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The exponential power distribution has been frequently employed in robustness

studies, and also as a prior distribution in various Bayesian models (see Box and

Tiao, 1973, for several examples).

If we start from (4.41), the density of X Â¼ exp Y is

& '

1 1

exp Ã€ r jlog x Ã€ mjr , 0 , x , 1:

f (x) Â¼ (4:43)

1=r s G(1 Ã¾ 1=r)

2xr rsr

r

Here em is a scale parameter and sr , r are shape parameters. The effect of the new

parameter r is illustrated in Figures 4.4 and 4.5.

Figure 4.5 suggests that the density becomes more and more concentrated on a

bounded interval with increasing r. This is indeed the case: For m Â¼ 0; r ! 1; the

limiting form is the distribution of the exponential of a random variable following a

uniform distribution on [Ã€1, 1]; as may be seen from the following argument given

132 LOGNORMAL DISTRIBUTIONS

Figure 4 From the log-Laplace to the lognormal distribution: m Â¼ 0, s Â¼ 1, and r Â¼ 1(0:1)2 (from

the bottom).

by Lunetta (1963). The characteristic function of the generalized error distribution

(4.41) is given by

& '

Ã°1

jxjr

1

f(t) Â¼ exp Ã€ r cos(tx) dx

sr r1=r G(1 Ã¾ 1=r) rsr

0

X

1

G[(2s Ã¾ 1)=r]t 2s

(Ã€1)s : (4:44)

Â¼

r2s=r s2s G(1=r)(2s)!

r

sÂ¼0

Now,

G[(2s Ã¾ 1)=r] G[1 Ã¾ (2s Ã¾ 1)=r] 1

,

lim Â¼ lim Â¼

2s=r s2s G(1=r) 2s=r s2s G(1 Ã¾ 1=r)

r!1 r r!1 (2s Ã¾ 1)r 2s Ã¾ 1

r r

yielding

1X 1

t 2sÃ¾1 sin t

(Ã€1)s

lim f(t) Â¼ ,

Â¼

t sÂ¼0 (2s Ã¾ 1)! t

r!1

133

4.10 GENERALIZED LOGNORMAL DISTRIBUTION

Beyond the lognormal distribution: m Â¼ 0, s Â¼ 1, and r Â¼ 2, 10, 100 (from left to right).

Figure 5

which can be recognized as the characteristic function of a random variable

distributed uniformly on the interval [Ã€1, 1]: The resulting distribution therefore

possesses the p.d.f.

1

eÃ€1

f (x) Â¼ , x e: (4:45)

2x

Interestingly, this is the p.d.f. of a doubly truncated Pareto-type variable, a

distribution considered by Bomsdorf (1977) which was brieï¬‚y mentioned in the

preceding chapter, under the name of prize-competition distribution.

The case where r Â¼ 1 (the log-Laplace distribution) was proposed as an income

distribution by Frechet as early as 1939. Here closed forms for the c.d.f. and quantile

Â´

function are available, namely,

8

m Ã€ log x

>1

> exp Ã€ for 0 , x , exp m,

> ,

<2 s1

F(x) Â¼

> log x Ã€ m

1

>

> 1 Ã€ exp Ã€

: for x ! exp m,

,

s1

2

and

&

exp{m Ã¾ s1 log(2u)}, for 0 , u , 0:5,

F Ã€1 (u) Â¼

exp{m Ã€ s1 log[2(1 Ã€ u)]}, for 0:5 u , 1:

134 LOGNORMAL DISTRIBUTIONS

In the general case, the c.d.f. may be written in the form (Pollastri, 1987b)

8

> G[1=r, B(x)],

> for x , exp m,

>

> 2G(1=r)

>

>

>

<

1

F(x) Â¼ (4:46)

for x Â¼ exp m,

,

>2

>

>

>

> 1 g[1=r, M (x)]

>

>Ã¾

: , for x . exp m,

2 2G(1=r)

where B(x) Â¼ [(m Ã€ log x)=sr ]r =r; M (x) Â¼ [(log x Ã€ m)=sr ]r =r; and G(n, x) Â¼

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