`

and Gokhale (1996). Both of these works use the term exponential power distribution.

An expression for the asymptotic covariances of the ML estimates is obtained by

inversion of the Fisher information; thus, for a sample of size n we obtain

8 9

sr2 r2=r G(s)

> >

> >

0 0

> >

> (r À 1)G(1 À 1=r) >

> >

> >

> >

> >

!

< =

2 2

1 sr B rBsr

À1

:

I (u) ¼ 0 1þ 0

sc 0 (s) À 1 >

n> r sc (s) À 1

> >

> >

> >

> >

> >

3

> >

rBsr r

> >

: ;

0 0 0

sc (s) À 1 sc (s) À 1

For the generalized normal distribution, two simulation studies have been

conducted in order to investigate the small sample behavior of the estimators.

139

4.10 GENERALIZED LOGNORMAL DISTRIBUTION

Rahman and Gokhale (1996) found that the method of moments (MM) and ML

estimators for m and sr perform similarly for r 2; whereas for r . 2 the ML

estimator of r seems to perform better than its MM counterpart for small samples.

For r , 2 the situation is reversed. Agro (1995) noted that for samples of size

`

n 100 there is sometimes no well-de¬ned optimum of the likelihood when r ! 3

and that r is frequently overestimated in small samples for r . 2:

In both of these works, random samples were obtained using a rejection method

following Tiao and Lund (1970). However, it is possible to generate simulated data

using methods that exploit the structure of the generalized normal distribution. The

following algorithm for the generation of samples from the generalized lognormal

distribution makes use of a mixture representation utilizing the gamma (Ga)

distribution and is adapted from Devroye (1986, p. 175):

Generate V $ U [À1, 1] and W $ Ga(1 þ 1=r, 1):

.

Compute Y :¼ r1=r sr VW 1=r þ m:

.

Obtain X ¼ exp(Y ):

.

Jakuszenkow (1979) and Sharma (1984) studied the estimation of the variance of

a generalized normal distribution. (In the terminology of inequality measurement,

this is the variance of logarithms of the generalized lognormal distribution.) Since

the variance of the distribution is a multiple of sr2 ; one may equivalently study the

estimation of sr2 : Sharma showed that the estimator

P

G[(n þ 2)=r]( n jxi jk )2=k

i¼1

G[(n þ 4)=k]

^

is Lehmann-unbiased for the loss function L(u) ¼ (u À u)uÀ2 and also admissible, for

¬xed r: Thus, the result may be best perceived as pertaining to the familiar special cases

of the generalized lognormal family, the lognormal and log-Laplace distributions.

Further results on estimation are available in the log-Laplace case (where r ¼ 1).

Moothathu and Christudas (1992) considered the UMVU estimation of log-Laplace

P

characteristics when m ¼ 0: They noted, that the statistic T ¼ n jlog xi j follows a

i¼1

gamma distribution and is complete as well as suf¬cient for s1 : An unbiased

estimator of s1p ; p ¼ 1, 2, . . . ; is given by

Tp

s1p

^ ,

¼

(n)p

where (n)p denotes Pochhammer™s symbol for the forward factorial function.

Furthermore, they showed that the UMVUE of the Gini coef¬cient is given by

!

n þ 1 n þ 2 T2

3T

^

G¼ 1 F2 1; , ; ,

n 2 2 16

where 1 F2 is a generalized hypergeometric function.

140 LOGNORMAL DISTRIBUTIONS

We conclude our brief survey of the generalized lognormal distribution by

reporting on some empirical applications. The distribution was ¬tted to Italian and

German income data with mixed success. Brunazzo and Pollastri (1986) used the

distribution for approximating Italian data of 1948 and obtained a shape parameter

r ¼ 1:4476; considerably below the lognormal benchmark value of r ¼ 2: Scheid

(2001) ¬t the distribution to 1993 German household incomes (sample size: 40,000).

Estimating by the method of moments and maximum likelihood, she found that the

distribution improves upon the two-parameter lognormal, but the estimate of r is

only slightly below 2. Although both likelihood ratio and score tests con¬rm the

signi¬cance of the difference, the standard as well as the generalized lognormal

distributions are empirically rejected using nonparametric goodness-of-¬t tests.

Inoue (1978) postulated a stochastic process giving rise to the log-Laplace

distribution and ¬t the distribution to British data for the period 1959 “1960 by the

method of maximum likelihood. He found that the ¬t is more satisfactory than for

the lognormal for this period.

Vianelli (1982a, 1983) brie¬‚y considered a family of generalized lognormal

distributions with bounded support. The p.d.f. is of the form

r 1

1 À r jlog x À mj ,

f (x) ¼

2sr (rq)1=r B(1=r, q þ 1)x sr

1=r 1=r

x mesr (rq) : The logarithm of a random variable following

where meÀsr (rq)

this distribution with r ¼ 2 can be viewed a Pearson type II distribution. For b ¼

sr (rq)1=r ! 1; we get the generalized lognormal distribution (4.43).

4.11 AN ASYMMETRIC LOG-LAPLACE DISTRIBUTION

The preceding section presented a one-parameter family of generalizations for the

lognormal distribution. Interestingly, an asymmetric variant of one of its members,

the log-Laplace distribution (a generalized lognormal distribution with r ¼ 1),

appears in a recent dynamic model of economic size phenomena proposed by Reed

(2001a,b, 2003).

He started from a continuous-time model with a varying but size-independent

growth rate. The probably most widely known dynamic model possessing this

property is a stochastic version of simple exponential growth, geometric Brownian

motion, that is de¬ned in terms of the stochastic differential equation

dXt ¼ mXt dt þ sXt dBt , (4:61)