equilibrium distributions and proposed a model in which the transition probabilities

or parameters of the distribution are allowed to change over time.

These models were often criticized by applied economists who favor models

based on human capital and the concept of economic man (Mincer, 1958; Becker,

1962, 1964). Some of them scorn size distribution of income and refer to them as

antitheories. Their criticism often goes like this:

Allowing a stochastic mechanism to be the sole determinant of the income distribution is

TO GIVE UP BEFORE YOU START. The deterministic part of a model (in econometrics)

is “what we think we know,” the disturbance term is “what we don™t know.” The

probabilistic approach allocates 100% variance in income to the latter.

In our opinion this type of argument shows a lack of understanding of the concept

of stochastic model and by extension of the probabilistic-statistical approach.

CHAPTER TWO

General Principles

Before embarking on a detailed discussion of the models for economic and actuarial

size phenomena, we will discuss a number of unifying themes along with several

tools that are required in the sequel. These include, among others, the ubiquitous

Lorenz curve and associated inequality measures. In addition, we present some

concepts usually associated with reliability and engineering statistics such as the

hazard rate and the mean residual life function that are known in actuarial science

under different names. Here these functions are often used for preliminary model

selection because they highlight the area of a distribution that is of central interest in

these applications, the extreme right tail.

We also brie¬‚y discuss systems of distributions in order to facilitate subsequent

classi¬cations, namely, the Pearson and Burr systems and the less widely known

Stoppa system. The largest branch of the size distributions literature, dealing with

the size distribution of personal income, has developed its own systems for the

generation of distributions; these we survey in Section 2.4.

Unless explicitly stated otherwise, we assume throughout this chapter that the

underlying c.d.f. F is continuous and supported on an interval.

2.1 SOME CONCEPTS FROM ECONOMICS

The literature on Lorenz curves, inequality measures, and related notions is by now

so substantial that it would be easy to write a 500-page volume dealing exclusively

with these concepts and their rami¬cations. We shall be rather brief and only present

the basic results. For re¬nements and further developments, we refer the interested

reader to Kakwani (1980b), Arnold (1987), Chakravarty (1990), Mosler (1994), or

Cowell (2000) and the references therein.

19

20 GENERAL PRINCIPLES

2.1.1 Lorenz Curves and the Lorenz Order

In June 1905 a paper entitled “Methods of measuring the concentration of wealth,”

written by Max Otto Lorenz (who was completing at that time his Ph.D. dissertation

at the University of Wisconsin and destined to become an important U.S.

Government statistician), appeared in the Journal of the American Statistical

Association.

It truly revolutionized the economic and statistical studies of income

distributions, and even today it generates a fertile ¬eld of investigation into the

bordering area between statistics and economics. The Current Index of Statistics (for

the year 1999) lists 13 papers with the titles Lorenz curve and Lorenz ordering. It

would not be an exaggeration to estimate that several hundred papers have been

written in the last 50 years in statistical journals and at least the same number in

econometric literature. It should be acknowledged that Lorenz™s pioneering work lay

somewhat dormant for a number of decades in the English statistical literature until it

was resurrected by Gastwirth in 1971.

To draw the Lorenz curve of an n-point empirical distribution, say, of household

income, one plots the share L(k=n) of total income received by the k=n Á 100% of the

lower-income households, k ¼ 0, 1, 2, . . . , n, and interpolates linearly.

In the discrete (or empirical) case, the Lorenz curve is thus de¬ned in terms of the

n þ 1 points

Pk

k xi:n

¼ Pi¼1 , k ¼ 0, 1, . . . , n,

L (2:1)

n

n i¼1 xi:n

where xi:n denotes the ith smallest income, and a continuous curve L(u), u [ [0, 1],

is given by

( )

X

bunc

1

L(u) ¼ xi:n þ (un À bunc)xbuncþ1:n , 0 u 1, °2:1aÞ

n

x i¼1

where bunc denotes the largest integer not exceeding un.

Figure 2.1 depicts the Lorenz curve for the (income) vector x ¼ (1, 3, 5, 11). By

de¬nition, the diagonal of the unit square corresponds to the Lorenz curve of a

society in which everybody receives the same income and thus serves as a

benchmark case against which actual income distributions may be measured.

The appropriate de¬nition of the Lorenz curve for a general distribution follows

easily by recognizing the expression (2.1) as a sequence of standardized empirical

Ð1

incomplete ¬rst moments. In view of E(X ) ¼ 0 F À1 (t) dt, where the quantile

function F À1 is de¬ned as

F À1 (t) ¼ sup{x j F(x) t [ [0, 1],

t}, (2:2)

21

2.1 SOME CONCEPTS FROM ECONOMICS

Figure 2.1 Lorenz curve of x ¼ (1, 3, 5, 11).

equation (2.1a) may be rewritten as

°u

1

F À1 (t) dt, u [ [0, 1]:

L(u) ¼ (2:3)

E(X ) 0

It follows that any distribution supported on the nonnegative hal¬‚ine with a ¬nite

and positive ¬rst moment admits a Lorenz curve. Following Arnold (1987), we shall

occasionally denote the set of all random variables with distributions satisfying

these conditions by L. Clearly, the empirical Lorenz curve can now be rewritten in

the form

°u

1 À1

u [ [0, 1],

Ln (u) ¼ Fn (t) dt, (2:4)

x 0

an expression that is useful for the derivation of the sampling properties of the

Lorenz curve.

22 GENERAL PRINCIPLES