Optimizing Optimization: The Next Generation of Optimization Applications and Theory (Quantitative Finance)

272 Optimizing Optimization

Appendix

Parameters of the Johnson density

Slifker and Shapiro (1980, pp. 243 – 246) report the following parameter esti-
mates for each of the four Johnson distributions using their method of quan-
tiles estimator. Interested readers should consult the original article for a full
derivation of these formulae.

Parameters estimates of the S U distribution

ηη γη









(^2) 
1
2
0,
2
1
z 1
m
p
n
p
n
p
m
p
cosh
() h
⎛
⎝
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
sin
mmn
p^2
12
 1
⎛
⎝
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟⎟
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
/
λ

 
21
22
2
1/2
p
mn
p
m
p
n
p
m
p
n
p
⎛
⎝
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟⎟
⎛
⎝
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟
⎛
⎝
⎜⎜
⎜⎜
⎞⎞
⎠
⎟⎟
⎟⎟
1/2 ()λ^0
ε




xx
p
n
p
m
p
m
p
n
p
zz
2
22
⎛
⎝
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟
⎛
⎝
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟

Parameters estimates of the S B distribution

η

 

z

p

m

p

n

1

1/2

1

2

cosh^11

⎛

⎝

⎜⎜

⎜

⎞

⎠

⎟⎟

⎟

⎛

⎝

⎜⎜

⎜

⎞

⎠

⎟⎟

⎟

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⎛

⎝

⎜⎜

⎜⎜

⎜

⎞

⎠⎠

⎟⎟

⎟⎟

⎟⎟

()η 0

γη



 1

11 4

sinh

p

n

p

m

p

m

p

n

⎛

⎝

⎜⎜

⎜

⎞

⎠

⎟⎟

⎟

⎛

⎝

⎜⎜

⎜

⎞

⎠

⎟⎟

⎟

⎛

⎝

⎜⎜

⎜

⎞

⎠

⎟⎟

⎟

⎡

⎣

⎢

⎢

⎤⎤

⎦

⎥

⎥

⎛

⎝

⎜⎜

⎜⎜

⎞

⎠

⎟⎟

⎟⎟⎟

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

1/2

2

21

p

mn