7. Fleishman distribution

We guess 1.5 is the lower bound of shape factor for most unbounded parametric distribution families. For example, for Fleishman distribution, by the empirical formula from [5], γ<sup>4</sup> > 1:738γ<sup>2</sup> <sup>3</sup> � 0:3544γ<sup>3</sup> þ 1:978, the minimum shape factor is 1.72213, larger than 1.5.

The lower bound of shape factor from unbounded distributions seems, in general, to be higher than bound distributions'. Outside of the latter's upper bound and near the former's lower bound, for a SF value slightly larger than 1.5, in practice, most parametric distributions have difficult matching both the kurtosis and skewness: the comparatively best one is selected for study in the next section.

## 8. Hyperbolic distribution

Taking a sequence of numerical minimization of the shape factor, for various values of fixed λ, we get the empirical minimum shape factor curve for generalized hyperbolic distribution (GH), HyperbolicDistribution½λ; α; β; δ; μ�, in Figure 24.

We observed that when λ > �0.6, the minimum shape factor is attained when α^2-β^2-> 0 and β-> 0, that is, it is attained by a skew hyperbolic t distribution [17–19]. When looking at the plot of shape factor with respect to λ, we feel that it must have some simple formula. So we utilize Mathematica symbolic calculation to expand the shape factor with asymptotic expansion for BesselK½λ; a�, or Kλð Þ a in [20], with respect to α^2-β^2 and then take the symbolic limit, Figure 25.


Figure 25. Derivation of the GH shape factor limit when λ > ˜2.

What Determines EP Curve Shape? DOI: http://dx.doi.org/10.5772/intechopen.82832

The semi-empirical formula for the minimum shape factor in this range thus obtained is very simple, Eq. (10–11), which has the global minimum of 1.5 when λ turns to 0.

$$\min\_{\alpha\_2, \beta\_2, \delta\_2, \mu} \text{SF} = \mathbf{1.5} + \mathbf{0.75}\lambda,\text{when }\lambda \ge \mathbf{0},\tag{10}$$

$$\min\_{\alpha,\beta,\delta,\delta,\mu} \text{SF} = \mathbf{1} + \frac{\mathbf{1}}{2 + \lambda}, \text{when } -0.6 \le \lambda \le 0 \tag{11}$$

(13)

When λ ≤ �0.65, however, the minimum shape factor is not attained when α^2 β^2-> 0. When λ is in the interval [�9,�0.65], the attainable smallest shape factor is between 3.15 and 1.74, with an empirical 10th order polynomial formula Eq. (12), or less accurately a mixed exponential and power function Eq. (13), found through the Mathematica FindFormula.

$$\begin{aligned} \min\_{a,b,b,\mu} SF &= 1.1130471668735116 - 1.6512030619809768\lambda - 1.6137376958833365\lambda^2 \\ &\quad - 1.148503817210114\lambda^3 - 0.542178585853132\lambda^4 \\ &\quad - 0.1709457883426522\lambda^5 - 0.03603744794749387\lambda^6 - 0.005000441043297472\lambda^7 \\ &\quad - 0.000437218954755793\lambda^3 - 0.000021791071048963054\lambda^9 \\ &\quad - 4.711954312790356 \times 10^{-7}\lambda^{10} \\\\ \min\_{a,b,\delta,\delta,\mu} SF &= 2.2104215691249425 - 0.6522131009473879\*1.6355318649123258^{\lambda} \\ &\quad + \frac{0.018965779149540653}{\lambda^3} - 0.1051542360603726\lambda \end{aligned} \tag{12}$$

So for each given K/S^2 value, there exists a permissible interval of λ, whose lower bound is calculated via Eq. (11–12) and upper bound is calculated via Eq. (10). When λ changes inside this interval, we noticed that the 0.99 TVaR of the first four moments matched generalized hyperbolic distribution will increase with respect to λ. If the lower bound still has 0.99 TVaR bigger than the input TVaR, then it is not possible to fit with moments matched HyperbolicDistribution. The opposite statement is also valid for the interval upper bound.

With this knowledge, the NATH permissible λ interval is [�0.8439,0.4454], and the left end point still have 0.99 TVaR larger than the input TVaR, but now only by 4.05%, better than the 5% error of GB2.

### 9. Conclusion and discussions

We proposed using the ratio of kurtosis by squared skewness as the best candidate for shape factor that can characterize the distribution asymmetry, as well as the PDF steepness. The closer this factor to 1, the more asymmetry and the steeper the PDF. The asymptotic approximation and symbolic limit is used to calculate the boundary of this factor for various distributions: the Beta, the Kumaraswamy, and the Hyperbolic Distributions, for example. This range information of the shape factor, with the surprisingly simple formulas in the three above examples (Eqs. 5–8, 10, 11), can be used to select or eliminate candidate distributions for fitting. The plot of the shape factor together with plot for skewness and kurtosis can aid in setting the initial value or parameter intervals when fitting distribution to data by numerical optimization, which usually would not work well without this information.

The idea of the shape factor and the careful study of each distribution for this shape factor is the preliminary for the numerical optimization that finally finds the best fit. The information provided by shape factor plot is rough but the numerical optimization's dependence on initial value or intervals is delicate, exemplified by GB2 case. The optimization function NMinimize and FindMinimum in Mathematica sometimes can only find a local optimum at best. As shown in [21, 22], the DyHF and the CMODE algorithms are the two best no-adjustment-needed global optimization algorithms. Now that the C<sup>2</sup> oDE algorithm is better than these two [23], it would be desirable to see how it works on the GB2 fit problem. With a foolproof universally applicable global optimization algorithm, the ado with shape factor and their boundaries will no longer be needed, or be used merely as some validations; but before that time, the hard earned knowledge about shape factor through CAS is still indispensable. This is a good topic for subsequent research.

Our shape factor idea is only a small step ahead of the skewness-kurtosis plot of Pearson [6] and McDonald et al [15, 24–26]. Or we just made the idea implicitly in their plot explicit. But with this clearly defined form, anyone can readily start calculating it for any interested candidate distribution.

Our formula Eq. (5) is not new, since Beta distribution has the same range of skewness, kurtosis, and shape factor as the scaled Beta distribution, the B1 distribution in [15]. Our presentation is an example of how our method can be used to easily arrive at those formulas. Theoretically equivalent expressions are not equivalent in application. With data distributions usually not having small skewness, Eq. (5) says that the Beta distribution has a shape factor roughly in the range of (1, 1.5), this not only reveals an intrinsic property of Beta distribution, but is also more easily applicable in practice than the skewness-kurtosis plot.

The residual error of all the distributions tested so far indicates that the power function or simple exponential function PDF is not enough to provide the additional freedom of shifting for the EP curve on the condition of matched first four moments. Other forms such as mixtures, combinations, or transformations of distributions may need to be considered. A previous study indicated the following transformations are good candidates [4, 27–32]: EWGU, KGG, EG, EWED, LIG, THT. Further research will be done along these lines.
