dt; g(n, x) ¼ 0 eÀt t nÀ1 dt are the incomplete gamma functions.

xet

The moments of the generalized lognormal distribution are (see, e.g., Brunazzo

and Pollastri, 1986)

exp(km) X (ksr )2i 2i=r 2i þ 1

1

k

:

rG

E(X ) ¼ (4:47)

G(1=r) i¼0 (2i)! r

For r ¼ 1; that is, for the log-Laplace distribution, the in¬nite sum on the r.h.s.

converges only for jks1 j , 1; in that case, it equals (Vianelli, 1982b)

exp(km)

E(X k ) ¼ : (4:48)

1 À (ks1 )2

2

In particular, for k ¼ 1 we get E(X ) ¼ exp m=(1 À s1 ): For generalized lognormal

distributions with r . 1; all the moments exist. Brunazzo and Pollastri (1986) noted

that the mean, variance, and standard deviation are increasing in sr and decreasing

in r; whereas for Pearson™s coef¬cient of skewness the opposite behavior is

observed. Thus, the distribution becomes more symmetric as sr increases. Pollastri

(1997) investigated the kurtosis of the distribution utilizing Zenga™s (1996) kurtosis

diagram and found that for ¬xed median and mean deviation from the median the

kurtosis decreases as r and/or sr increase.

Since the exponential power distribution (4.41) is symmetric about m; the median

of the generalized lognormal distribution is given by

xmed ¼ exp(m) (4:49)

and the mode equals

xmode ¼ exp[m À srr=(rÀ1) ] for r . 1, (4:50)

while

xmode ¼ exp m for r ¼ 1, s1 , 1: (4:51)

The distribution is thus unimodal, with a cusped mode in the log-Laplace case.

135

4.10 GENERALIZED LOGNORMAL DISTRIBUTION

From (4.47), (4.49), and (4.50) it follows that the generalized lognormal

distribution satis¬es the mean-median-mode inequality, namely,

exp m X sr2i 2i=r 2i þ 1

1

. exp m . exp[m À srr=(rÀ1) ]:

rG

G(1=r) i¼0 (2i)! r

The entropy of the distribution is (Scheid, 2001)

1 c

E[Àlog f (X )] ¼ À log þ m, (4:52)

sr

r

where c ¼ [2r1=r G(1 þ 1=r)]À1 ; which simpli¬es to

E[Àlog f (X )] ¼ 1 þ log(2s1 ) þ m (4:53)

for the log-Laplace distribution. It can be shown that the entropy is a decreasing

function of r:

The Lorenz and Zenga curves may be obtained numerically in terms of the ¬rst-

moment distribution; Pollastri (1987b) provided some illustrations. She observed

that inequality, as measured by the Lorenz curve, is decreasing in r for ¬xed sr and

increasing in sr for ¬xed r:

Expressions for inequality measures of the generalized lognormal are often

somewhat involved. For the Gini coef¬cient there does not seem to be a simple

expression for a general r: Pollastri (1987b) suggested evaluating numerically

X

k

G ¼1À [F(xi ) À F(xiÀ1 )][F(1) (xi ) þ F(1) (xiÀ1 )],

i¼1

where F(x0 ) ¼ F(1) (x0 ) ¼ 0. However, in the log-Laplace case there is a simple

closed form for the Gini index (Moothathu and Christudas, 1992)

3sÀ1

1

:

G¼ (4:54)

À2 À 1)

(4s1

The formulas of Theil™s inequality measures were derived by Scheid (2001). The

(¬rst) Theil coef¬cient is given by

exp(m) X (sr )2iþ2 (2iþ2)=r 2i þ 3

1

G

T1 ¼ r

G(1=r)E(X ) i¼0 (2i þ 1)! r

À log {E(X )= exp m}, (4:55)

136 LOGNORMAL DISTRIBUTIONS

and Theil™s second measure equals

T2 ¼ log[E(X )=exp m]: (4:56)

Again, in the log-Laplace case these expressions are simpli¬ed and reduced to

2

2s1

2

s1 )

T1 ¼ log(1 À (4:57)

þ 2

1 À s1

and

2

T2 ¼ Àlog(1 À s1 ), (4:58)

respectively, provided s1 , 1:

Variants of the ¬rst measure with the median and mode replacing the mean, as

suggested by Shimizu and Crow (1988) for the standard lognormal distribution, are

(Scheid, 2001)

sr 2=r X (sr )2iþ1 2i=r 2i þ 3

1

rG

Tmed r ,

¼

G(1=r) (2i þ 1)! r

i¼0

and, for r . 1;

sr exp [srr=(rÀ1) ] X (sr )2i 2i=r

1

sr 2=r 2i þ 3

rG

Tmode r

¼