exceptional.

More recently, Campano (1987), a statistician with the United Nations in Geneva,

reconsidered Champernowne™s own results for the four-parameter version (7.32) for

U.S. income data of 1947. He also compared Champernowne and Dagum type I

¬ttings for 1969 U.S. family incomes (previously investigated by Dagum, 1977) and

found the Champernowne distribution to ¬t slightly better. (However, this version of

the Champernowne distribution has one additional parameter compared to the

Dagum type I.)

7.4 BENKTANDER DISTRIBUTIONS

Starting from the observation that empirical mean excess functions point to

distributions that are intermediate between the Pareto and exponential distributions,

the Swedish actuary Gunnar Benktander (1970) discussed two new loss models.

Whereas for the exponential distribution the mean excess function is given by

e(x) ¼ l, x ! 0, (7:33)

248 MISCELLANEOUS SIZE DISTRIBUTIONS

l being the exponential scale parameter, in the Pareto case we have

x

, x ! x0 :

e(x) ¼ (7:34)

aÀ1

However, empirically one observes mean excess functions that are increasing but at a

decreasing rate. Two intermediate versions are therefore de¬ned by

x

, a . 0, b ! 0,

eI (x) ¼ (7:35)

a þ 2blog x

and

x1Àb

, a . 0, 0 , b

eII (x) ¼ 1: (7:36)

a

It follows from Watson and Wells (1961) that the ¬rst form is of the lognormal type,

whereas the second asymptotically resembles the mean excess function of a Weibull

distribution (Beirlant and Teugels, 1992).

Using the relation between the mean excess function and the c.d.f.,

& °x '

e(x0 ) 1

F(x) ¼ 1 À dt , x ! x0 ,

exp À

e(x) x0 e(t)

we obtain after some calculations

& '

2b

FI (x) ¼ 1 À xÀ(1þaþblog x) 1 þ log x , 1 x, (7:37)

a

where a . 0, b ! 0, and

& '

a axb

À(1Àb)

FII (x) ¼ 1 À x , 1 x, (7:38)

exp À

b b

where a . 0, 0 , b 1. Equation (7.37) de¬nes the Benktander type I and (7.38)

de¬nes the Benktander type II distribution. We note that although the distributions

are commonly known under these names (see Beirlant, Teugels, and Vynckier, 1996;

Embrechts, Kluppelberg, and Mikosch, 1997), they seem to have appeared for the

¨

¬rst time in Benktander and Segerdahl (1960).

The densities are given by

& '

2blog x 2b

fI (x) ¼ xÀ2ÀaÀblog x 1 þ (1 þ a þ 2blog x) À , 1 x, (7:39)

a a

249

7.4 BENKTANDER DISTRIBUTIONS

and

& '

a axb À2þb

{(1 À b) þ axb },

fII (x) ¼ exp À x 1 x, (7:40)

b b

respectively.

Using the relation between the mean excess function and the hazard rate,

1 þ e0 (x)

r(x) ¼ ,

e(x)

the hazard rates are easily obtained as

a þ 1 þ 2blog x 2b

rI (x) ¼ , 1 x, (7:41)

À

x x(a þ 2blog x)

and

a 1Àb

rII (x) ¼ , 1 x, (7:42)

þ

1Àb

x x

from which it is again obvious that the Pareto distribution is a limiting case (for

b ¼ 0 in both cases).

It follows directly from the expressions for the mean excess functions that the

means of both Benktander distributions are equal to

1

E(X ) ¼ 1 þ

a

and are independent of b.

Kluppelberg (1988) has shown that both distributions, as well as the associated

¨

integrated tail distributions

°x

1

Fint (x) ¼ F ( y) dy, x ! 0, (7:43)

E(X ) 0

belong to the class of subexponential distributions, that is, they satisfy the condition

Ã2

F (x)

¼ 2,

lim

x!1 F (x)

where F Ã2 stands for the convolution product F * F. This property is of importance

in risk theory (and other ¬elds) since it allows for a convenient treatment of

problems related to convolutions of heavy-tailed distributions. It is interesting that

the Benktander I survival function is proportional to the Benini density (see

250 MISCELLANEOUS SIZE DISTRIBUTIONS