Xi:n

This is essentially an “exponentiated” version of a classical exponential charac-

terization in terms of spacings due to Fisz (1958). (The quantities Xiþ1:n =Xi:n are

often referred to as geometric spacings.) Re¬nements and related results may be

found in Galambos and Kotz (1978) and Arnold (1983).

Several characterizations of the Pareto distributions are based on the linearity of

the mean excess (or mean residual life) function, although sometimes in heavily

disguised form. This essentially goes back to Hagstrm (1925) and D™Addario

(1939), however rigorous proofs are of a more recent date (Arnold, 1971; Huang,

1974).

A closely related class of characterizations can be subsumed under the heading

“truncation equivalent to rescaling,” a line of research initiated by Bhattacharya

(1963). The basic result is that

z

P(X . y j X . z) ¼ P X . y , for all y . z ! x0 , (3:45)

x0

characterizes the Par(x0 , a) distribution. If one supposes that moments exist, this

characterization is reduced to the linearity of the mean excess function as a function

of z (e.g., Kotz and Shanbhag, 1980). Note that income distributions arising from

75

3.4 CHARACTERIZATIONS

income tax statistics are generally of the truncated form, and Lorenz curves and

inequality measures are quite often calculated for such data. The preceding

characterization implies that only if the entire distribution is of the Pareto form, the

truncated distribution can be safely used for inferences about the inequality

associated with the entire distribution. Otherwise, reported inequality statistics will

often be too low. See Ord, Patil, and Taillie (1983) for an additional discussion of the

effect. Further variations on this theme are the truncation invariance of the Gini and

generalized entropy indices (Ord, Patil, and Taillie, 1983) and the truncation

invariance, both from above and from below, of the Lorenz curve of a two-

component Pareto-power function mixture (Moothathu, 1993). (Recall that the

power function distribution is the inverse Pareto distribution.)

Several characterizations of the Pareto distribution have been cast in the

framework of income underreporting, although it is dif¬cult to believe that the

proposed mechanisms describe the actual underreporting process.

Krishnaji (1970) provided widely quoted characterization assuming that reported

income Y is related to true (but unobservable) income X through a multiplicative

error

d

Y ¼ RX ,

where R, X are independent. In order to be meaningful as an underreporting factor, it

is clearly necessary that R take values in [0, 1]. (In an actuarial context, the

appropriate framework would be the overreporting of insurance claims. If we set

d

Y ¼ X =R, the characterizations given below apply with minor modi¬cations.)

Assuming that R possesses a power function distribution (a special case of the beta

distribution) with the density

f (r) ¼ pr pÀ1 , 0 r 1, (3:46)

where p . 0, Krishnaji obtained the following characterization: If P(X . x0 ) ¼ 1,

for some x0 . 0, and P(RX . x0 ) . 0 with R, X independent, then

P(RX . y j RX . x0 ) ¼ P(X . y), y . x0 , (3:47)

if and only if X $ Par(x0 , a). However, there is nothing special about the distribution

(3.46) of R. Fosam and Sapatinas (1995) have shown that the result holds true for

any R supported on (a subset of) (0, 1) such that the distribution of log R is non-

arithmetic. Indeed, (3.47) may be rewritten as

& ' & '

X

X X y

. yÃ R Ã

yÃ ¼

x .1 ¼P x .y , . 1,

PR

x0 x0

0 0

76 PARETO DISTRIBUTIONS

which is seen to be equivalent to

& ' & '

X X X

. À log R þ x log . À log R ¼ P log .x ,

P log

x0 x0 x0

x ¼ log yÃ . 0:

The conclusion follows therefore directly from the (strong) lack of memory

property of the exponential distribution; hence, this characterization is a

consequence of (3.42).

The result remains unaffected if condition (3.47) is replaced by the condition

E(Y À x j Y . x) ¼ E(X À x j X . x), x . x0 . 0, (3:48)

with E(X þ ) , 1, as follows from Kotz and Shanbhag (1980).

An interesting alternative characterization in the framework described above is as

follows. If there exists a random variable Z such that the regression E(Z j X ¼ y) is

linear, then, under some smoothness conditions, E(Z j RX ¼ y) is also linear if and

only if X follows a Pareto distribution (Krishnaji, 1970). The result was extended by

Dimaki and Xekalaki (1990) and further generalized by Fosam and Sapatinas (1995)

who weakened (3.46) to R $ beta( p, q), q [ IN, requiring only E(Z j X ¼ y) ¼

d þ bxa , for some positive a.

Revankar, Hartley, and Pagano (1974) provided a second characterization in

terms of underreported incomes. However, in contrast to Krishnaji™s approach

utilizing a multiplicative reporting error, they postulated an additive relation between

true and reported income. Let the random variables X , Y , and U denote the actual

(unobserved) income, reported income, and the underreporting error, respectively,

and de¬ne

where 0 , U , max{0, X Àm},

Y ¼ X À U,

where m is the tax-exempt level. Under the assumptions that (1) the average amount

of underreported income from a given X ¼ x . m is proportional to x À m, that is,

E(U j X ¼ x) ¼ b(x À m) ¼ a þ bx, (3:49)

0 , b , 1 and a ¼ Àbm, and (2) E(X ) , 1, it follows that

E(U j X . y) ¼ a þ by, (3:50)

with b . b . 0, if and only if X follows a Pareto distribution with a ¬nite ¬rst

moment. In particular, for a ¼ a the Par(x0 , a) distribution is obtained. [For a , a

we obtain a Pareto (II) distribution, while b ¼ b yields a characterization of the

exponential distribution. The case b , b was investigated by Stoppa (1989).]

77

3.5 LORENZ CURVE AND INEQUALITY MEASURES

The classical Pareto distribution is also the maximum entropy density on [x0 , 1)

subject to the constraint of a ¬xed geometric mean (e.g., Naslund, 1977; Kapur,

¨

1989, p. 56). This may also be considered a restatement of the corresponding

exponential characterization: The exponential distribution is the maximum entropy

distribution on [0, 1) when the ¬rst moment is prescribed (Kapur, 1989, p. 56).

Thus, the Pareto result follows from (3.42) together with (3.32).

Nair and Hitha (1990) provided a characterization in terms of the “equilibrium

distribution” de¬ned via the p.d.f. fZ (x) ¼ 1=E(X ){1 À FX (x)}, a concept of special

signi¬cance in renewal theory. They showed that the condition E(X ) ¼ kE(Z), for

some k . 1, characterizes the Pareto distribution within a subclass of the one-

parameter exponential family speci¬ed by f (x) ¼ u(u)v(x) exp{Àu log x}, where u