entropy distribution under the constraints of a ¬xed ¬rst moment and a ¬xed

geometric mean. Similarly, the inverse gamma distribution is the maximum entropy

distribution if the ¬rst moment and harmonic mean are prescribed (Ord, Patil, and

Taillie, 1981).

Regular variation of the p.d.f. also gives us the basic asymptotic property of the

hazard rate and mean excess function

r(x) [ RV1 (À1) (5:70)

and

e(x) [ RV1 (1): (5:71)

173

5.4 INVERSE GAMMA (VINCI) DISTRIBUTION

5.4.2 Inequality Measurement

From (5.49) we know that the gamma distributions are Lorenz-ordered with respect

to the shape parameter p, with diminishing inequality being associated with

increasing p. Unfortunately, this does not translate directly into a corresponding

result for the inverse gamma distribution. In order to obtain such a result, a stronger

ordering concept than the Lorenz ordering is required. As mentioned in Chapter 2,

the star-shaped ordering implies Lorenz ordering, and van Zwet (1964) showed

that the gamma distribution is ordered according to the convex transform ordering

and therefore also in the sense of the weaker star-shaped ordering. Since the star-

shaped ordering is closed under inversion, in the sense that (e.g., Taillie, 1981)

1 1

X1 !Ã X2 ( , (5:72)

) !Ã

X1 X2

these results translate into

X1 !L X2 ( p1 p2 , (5:73)

)

for the inverse gamma distributions, provided p . 1 [in order to assure the existence

of E(X ) and therefore that of the Lorenz curve].

5.4.3 Estimation

In view of the genesis of the distribution, estimation can proceed by considering the

“inverted” data 1=xi , i ¼ 1, . . . , n, and using methods appropriate for the gamma

distribution. Alternatively, the likelihood equations

np X 1

n

¼ 0, (5:74)

À

b xi

i¼1

X

n

nlog b À nc( p) À log xi ¼ 0 (5:75)

i¼1

can be solved directly.

From (5.74) and (5.75) we get the Fisher information on u ¼ (b, p)`

2p 13

À

6 b2 b7

I (u) ¼ 6 7: (5:76)

41 5

0

c ( p)

À

b

5.4.4 Empirical Results

Although the Vinci distribution was proposed as an income distribution some 80

years ago, we have not been able to track down ¬ttings to income data in the

174 GAMMA-TYPE SIZE DISTRIBUTIONS

literature available to us. In the actuarial literature, Cummins et al. (1990) used the

inverse gamma distribution for approximating the ¬re loss experiences of a major

university. The distribution turns out to be one of the best two-parameter models; in

fact, the data are appropriately modeled by the one-parameter special case where

a ¼ 1, an inverse exponential distribution.

5.5 WEIBULL DISTRIBUTION

All that Waloddi Weibull, a Swedish physicist, did in his pioneering reports No.™s

151 and 153 for the Engineering Academy in 1939 was to add a “small a” to the

c.d.f. of the exponential distribution, and what a difference it did cause! Nowadays

we refer to this operation as Weibullization. The Weibull distribution has no doubt

received maximum attention in the statistical and engineering literature of the last ten

years and is still going strong. In economics it is probably less prominent, but

D™Addario (1974) noticed its potentials for income data and Hogg and Klugman

(1983) for insurance losses.

Even the strong evidence that the Weibull distribution is indeed due to Weibull is

shrouded in minor controversy. Rosin and Rammler in 1933 used this distribution in

their paper “The laws governing the ¬neness of powdered coal.” Some Russian

sources insist that it should be called Weibull “ Gnedenko (or preferably Gnedenko “

Weibull!) since it turns out to be one of the three types of extreme value limit

distributions established rigorously by Gnedenko in his famous paper in the Annals

of Mathematics, published during World War II. And the French would argue that

this is nothing else but Frechet™s distribution, who initially identi¬ed it in 1927 to be

´

an extreme-value distribution in his “Sur la loi de probabilite de l™ecart maximum.”

´ ´

Here we shall follow the same pattern as in the gamma distribution section and

concentrate on the income and size applications of the Weibull distribution, some of

them quite recent.

5.5.1 De¬nition and Basic Properties

Being a generalized gamma distribution with p ¼ 1, the Weibull density is given

by

a x aÀ1 À(x=b)a

x . 0,

f (x) ¼ e , (5:77)

bb

where a, b . 0. Unlike the c.d.f. of the classical and generalized gamma

distributions, the c.d.f. of the Weibull distribution is available in terms of

elementary functions; it is simply

a

F(x) ¼ 1 À eÀ(x=b) , x . 0: (5:78)

175

5.5 WEIBULL DISTRIBUTION

We note that irrespective of the value of a,

F(b) ¼ 1 À eÀ1 :

From (5.78) we see that even the quantile function is available in closed form

F À1 (u) ¼ b{À log(1 À u)}1=a , 0 , u , 1, (5:79)

a property that facilitates the derivation of Lorenz-ordering results (see below).

From (5.79) we determine that the median of the distribution is

xmed ¼ b(log 2)1=a : (5:80)

A simple argument leading to a Weibull distribution as the distribution of ¬re

loss amount was given by Ramachandran (1974). Under the two assumptions that

(1) the hazard rate of the ¬re duration T is given by l(t) ¼ exp(at), a . 0, and (2)

that the resulting damage X is an exponential function of the duration,

X ¼ x0 exp(kT ), for some x0 , k . 0, the c.d.f. is given by (5.78).

The inverse Weibull distribution, that is, the distribution of 1=X for

X $ Wei(a, b), is discussed in the following section, under the name of the log-