b

where C ¼ 0:577216. . . is Euler™s constant. If a , 1, the density is unbounded.

5.5.4 Empirical Results

As noted in the introduction to this chapter, the Weibull distribution was apparently

used only sporadically as an income or size distribution and most applications are of

comparatively recent date.

Incomes

Bartels (1977) ¬t the distribution to 1969 ¬scal incomes in three regions of the

Netherlands. Here the model seems to provide a rather unsatisfactory ¬t; it is

178 GAMMA-TYPE SIZE DISTRIBUTIONS

outperformed by (generalized) gamma and Champernowne distributions and

variants of it. For the French wages strati¬ed by occupation for 1970 “1978

the three-parameter distribution is also not satisfactory, being outperformed by the

Dagum type II, a Box “ Cox-transformed logistic, the Singh “Maddala, and the three-

parameter lognormal distributions (Espinguet and Terraza, 1983). However, it does

better than a four-parameter beta type I distribution for these data.

McDonald (1984) applied the Weibull distribution for 1970, 1975, and 1980 U.S.

family incomes. The distribution is outperformed by three- and four-parameter

models such as the (generalized) beta, generalized gamma, and Singh “ Maddala

distributions, but is superior to all other two-parameter models”notably the

lognormal and gamma distributions”for the 1980 data, where it ranks fourth out of

11 models considered.

Atoda, Suruga, and Tachibanaki (1988) considered grouped income data from the

Japanese Income Redistribution Survey for 1975, strati¬ed by occupation. Although

among the distributions they employed, the Singh “Maddala often appears to be the

most appropriate, the Weibull does only slightly worse than the more ¬‚exible

generalized gamma distribution, in one case even better when the selection criterion

is the AIC. In a later study employing individual data from the same source, the

Weibull distribution is again comparable to the generalized gamma distribution for

one stratum, and only slightly worse for the remaining ones (Tachibanaki, Suruga,

and Atoda, 1997).

Bordley, McDonald, and Mantrala (1996) ¬t the Weibull distribution to U.S.

family incomes for 1970, 1975, 1980, 1985, and 1990. It is outperformed by all

three- and four-parameter models”notably the GB2, Dagum, and Singh “ Maddala

distributions”and by the two-parameter gamma distribution for three data sets. For

the remaining years it is comparable to the gamma distribution.

Brachmann, Stich, and Trede (1996), in their study of German household

incomes over the period 1984 “1993, found the Weibull distribution to perform

better than the lognormal, but not nearly as well as the gamma distribution. However,

only the GB2 and Singh “ Maddala distribution seem to provide a satisfactory ¬t for

these data.

Actuarial Losses

In the actuarial literature, Hogg and Klugman (1983) ¬t the Weibull distribution to a

small data set (35 observations) of hurricane losses and found that it performs about

as well as the lognormal distribution. They also considered data for malpractice

losses, for which beta type II and Lomax (Pareto type II) distributions are preferable.

In the Cummins et al. (1990) study employing 16 loss distributions, the Weibull

distribution does not provide an adequate ¬t to the Cummins and Freifelder (1978)

¬re loss data. Speci¬cally, the data seem to require a model with heavier tails such as

an inverse Weibull distribution.

Nonetheless, from these works it is clear that the Weibull distribution often does

considerably better than the more popular lognormal distribution. Among the two-

parameter models it appears to be comparable to the gamma distribution.

179

5.6 LOG-GOMPERTZ DISTRIBUTION

5.6 LOG-GOMPERTZ DISTRIBUTION

The Gompertz distribution was introduced some 120 years before the Weibull one by

Benjamin Gompertz in 1825 in the Philosophical Transactions of the Royal Society,

to ¬t mortality tables. It is usually de¬ned for positive values. Nowadays it is used in

actuarial statistics and competing risks, in the early 1970s it attracted attention in

Applied Statistics (Garg, Rao, and Redmond, 1970; Prentice and El Shaarawi, 1973).

When de¬ned over the real line, the Gompertz c.d.f. is

F( y) ¼ exp (ÀaeÀy=b ), À1 , y , 1, (5:92)

where a, b . 0, which is a type I extreme value distribution.

5.6.1 De¬nition and Basic Properties

The log-Gompertz distribution appears to be used mainly in income and size

distributions and was noticed by Dagum (1980c) in this connection. It is a member

of Dagum™s (1980c, 1990a, 1996) generating system; see Section 2.4.

From (5.92), the c.d.f. of X ¼ exp Y is

& Àa '

x

x . 0,

F(x) ¼ exp À , (5:93)

b

where b . 0 and a . 0. This yields the log-Gompertz density

Àa

f (x) ¼ aba xÀaÀ1 eÀ(x=b) , x . 0, (5:94)

which is easily recognized as the p.d.f. of an inverse Weibull distribution. The case

where a ¼ 1, the inverse exponential distribution, is also a special case of the inverse

gamma (Vinci) distribution discussed in Section 5.4.

As in the Weibull case, the quantile function is available in closed form, being

F À1 (u) ¼ b{(À log u)À1=a }, 0 , u , 1: (5:95)

The median therefore occurs at

xmed ¼ b( log 2)À1=a : (5:96)

The moments exist only for k , a; in that case, they are given by

k

E(X k ) ¼ bk G 1 À : (5:97)

a

180 GAMMA-TYPE SIZE DISTRIBUTIONS

Speci¬cally,

1

E(X ) ¼ bG 1 À (5:98)

a

and

& '

2 1

2

2

var(X ) ¼ b G 1 À :

ÀG 1À (5:99)

a a

The mode is at

1=a

a

¼b :

xmode (5:100)

aþ1

We see that in contrast to the Weibull distribution, there is always an interior mode.

The hazard rate and mean excess function were studied by Erto (1989) in an

Italian publication dealing with lifetime applications. The hazard rate is

Àa

aba xÀaÀ1 eÀ(x=b)

r(x) ¼ , x ! 0: (5:101)

1 À exp{À(x=b)Àa }

It is noteworthy that, irrespectively of the value of the shape parameter a, r(0) ¼ 0

and limx!1 r(x) ¼ 0; in fact, r(x) is a unimodal function (similar to the lognormal

hazard rate). There is no simple expression for the abscissa of the mode, but it can be

bounded: From the derivative of log r(x) we obtain the ¬rst-order condition