , x0

f (x) ¼ x x, (3:122)

log

G(b) x0

where b ! 1, a . 1, a model to be discussed in Section 5.4. It appears that the latter

distribution is perhaps more tractable, especially since there is a well-developed

theory of estimation for the gamma distribution.

101

3.11 STABLE DISTRIBUTIONS

3.11 STABLE DISTRIBUTIONS

In a series of pioneering contributions around 1960, Mandelbrot (1959, 1960, 1961)

´

argued that incomes follow what he calls a Pareto “Levy distribution, that is, a

maximally skewed stable distribution with a characteristic exponent a between 1 and

2. Here stability means that if X1 and X2 are independent copies of X and bi, i ¼ 1, 2,

are positive constants, then

d

b1 X1 þ b2 X2 ¼ bX þ a

for some positive b and real a. In other words, the shape of X is preserved under

addition, up to location and scale.

There is by now a very substantial literature on stable distributions, partially

furnished by successful applications, notably in ¬nance, and partially due to recent

advances in statistical computing. We refer the interested reader to the classical text

by Zolotarev (1986) for probabilistic properties of stable laws. Their main attraction

is that they represent the only limits of properly normalized sums of i.i.d. random

variables and therefore generalize the normal distribution. Indeed, nonnormal stable

laws arise in the (generalized) central limit problem for i.i.d. random variables if

these variables do not possess ¬nite variances. In that case, the limit distribution does

not have ¬nite variance either, its right tail being regularly varying and therefore of

the Pareto type (without following an exact Pareto distribution), thus justifying the

inclusion of stable distributions here.

Mandelbrot argued that the total income of any income recipient is obtained by

adding incomes from different sources (e.g., family income is obtained as the sum of

the incomes of all family members). If all these types of income follow the same

type of distribution and one is willing to make an independence assumption, one

expects that total income will be adequately approximated by a stable distribution.

Although that sounds very attractive, the drawback is that stable laws must

generally be de¬ned in terms of their characteristic functions, as only in three

exceptional cases”the normal and Cauchy distributions and a special case of the

inverse gamma distribution (see Section 5.4)”are the densities and/or distribution

functions are available in terms of elementary functions. The characteristic function

of the stable laws is

np o

a

itx

log f(t) ¼ log E(e ) ¼ imt À ljtj exp ig sign(t) : (3:123)

2

Here 0 , a , 1 or 1 , a 2. For a ¼ 1, comprising a small family of Cauchy-

type distributions, a slightly different form must be used. The parameter a is usually

referred to as the characteristic exponent or index of stability; in a certain sense, it is

the most important of the four parameters since it governs the tail behavior of the

distribution and therefore the existence of moments. The kth absolute moment EjX jk

exists iff k , a. The parameter m [ IR is a location parameter, l . 0 is a scale,

and g is a skewness parameter, with jgj 1 À j1 À aj. Clearly, a ¼ 2 yields the

normal distribution. [We warn our readers that there are a multitude of alternative

102 PARETO DISTRIBUTIONS

parameterizations of the stable laws! The form (3.123) of the characteristic function

is a variant of Zolotarev™s (1986, p. 12) parameterization (B).]

The connection with income distributions is as follows: If 1 , a 2, which is

broadly consistent with Pareto™s original a ¼ 1:5, we have jgj 2 À a. Thus, the

distribution is maximally skewed to the right if g ¼ 2 À a. Mandelbrot suggested

that this is the relevant case for applications to income data because otherwise

the probability of negative incomes might become too large. [Stable densities are

positive on the whole real line unless a , 1 and the distribution is maximally

positively (negatively) skewed, in which case the support is the positive (negative)

hal¬‚ine.] Thus, the maximally skewed stable laws de¬ne a three-parameter

subfamily of all stable distributions. In view of jg j 1 À j1 À aj for all stable

laws, the parameter g becomes less and less meaningful (as well as harder to

estimate) as a approaches 2. This means that in the maximally skewed stable case a

must stay well below 2 in order to retain a reasonably skewed distribution.

It took almost 20 years after Mandelbrot™s discovery for an attempt to be made to

¬t a maximally skewed stable distribution to income data. Van Dijk and Kloek

(1978, 1980) addressed the problem of estimation from grouped data. They

employed multinomial maximum likelihood and minimum x2 estimates that are

based on numerical inversion of the characteristic function, followed by numerical

integration of the resulting densities. (This was a somewhat burdensome com-

putational procedure in 1980.) Van Dijk and Kloek™s estimates of a, for Australian

family disposable income in 1966 “1968 and for the Dutch gross incomes in 1973,

ranged from 1.17 to 1.72.

These authors also considered log-stable distributions, assuming that not income

itself but rather its logarithm (income power) follows a stable distribution. This is

quite natural since the normal distribution is a particular stable distribution and its

offspring, the lognormal, is one of the classical size distributions. The appearance of

the stable distributions in connection with income data could therefore be justi¬ed

by appealing to a generalized form of Gibrat™s law of proportionate effect (leading to

the lognormal distribution; see Section 4.2). Although this approach is arguably best

discussed within the framework of the following chapter, dealing with lognormal

distributions, we mention it here since the estimation of stable and log-stable models

is completely analogous. In a log-stable framework it is no longer necessary to

con¬ne ourselves to the three-parameter subclass of Pareto-Levy distributions, and

´

consequently the unrestricted four-parameter family of stable distributions can be

used. Van Dijk and Kloek obtained estimates of a between 1.53 and 1.96 for the

logarithmic incomes.

Van Dijk and Klock preferred log-stable distributions over stable ones on the

grounds of some pooled tests. We are however somewhat skeptical as far as the

usefulness of the log-stable model is concerned. From Kleiber (2000b) the log-stable

densities lack moments of any order and moreover they exhibit a pole at the origin, a

feature that does not seem to be consistent with the data. (Of course, the latter feature

cannot be veri¬ed from grouped data.)

Van Dijk and Kloek reported that both the maximally skewed stable and log-

stable family perform generally better than the log-t and Champernowne

103

3.12 FURTHER PARETO-TYPE DISTRIBUTIONS

distributions. However, they also noted that “the data considered were not suf¬cient

to settle the dispute about the question what is the correct model to describe the

right-hand tail of an income distribution” (van Dijk and Kloek, 1978, p. 19). In fact,

no data however abundant can settle this basic dispute.

On the actuarial side, Holcomb (1973; see also Paulson, Holcomb, and Leitch, 1975,

p. 169) asserted that the claim experience for a line of nonlife insurance is a mixture of

independent random variables, from nonidentical distributions all lying within the

domain of attraction of a stable law with a support that is bounded from below (a , 1).