Holcomb considered three data sets. The ¬rst comprised data for the period

1965 “1970, detailing 2,326 national losses due to burglaries experienced by a U.S.

chain of retail stores. The second contained 2,483 records of vandalism for the same

chain and the same period, whereas the third comprised 1,142 records constituting a

different chain™s combined history of insured losses due to all property and liability

perils. Holcomb obtained estimates of 0.82, 0.88, and 0.67, respectively, for a.

A problem with this approach is that the employed distributions possess in¬nite

means that are dif¬cult to reconcile with common actuarial premium principles, most

of which require a ¬nite ¬rst moment. This may explain to some extent why

maximally skewed stable distributions have not been used much in actuarial

applications.

3.12 FURTHER PARETO-TYPE DISTRIBUTIONS

Krishnan, Ng, and Shihadeh (1990) proposed a generalized Pareto distribution by

introducing a more ¬‚exible (polynomial) form for the elasticity

Xk

@ log{1 À F(x)}

b i xi : (3:124)

¼ Àa À

@ log x i¼1

This approach yields c.d.f.™s of the type

( )

X

k

bi x i ,

1 À F(x) / xÀa exp À (3:125)

i¼1

called polynomial Pareto curves by Krishnan, Ng, and Shihadeh and comprising the

Pareto type I (for b1 ¼ Á Á Á ¼ bk ¼ 0) distribution as a special case. When we apply

their linear speci¬cation (b2 ¼ Á Á Á ¼ bk ¼ 0), which is very close to Pareto™s third

proposal (3.7), to two data sets, it turns out that the estimates of the new parameter

b1 are rather small, con¬rming Pareto™s work.

Perhaps somewhat unexpectedly, the size distribution of prizes in many popular

lotteries and prize competitions is also intimately related to the Pareto distribution.

Observing that the prizes in German lotteries are often characterized by the fact that

the number of prizes ni of a certain class i is inversely proportional to their values xi ,

104 PARETO DISTRIBUTIONS

Bomsdorf (1977) was led to a study of the discrete distributions with probability

mass function

p

i ¼ 1, . . . , k,

f (x) ¼ , xi ¼ di Á a,

di

P

where a . 0, d1 ¼ 1, di , diþ1 , p ¼ 1= k diÀ1 , which he called the prize-

i¼1

competition distribution. A continuous analog is clearly given by the density

c

a2 :

f (x) ¼ , a1 x (3:126)

x

If we rewrite the supporting interval in the form [an , anþ1 ], where n ¼ loga a1 [ IR

and a ¼ a2 =a1, the normalizing constant is found to be c ¼ loga e. Thus for the case

where a ¼ e, matters simplify to the hyperbolic function

1

en enþ1 ,

f (x) ¼ , x (3:127)

x

with the c.d.f.

en enþ1 ,

F(x) ¼ log x À n, x

an expression that is seen to describe a doubly truncated Pareto-type distribution

corresponding to a ¼ 0 in (3.2). [Note that in (3.2) a ¼ 0 is inadmissible because on

an unbounded support the resulting function would not be a distribution function.]

The moment generating function of the prize-competition distribution (3.127) is

given by the formula

X einþi X ein

1 1

i

ti

m(t) ¼ 1 þ t ,

À

i Á i! i¼1 i Á i!

i¼1

from which the ¬rst moment and the variance are found to be

E(X ) ¼ enþ1 À en

and

e2nþ2 3e2n

var(X ) ¼ 2 Á e2nþ1 À :

À

2 2

The Bomsdorf (1977) distribution (the continuous version) has been extended to

the distribution with c.d.f.

( log x)b

e1=b ,

F(x) ¼ k Á x , b = À1, 1 x

bþ1

105

3.12 FURTHER PARETO-TYPE DISTRIBUTIONS

by Stoppa (1993) and constitutes family 6 in the ¬rst period of his extensive

table of distributions based on a differential equation for the elasticity described in

Section 2.3. This is an L-shaped distribution with the mode at zero for b . 1 and at

e(1Àb)=b for b 1.

A so-called generalized Pareto distribution with c.d.f.

8

< 1 À 1 À cx 1=c , if c = 0,

b

Fc (x) ¼ (3:128)

: Àx

1Àe , if c ¼ 0,

where

x!0 if c ! 0,

1

if c , 0,

0 x À

c

is of great importance in the analysis of extremal events (see, e.g., Embrechts,

Kluppelberg, and Mikosch, 1997, or Kotz and Nadarajah, 2000). In the context

¨

of income distributions, this model was ¬tted to the 1969 personal incomes in 157

counties of Texas, and to lifetime tournament earnings of 50 professional golfers

through 1980 by Dargahi-Noubary (1994). [The data are given in Arnold (1983),

Appendix B.]

Colombi (1990) proposed a “Pareto-lognormal” distribution that is de¬ned as the

distribution of the product X :¼ Y Á Z of two independent random variables, one

following a Par(1, a) distribution and the other following a two-parameter lognormal

distribution (to be described in detail in the next chapter). The p.d.f. of the product is

given by

& '

us 2

u