Note that the latter expression does not contain any terms involving the gamma

function. This re¬‚ects the fact that the log-gamma distribution is closely related to

the classical Pareto distribution, whose moments are obtained for p ¼ 1.

From (5.59) we get

p

b

E(X ) ¼ (5:60)

bÀ1

and

p

b 2p

b

b . 2:

var(X ) ¼ , (5:61)

À

bÀ2 bÀ1

The mode and geometric mean are given by

& '

pÀ1

xmode (5:62)

¼ exp

bþ1

and

p

xgeo , (5:63)

¼ exp

b

respectively. We see that xmode , xgeo , E(X ).

The basic reproductive property of the distribution is a direct consequence of the

well-known reproductive property of the gamma distribution (5.41). Since the latter

is closed under addition, if the scale parameters are equal, we determine that the

log-gamma distribution is closed with respect to the formation of products: if

Xi $ log Ga( pi , b), i ¼ 1, 2, are independent, we have

X1 Á X2 $ log Ga( p1 þ p2 , b): (5:64)

Regarding estimation, we can be brief, because in view of its genesis, we may

translate this problem to that of parameter estimation for a two-parameter gamma

distribution and make use of the extensive literature devoted to that topic.

Speci¬cally, the Fisher information on u ¼ (b, p)` is identical to the Fisher

˜ ˜

information I (u) of the classical gamma distribution with parameters u ¼ (1=b, p)` ;

see (5.55). (Recall that for the log-gamma distribution the p.d.f. was reparameterized

via b ! 1=b:) The Fisher information I (u) of the reparameterization is thus

˜

obtained using the relation I (u) ¼ JI (u)J ` , where J is the Jacobian of the inverse

171

5.3 LOG-GAMMA DISTRIBUTION

transformation (see, e.g., Lehmann and Casella, 1998, p. 125), and therefore

given by

2p 13

À

6 b2 b7

I (u) ¼ 6 7: (5:65)

4 À1 5

0

c ( p)

b

5.3.2 Lorenz Curve and Inequality Measures

The Lorenz curve of the log-gamma distribution is not available in a simple closed

form. It can be expressed in terms of a moment distribution in the form

& '

g( p, by) g[ p, (b À 1)y]

, where y ¼ log x :

(u, v) j u ¼ ,v¼ (5:66)

G( p) G( p)

Here g(Á, Á) is the incomplete gamma function.

The Gini coef¬cient of the two-parameter log-gamma distribution is (Bhattacharjee

and Krishnaji, 1985)

bÀ1

:

G ¼ 1 À 2Ba ( p, p), a¼ (5:67)

2b À 1

This shows that, for a ¬xed p, the Gini coef¬cient decreases with increasing b, with a

(positive) lower bound that depends on the second shape parameter p.

5.3.3 Empirical Results

Bhattacharjee and Krishnaji (1985) showed that the landholdings in 17 Indian states

for 1961 “ 1962 are better approximated by a log-gamma than by a lognormal

distribution. A gamma distribution with a decreasing density provides a comparable

¬t.

In the actuarial literature, Hewitt and Lefkowitz (1979) estimated a two-

component gamma-loggamma mixture for automobile bodily injury loss data. This

mixture model does considerably better than a two-parameter lognormal distribution.

Also, Ramlau-Hansen (1988) reported that the log-gamma distribution provides a

satisfactory ¬t when modeling ¬re losses for a portfolio of single-family houses and

dwellings in Denmark for the period 1977“ 1981. His estimated tail index b

(essentially Pareto™s alpha) falls in the vicinity of 1.4.

172 GAMMA-TYPE SIZE DISTRIBUTIONS

5.4 INVERSE GAMMA (VINCI) DISTRIBUTION

5.4.1 De¬nition and Basic Properties

Some authors, especially those dealing with reliability applications such as Barlow

and Proschan (1981), call this distribution the inverted gamma. Indeed, if Z is a

gamma variable, then X ¼ 1=Z is a variable with the density

b p ÀpÀ1 À(b=x)

, x . 0,

f (x) ¼ x e (5:68)

G( p)

where p, b . 0. (As in the preceding section, we have reparameterized the density

using b ! 1=b.)

The distribution is encountered in Bayesian reliability applications. It is also

hidden among the Pearson curves, speci¬cally Pearson V, and Vinci (1921) should

be credited for his income distribution applications.

Distribution (5.68) is a special case of the inverse generalized gamma distribution

(a ¼ À1); for p ¼ 1 we obtain a special case of the log-Gompertz distribution to be

´

discussed below. The case where p ¼ 1=2, more widely known as the Levy

distribution, is of special interest in probability theory: It is one of the few stable

distributions for which an expression of the density in terms of elementary functions

is available and arises as the distribution of ¬rst-passage times in Brownian motion

(see, e.g., Feller, 1971).

Since the density (5.68) is regularly varying (at in¬nity) with index ÀpÀ1, the

moments exist only for k , p. They are given by [compare with (5.12)]

G( p)

E(X k ) ¼ bk : (5:69)

G( p À k)

The mode occurs at ( p þ 1)b.