252 Optimizing Optimization

S B , is a bounded distribution that has been called the four-parameter lognormal
distribution; and finally, S U , is an unbounded distribution based on an inverse
hyperbolic sine transform. 8 Each of the three distributions in the Johnson
family employs a transformation of the original variable to yield a standard
Gaussian variate, i.e.,

Z

X





γη

ε

λ

g

⎛

⎝

⎜⎜

⎜

⎞

⎠

⎟⎟

⎟

(11.1)

where Z is a standard normal random variable, γ and η are shape parameters, λ
is a scale parameter, ε is a location parameter, and g (.) is one of the following:

gx

xS

xS

x

x

N

L

()

ln( )

 ln



Normal Family:

Lognormal Family:

1

⎛

⎝

⎜⎜

⎜

⎞

⎠⎠

⎟⎟

⎟

⎧

⎨

⎪⎪

⎪⎪

⎪

Bounded Family:

Unbounded Family:

S

xx S

B

ln( ) U

(^21)
⎪⎪⎪
⎪
⎩
⎪⎪
⎪⎪
⎪⎪
⎪⎪
Furthermore, without loss of generality, we assume that η 0 and λ 0 and we
observe the standard convention that λ  1 and ε  0 for S N and λ  1 for S L.
From Equation (11.1), it follows that the probability density function (pdf)
for X is given by:
fx g
X
g
X
() exp


η 
λπ
ε
λ
γη
ε
2 λ
1
2
′
⎛
⎝
⎜⎜
⎜
⎞
⎠
⎟⎟
⎟
⎛
⎝
⎜⎜
⎜
⎞
⎠
⎟⎟
⎟
⎛
⎝
⎜⎜
⎜⎜
⎞
⎠
⎟⎟⎟
⎟⎟
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
2
(11.2)
∀ x  Ω , where g (.) is the first derivative of the function g(.), given by:
gx
S
xS
xx
N
L
′()
[( ) ]

−
−
−
1
1
1
1
Normal Family:
Lognormal Family:
Bounnded Family:
/ Unbounded Family:
S
xS
B
() U
(^212) + 1
⎧
⎨
⎪⎪
⎪⎪
⎪⎪
⎩
⎪⎪
⎪⎪
⎪⎪⎪ −
and the support of the distribution is:
Ω



(, )
[, ]
[, ]
∞∞
∞
Normal Family:
Lognormal Family:
B
S
S
N
ε L
εε λ oounded Family:
Unbounded Family:
S
S
B
(, )∞∞ U
⎧
⎨
⎪⎪
⎪⎪
⎪⎪
⎩
⎪⎪
⎪⎪
⎪⎪
8 The three-parameter lognormal distribution differs from the more common two-parameter ver-
sion in that the random variable is defined on the interval [ α , ) rather than [0, ).