It is perhaps worthwhile to examine brie¬‚y the actuarial motivation for (7.2).

Ramachandran (1969) and Shpilberg (1977) searched for a model for the probability

distribution of an individual ¬re loss amount and started from the hazard rate of ¬re

duration, assuming that, for a homogeneous group of risks, the ¬re loss increases

exponentially with the duration of the ¬re. They observed that an exponential

distribution of ¬re duration leads to a Pareto distribution as the distribution of the ¬re

loss amount (see Section 3.2) and claimed that a model which re¬‚ects a gradual

decrease in the probability of survival of the ¬re (as implied by an increasing failure

rate distribution) would be more plausible empirically. The simplest functional form

for the hazard rate with the required properties is the linear function

r(t) ¼ a þ bt, (a, b . 0), (7:4)

where t is the duration of the ¬re, resulting in a ¬re duration distribution with the

c.d.f.

12

F(t) ¼ 1 À exp Àat À bt :

2

This distribution was previously discussed by Flehinger and Lewis (1959) in a

reliability context. Under the assumption that x=x0 / ekt, k . 0, that is, the ratio of

237

7.1 BENINI DISTRIBUTION

the ¬re loss amount x to the minimum discernible loss x0 increases exponentially

with the ¬re duration, one directly obtains the Benini c.d.f. in the form (7.1).

The density of the three-parameter Benini distribution (7.2) is

( !2 )&

'

a 2b log (x=x0 )

x x

À b log

f (x) ¼ exp Àa log , x0 x, (7:5)

þ

x0 x0 x x

whereas in the two-parameter case (7.3) we get

( !2 )

x x

À1

f (x) ¼ 2bx Á exp Àb log , x0 x: (7:6)

Á log

x0 x0

Here x0 is a scale and a and b are shape parameters. For a ¼ 0, that is, Benini™s

original form, it moreover follows from (7.3) that the quantile function is available in

a closed form, namely,

s¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬

1

F À1 (u) ¼ x0 exp À log (1 À u), for 0 , u , 1: (7:7)

b

This is an attractive feature for simulation purposes.

Figure 7.1 depicts several two-parameter Benini densities.

Figure 7.1 Benini densities: x0 ¼ 1, a ¼ 0, and b ¼ 1(1)5 (from right to left).

238 MISCELLANEOUS SIZE DISTRIBUTIONS

For a ¼ 0 and x0 ¼ 1, the distribution can also be recognized as a log-Rayleigh

distribution [namely, if Y follows a Rayleigh distribution, with c.d.f. F( y) ¼

1 À exp(Ày2 ), y . 0, then X :¼ exp(Y ) has the distribution with c.d.f. (7.3),

where b ¼ 1 and x0 ¼ 1]. Hence, the log-Weibull family, with c.d.f. F(x) ¼

1 À exp{À( log x) p }, can be viewed as a natural generalization of the Benini

distribution. Indeed, Benini (1905, p. 231) discussed rather brie¬‚y this model and

reported that, for his data, when p ¼ 2:15, the ¬t is superior to that using model

(7.3).

The Benini distribution is a member of Dagum™s generating system of income

distributions (Dagum 1980a, 1983, 1990a); see Section 2.4.

An attractive feature of the three-parameter Benini distributions is that this family

contains the Pareto distribution as a special case (for b ¼ 0). This led DuMouchel

and Olshen (1975) to suggest a method of testing Pareto vs. Benini distributions.

Their test is based on the idea that, if X follows a Par(x0 , a) distribution, then

log(X =x0 ) is an exponentially distributed variable and hence the mean and standard

deviation of log(X =x0 ) coincide. Therefore, an appropriate test of H0 : b ¼ 0 vs.

H1 : b . 0 depends on the ratio of the sample standard deviation to the sample mean.

Their test statistic is

r¬¬¬

s2

n

1À 2 , (7:8)

8

y

where n is the sample size, y the sample mean, and s2 the sample variance of the

logarithms of the observations. This test is derived by using the Neyman™s C(a)

principle and is therefore locally asymptotically most powerful. Under H0, the test

statistic is asymptotically standard normal and the test rejects the null hypothesis for

large values of (7.8).

Since the Benini distribution is quite close to the lognormal distribution, the

DuMouchel-Olshen test may be viewed as a test of the Pareto distribution vs. the

lognormal, that is, it enables one to choose at least approximately between the two

classical size distributions.

Benini (1905, 1906) estimated the parameters of his distribution via regression

in the Pareto diagram using Cauchy™s method that is seldom used nowadays

(see, e.g., Linnik, 1961); later authors (Winkler, 1950; Head, 1968) preferred

ordinary least-squares.

7.2 DAVIS DISTRIBUTION

Harold T. Davis (1892 “ 1974), in his 1941 monographs The Theory of Econometrics

and Analysis of Economic Time Series, proposed another income size distribution,

which is a generalization of the so-called Planck™s law of radiation from statistical

physics.

Davis was one of the pioneers of econometrics in the United States; he helped to

found the professional journal Econometrica and served as its Associate Editor for

239

7.2 DAVIS DISTRIBUTION

26 years. He also worked on the staff of the Cowles Commission in its early days in

Colorado Springs. See Farebrother (1999) for a recent account of Davis™s life.

In an attempt to derive an expression that would represent not merely the upper

tail of the distribution of income, Davis required an appropriate model with the

following properties:

f (x0 ) ¼ 0, for some x0 . 0, that may be interpreted as the subsistence income.

.

In Davis™s words, x0 represents the wolf point, since below this point “the wolf,

which lurks so close to the doors of those in the neighborhood of the modal

income [here he assumes that the distribution is highly skewed with a mode

close to the lower bound of the support, our addition], actually enters the

house” (1941a, p. 405).

A modal income exists.

.

For large x the distribution approaches a Pareto distribution

.

f (x) v A(x À x0 )ÀaÀ1 :

Davis then postulated that f is of the form

C 1

n . 1,

fD (z) ¼ ,

znþ1 e1=z À 1

where z ¼ x À x0 . This is, for x0 ¼ 0 and n ¼ 4, the distribution of 1=V , where

V follows the Planck distribution mentioned above. Clearly, fD (0) ¼ 0 and the

density exhibits an interior mode. It remains to determine the normalizing constant

C. From, for example, Prudnikov et al. (1986, Formula 2.3.14.6), we have

°1