xg ¼ x0 exp , (3:32)

a

and the harmonic mean xh ¼ {E(X À1 )}À1 , which equals

xh ¼ x0 (1 þ aÀ1 ): (3:33)

A comparison with (3.3) shows that the geometric mean is equal to the (1 À eÀ1 )th

quantile. Also, formulas (3.27), (3.32), and (3.33) provide an illustration of the

familiar inequalities E(X ) ! xg ! xh .

It is easy to see that for a1 a2 and x01 ! x02, the c.d.f.s™ of two Par(x0i , ai )

distributions, i ¼ 1, 2, do not intersect, so under these conditions Pareto distributions

are stochastically ordered, with X1 !FSD X2 . This implies, among other things, that the

moments are also ordered (provided they exist), in particular E(X1 ) ! E(X2 ).

The density is decreasing; thus, the mode of this distribution is at x0 . Figure 3.1

provides some examples of Pareto densities.

From (3.3), the median is F À1 (0:5) ¼ 21=a x0 .

72 PARETO DISTRIBUTIONS

Figure 1 Pareto densities: x0 ¼ 1 and a ¼ 1(1)5 (from bottom left).

The Pareto distribution is closed with respect to minimization, in the sense that

X $ Par(x0 , a) ¼ X1:n $ Par(x0 , na): (3:34)

)

However, other order statistics do not possess Pareto distributions. In particular,

the distribution of the sample maximum is given by

Àa !n

x

, 0 , x0

Fn:n (x) ¼ 1 À x: (3:35)

x0

Fairly recently, Stoppa (1990a,b) proposed this distribution, for a general n [ IR,

as a model for the size distribution of personal income. It will be discussed in

Section 3.8 below in some detail.

The Pareto distribution is also closed with respect to the formation of moment

distributions and with respect to power transformations: if

F $ Par(x0 , a) ¼ F(k) $ Par(x0 , a À k), (3:36)

)

provided a , k, and, for a . 0,

aa

a

X $ Par(x0 , a) ¼ X $ Par x0 , : (3:37)

)

a

73

3.3 MOMENTS AND OTHER BASIC PROPERTIES

Also, if X follows a standard Pareto distribution, then W ¼ X À1 has the density

f (w) ¼ axa waÀ1 , 0 , w , xÀ1 : (3:38)

0 0

This is a power function distribution and thus a special case of the Pearson type I

distribution that will be encountered in Chapter 6.

The hazard rate is given by

a

, x . x0 ,

r(x) ¼ (3:39)

x

which is monotonically decreasing. Its slow decrease re¬‚ects the heavy-tailed nature

of the Pareto distribution. The mean excess function is

x

, x ! x0 :

e(x) ¼ (3:40)

aÀ1

Thus, the Pareto distribution obeys van der Wijk™s law (1.13). (Since the mean

excess function characterizes a distribution, the Pareto distribution is, in fact,

characterized by van der Wijk™s law within the class of continuous distributions.)

From (3.39) and (3.40) it follows that the product of the hazard rate and the mean

excess function is constant

a

:

r(x) Á e(x) ¼ (3:41)

aÀ1

The distribution theory associated with samples from Pareto distributions is

generally somewhat complicated. From general results in, for example, Feller (1971,

p. 279), it follows that convolutions of Pareto distributions also exhibit Paretian tail

behavior; however, expressions for the resulting distributions are quite involved.

Nonetheless, asymptotically the situation is under control, an important probabilistic

property of the Pareto distribution being associated with the central limit theorem:

properly normalized partial sums of i.i.d. Par(x0 , a) random variables are asymp-

totically normally distributed only if a ! 2, for a , 2 nonnormal stable

distributions arise (see, e.g., Zolotarev, 1986). The Pareto distribution is perhaps

the simplest distribution with this property.

A basic distributional property of the standard Par(x0 , a) distribution is its close

relationship with the exponential distribution

Y

d

X ¼ x0 exp , (3:42)

a

where Y is a standard exponential variable, that is, fY ( y) ¼ exp (Ày), y . 0. The

classical Pareto distribution may therefore be viewed as the “log-exponential”

distribution. This yields, for example, the following relationship between Pareto and

74 PARETO DISTRIBUTIONS

gamma distributions: If Xi , i ¼ 1, . . . , n, are i.i.d. with Xi $ Par(x0 , a), then

YXi

X

n n

Xi

a log ¼a ¼ Ga(n, 1): (3:43)

log

x0 x0

i¼1 i¼1

More general results on distributions of products and ratios of Pareto variables with

possibly different parameters were derived by Pederzoli and Rathie (1980) using

Mellin transforms.

In general, the intimate relationship between the Pareto and exponential

distributions implies that one can obtain many properties of the former from

properties of the latter, in particular characterizations to which we now turn.

3.4 CHARACTERIZATIONS

A large number of characterizations of Pareto distributions are based on the behavior

of functions of order statistics, which is a consequence of the vast literature on

characterizations of the exponential distribution in terms of these functions. Since we

are unaware of meaningful interpretations in connection with size phenomena, we

only present the most prominent example to illustrate the ¬‚avor of the results.

The example under consideration is

Xiþ1:n

Xi:n , independent ¼ X $ Pareto: (3:44)