x Àa 1 1 a x

,

exp À ¼À

b x x b(a þ 1) b

which is of the form

u(x) ¼ v(x):

It is not dif¬cult to see that both functions u(x) and v(x) are increasing up to a point

xn ¼ ba1=a ,

which is to the right of the mode (5.100), and decreasing thereafter. Also,

limx!0 u(x) ¼ 0 and v(xmode ) ¼ 0. This yields

u(xmode ) . v(xmode ) ¼ 0, u(xn ) , v(xn ), xmode , xn ,

from which it follows that the point of intersection of u(x) and v(x)”de¬ning the

mode of r(x)”is contained in the interval (xmode , xn ).

181

5.6 LOG-GOMPERTZ DISTRIBUTION

Since the log-Gompertz density is regularly varying at in¬nity, we determine that

the mean excess function is asymptotically linearly increasing [see (2.67)]

e(x) [ RV1 (1): (5:102)

5.6.2 Estimation

Being the inverse Weibull distribution, parameter estimation for the log-Gompertz

distribution proceeds most easily by considering the reciprocal observations 1=xi ,

i ¼ 1, . . . , n, and using methods appropriate for Weibull data.

The relationship with the Weibull distribution also yields the Fisher information

on u ¼ (a, b)`

2 3

6(C À 1)2 þ p 2 1ÀC

6 b7

6a2

6 7

I (u) ¼ 6 7, (5:103)

4 a2 5

1ÀC

b2

b

which coincides up to the sign of the off-diagonal elements with (5.91).

Erto has suggested a simple estimator utilizing a linearization of the survival

function. He proposed estimating parameters in the equation

!

1 1 1

þ log b (5:104)

log ¼ log log

x a 1 À F(x)

by least squares.

5.6.3 Inequality Measurement

As in the Weibull case, Lorenz ordering relations are easily obtained using the star-

shaped ordering (see Section 2.1.1). Speci¬cally, we have for Xi $ logGomp(ai , 1),

i ¼ 1, 2, using (5.95),

F1 (u) (À log u)À1=a1

À1

0 , u , 1,

,

¼

F2 (u) (À log u)À1=a2

À1

which is seen to be increasing in u if and only if a1 a2 . Since the star-shaped

ordering implies the Lorenz ordering”provided ai . 1, so that E(Xi ) , 1”we

have

a2 :

X1 ! L X2 ( a1 (5:105)

)

Hence, the log-Gompertz family is another family of distributions within which the

Lorenz ordering is linear.

182 GAMMA-TYPE SIZE DISTRIBUTIONS

5.6.4 Empirical Results

Cummins et al. (1990), in their comprehensive study employing 16 loss

distributions, ¬t the log-Gompertz (under the name of inverse Weibull) distribu-

tion to the annual ¬re loss experiences of a major university. The distribution turns

out to be the best two-parameter model; however, the data are appropriately

modeled by the one-parameter special case where a ¼ 1, an inverse exponential

distribution.

CHAPTER SIX

Beta-type Size Distributions

Beta distributions (there are two kinds of this distribution) are members of the

celebrated Pearson system and have been widely utilized in all branches of

sciences”both soft and hard. They are intrinsically related to the incomplete beta

function ratio

°x

1

upÀ1 (1 À u)qÀ1 du,

Ix ( p, q) ¼ 0 x 1, (6:1)

B( p, q) 0

and the incomplete beta function

°x

upÀ1 (1 À u)qÀ1 du, 0

Bx ( p, q) ¼ x 1: (6:2)

0

A historical account of these functions was provided by Dutka (1981), who traced

them to a letter from Isaac Newton to Henry Oldenberg in 1676. Needless to say, as

Thurow (1970) put it, “using a beta distribution is not meant to imply that God is a

beta generating function.”

6.1 (GENERALIZED) BETA DISTRIBUTION OF THE

SECOND KIND

For our purposes, the pivotal distribution in this family is the so-called generalized

beta distribution of the second kind (hereafter referred to as GB2). We should note

the contributions of McDonald (1984) and his associates in the development of GB2

distributions as an income distribution and in unifying the various research activities

in closely related ¬elds. We could mention as an example the multiauthor paper by

Cummins et al. (1990), which is a combination of two manuscripts by McDonald

183

184 BETA-TYPE SIZE DISTRIBUTIONS

and Pritchett (the Brigham Young University School) and Cummins and Dionne (the

University of Pennsylvania School) dealing with applications of the GB2 family of

distributions to insurance losses submitted independently to Insurance: Mathematics

and Economics in early 1988 and resulted in a uni¬ed treatment.

The interrelations between particular cases of the GB2 distributions and other

distributions known in the literature are somewhat confusing, but a natural

consequence of the independent uncoordinated research that has been so prevalent