Heuristic portfolio optimization: Bayesian updating with the Johnson family of distributions 255
3 z , the corresponding quantile of the data must be estimated. This is usually
done by solving the equation ( i 0.5)/ n Φ (.) for the value of i , where Φ (.)
corresponds to one of the four selected points. Usually, the value of i obtained
from the above equation will not be an integer, and therefore it is necessary to
interpolate. The four data values obtained correspond to x (^) 3 z , x (^) z , x z , and x (^3) z ,
which are no longer equally spaced.
It is these values of x that allow us to distinguish between the functional
forms of the Johnson distribution. For instance, Slifker and Shapiro (1980)
proved that, for a bounded symmetrical Johnson distribution, the distances
between each of the outer and inner points would be smaller than the distance
between the two inner points, and that the converse would be true for the
unbounded case. Consequently, by letting:
mxx
nxx
pxx
zz
zz
zz
3
3
it follows that each of the four types of Johnson density are described by the
following inequalities:
mn
p
S
mn
p
S
mn
p
U
B
1
1
1
for any distribution
for any distribution
andd for any distribution
and for any distribut
m
p
S
mn
p
m
p
S
L
N
1
11 iion
The proof of this proposition can be found in Slifker and Shapiro (1980,
pp. 243 – 246). After the functional form of the distribution has been determined, the
parameters of the corresponding distribution are then derived by solving a system of
equations. Expressions for the parameters are described in Appendix 11.9.1. 11
It remains to determine the choice of z 0. Slifker and Shapiro (1980) sug-
gest that this value should be dependent on the size of the data set, although
they conclude that a value around z 0.5 is reasonable in most cases. 12 Our
11 It should be noted that since the x ’s are continuous random variables, the probability that mn / p 2 1
is virtually zero and so it is necessary to define a tolerance level about unity.
12 The value is usually less than 1 because a choice of z 1 would make it very difficult to esti-
mate the quantiles corresponding to 3 z with any degree of precision.