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

Figure 6. Plot of Beta distribution min shape factor for given cv.

Figure 7.

Formula for minshape obtained using Mathematica.

From the curve we know when CV = 1.28, the minimal shape factor is 1.88, larger than 1.83 of NATH. In the best effort to match the input, we may elect to relax CV, for example, to 1.3, then the minimum shape factor is 1.85. With the constraint of a given CV, the minimum shape factor of the Beta Distribution may be significantly larger than its global minimum 1, so that it cannot attain to the wanted SF value.

Figure 8.

Contour plot of Beta distribution β parameter. The horizontal axis is the skewness and the vertical axis is the kurtosis.

#### 3.2 Shape factor range for given skewness

By solving Beta distribution parameters α and β through skewness sk and kurtosis kt, and examining the contour plot of β, we can see the allowable region is bound by two parabolas, Figure 8.

For a fixed skewness, α is monotonic increasing with respect to kurtosis; on the other hand, β has a singular point in some kurtosis, below that kurtosis is positive and monotonic increasing(in the region where α is positive), Figure 9.

Solving for that singular point we get the permissible kurtosis upper bound <sup>3</sup> <sup>þ</sup> <sup>3</sup> sk<sup>2</sup> , and solve for <sup>β</sup> <sup>¼</sup> <sup>0</sup> get the permissible kurtosis lower bound <sup>1</sup> <sup>þ</sup> sk<sup>2</sup> . <sup>2</sup>

Observe that the upper bound is when β turns to infinity, we can also get a simpler derivation of the upper bound by representing skewness and shape factor in α and β, letting β ! ∞, and then eliminating α to get shape factor as a function of skewness (Mathematica cannot solve equation for skewness which includes square root expression, we get around that by solving equation for the square of skewness, and then abandoning the negative solution introduced by this square).

A third way of more tedious calculation is through solving α by skewness and β, substituting the real solution into shape factor, and then take the limit for β ! ∞.

All three methods get the same upper bound of SF <sup>¼</sup> <sup>3</sup> <sup>þ</sup> <sup>3</sup> <sup>2</sup> sk<sup>2</sup> :

Figure 9. Plots of Beta distribution β parameter and α parameter vs. kurtosis for a given skewness 5.99378.

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

So for Beta distribution, the allowable region of skewness and kurtosis is bound below by kurtosis = skewness^2 + 1 where β ! 0, and above by kurtosis = 3 + 1.5\*skewness^2 where β ! ∞:

$$\mathbf{1} + \frac{1}{\mathbf{S}^2} \le \mathbf{S} \mathbf{F} \le \mathbf{1}. \mathbf{5} + \frac{\mathbf{3}}{\mathbf{S}^2}. \tag{5}$$

For the given skewness of 5.99378 of NATH, the maximum allowable kurtosis is 56.88813, less than the wanted 65.8902. So NATH cannot be fitted by any affine transformation of Beta distribution, certifying NATH as a trying case for distribution fitting. We will use it to test many of the well-known distributions in later sections. We also see surprisingly that unlike many of the other distribution families whose shape factors are too high, the Beta distributions have the shape factor range too low, or too close to 1. This suggests us to search for distributions with shape factors ranges in between.
