254 Optimizing Optimization
S B and S U distributions. Although a full treatment is beyond the scope of this
chapter, we nevertheless note a number of their more important features. 9 First,
both are symmetric about their respective means if and only if γ 0 and they
exhibit negative (positive) skewness if γ 0 ( γ 0). Moreover, for a fixed
γ , kurtosis is increasing in η. Second, the S U distribution is unimodal for all
parameter values, whereas the S B distribution may be either unimodal or bimo-
dal depending on the choice of γ and η. Finally, we note that for both S U and
S B , the density f ( x ) and all of its derivatives tend to zero as x tends to extreme
values in its support Ω. This means that the density is a perfectly smooth, infi-
nitely differentiable, function of x for all real values of x.
11.3.2 Density estimation
Aside from providing an insight into the flexibility of the Johnson system,
Figure 11.1 has a more practical application in that it is often the initial stage
for many of the popular estimation methodologies used to fit Johnson densities
( Johnson, 1949 ). For instance, Hill, Hill, and Holder (1976) use estimates of
the sample skewness and kurtosis to identify the appropriate functional form
of the Johnson density before implementing their method of moments proce-
dure. Unfortunately however, this approach is infeasible for our portfolio opti-
mization problem because the parameter estimates can only be obtained via
computationally demanding recursive formulae.
With this in mind, we choose an alternative estimation procedure based on
the method of quantile estimation introduced by Slifker and Shapiro (1980) ;
see also Wheeler (1980). This method is not only easier to implement than the
method of moments estimator, but simulation studies have also shown that it
produces superior estimates of the parameters for the S B and S U distributions;
see Wheeler (1980, p. 727).^10 The theoretical justification for this approach
is that each member of the Johnson system can be uniquely identified by the
distances between the tails and in the central portions of the distribution, much
in the same way that they can also be indentified by their skewness – kurtosis
values. However, in this case, computationally convenient formulae for each
of the parameters are readily available and so estimation is much quicker than
alternative approaches.
Mathematically , the Slifker and Shapiro (1980) algorithm begins by fixing
a number z 0 which implicitly defines three intervals of equivalent distance
between the values 3 z , z , z , and 3 z. After choosing z , the next step is to
find the cumulative probabilities of the normal distribution corresponding to
each of the selected z values. For example, if 3 z 1.645, then Φ (3 z ) 0.95.
After obtaining the probabilities for each of the four points, 3 z , z , z , and
10 The algorithms proposed by Slifker and Shapiro (1980) and Wheeler (1980) are almost identi-
cal, aside from the fact that the former uses four quantile points whereas the latter uses five.
9 We refer the reader to Stuart and Ord (1994) or Hahn and Shapiro (1967) for a more compre-
hensive treatment.