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


#### Figure 14. Derivation of Kumaraswamy distribution skewness upper bound for given α.

Figure 15.

Derivation of Kumaraswamy distribution kurtosis upper bound for given α and shape factor boundary value for given α when β ! ∞.

factor will converge to its minimum. We use Mathematica to calculate the asymptotic expansion of the Gamma function and the quotient of Gamma function at infinity for orders up to 4 or 2, take the symbolic limit for β ! ∞, to get these boundary values, Figures 14 and 15.

We thus have a simple formula for boundary value of Kumaraswamy distribution shape factor:

$$\underset{\beta \to \infty}{\text{limit } S} = \frac{2 \text{Gamma} \text{ma} \text{ma} \begin{bmatrix} \frac{1}{a} \end{bmatrix}^{\beta} - 6a \text{Gamma} \text{ma} \begin{bmatrix} \frac{1}{a} \end{bmatrix} \text{Gamma} \text{ma} \begin{bmatrix} \frac{2}{a} \end{bmatrix} + 3a^2 \text{Gamma} \text{ma} \begin{bmatrix} \frac{2}{a} \end{bmatrix}}{\left( a \left( -a \text{Gamma} \text{ma} \left[ 1 + \frac{1}{a} \right]^2 + 2 \text{Gamma} \text{ma} \left[ \frac{2}{a} \right] \right) \right)^{3/2}},\tag{6}$$

˜ °˛ ˜ ° ˜ ° ˜ ° ˜ °˝ ˜ ° <sup>3</sup> �3Gamma <sup>1</sup> Gamma <sup>1</sup> � <sup>4</sup>αGamma <sup>1</sup> Gamma <sup>2</sup> <sup>þ</sup> <sup>4</sup>α2Gamma <sup>3</sup> <sup>þ</sup> <sup>α</sup>4Gamma <sup>4</sup>þ<sup>α</sup> <sup>α</sup> <sup>α</sup> <sup>α</sup> <sup>α</sup> <sup>α</sup> <sup>α</sup> limit <sup>K</sup> <sup>¼</sup> ˜ °<sup>4</sup> ˜ °<sup>2</sup> ˜ ° ˜ °<sup>2</sup> , <sup>β</sup>!<sup>∞</sup> Gamma <sup>1</sup> � <sup>4</sup>αGamma <sup>1</sup> Gamma <sup>2</sup> <sup>þ</sup> <sup>α</sup>4Gamma <sup>2</sup>þ<sup>α</sup> α α α α (7) <sup>K</sup> limit <sup>¼</sup> <sup>β</sup>!<sup>∞</sup> <sup>S</sup><sup>2</sup> ˛ ˝ ˛ ˛ ˝ ˝ ˜ ° ˜ ° <sup>3</sup> ˜ ° ˜ ° ˜° ˜° ˜ ° ˜ ° <sup>2</sup> <sup>3</sup> <sup>α</sup><sup>3</sup> �αGamma <sup>1</sup> <sup>þ</sup> <sup>1</sup> <sup>þ</sup> 2Gamma <sup>2</sup> �3Gamma <sup>1</sup> Gamma <sup>1</sup> � <sup>4</sup>αGamma <sup>1</sup> Gamma <sup>2</sup> <sup>þ</sup> <sup>4</sup>α2Gamma <sup>3</sup> <sup>þ</sup> <sup>α</sup>4Gamma <sup>4</sup>þ<sup>α</sup> α α α α α α α α ˛ ˝ ˛ ˝ : ˜ ° ˜ ° ˜ ° ˜ ° <sup>2</sup> ˜ ° ˜° ˜° ˜ ° <sup>3</sup> <sup>4</sup> <sup>2</sup> <sup>2</sup> 2Gamma <sup>1</sup> � <sup>6</sup>αGamma <sup>1</sup> Gamma <sup>2</sup> <sup>þ</sup> <sup>3</sup>α2Gamma <sup>3</sup> Gamma <sup>1</sup> � <sup>4</sup>αGamma <sup>1</sup> Gamma <sup>2</sup> <sup>þ</sup> <sup>α</sup>4Gamma <sup>2</sup>þ<sup>α</sup> α α α α α α α α (8)

Its plot Figure 16 has two branches, the dividing point is α ! 3:602349425719043 where the skewness is zero, and below it is mainly the positive skewness region while above it is the negative skewness region.

The minimum value at the left branch of Figure 16 is 1.91227 and arrived at α = 0.641149. When α > 1000 the numerical value for that boundary can be negative and is thus unreliable. The value 1.91227 is not the global minimum of the shape factor: for α = 0.641149 the shape factor plot Figure 17 with respect to β decreases first, at the point 10.6095 arriving at the minimum value of 1.80935, and increasing after the point 10.6095.

In principle, the extreme value of the shape factor for a given skewness will arrive either at the upper boundary where β ! ∞ or at the lower boundary where

Figure 16.

Plot of Kumaraswamy distribution shape factor boundary value for given α when β ! ∞.

Figure 17. Plot of Kumaraswamy distribution shape factor for given α = 0.641149, β in the range 0.3–1 and 1–300.

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

Figure 18. Plot of Eq. (6)–(8) and plot of Kumaraswamy distribution maximum shape factor for given skewness.

α ! 0, or at some middle point where the contour plot of the skewness and the contour plot of the kurtosis will be tangent to each other. The Mathematica contour plot does not work for a very small α, but by numerical minimization we know the global minimum of the Kumaraswamy distribution shape factor is 1.03709 when <sup>α</sup> <sup>¼</sup> <sup>1</sup>:80143∗10�<sup>9</sup>, <sup>β</sup> <sup>¼</sup> <sup>0</sup>:247044. The conditional minimum of the shape factor when skewness <sup>=</sup> 5.99378 is about 1.04753 when <sup>α</sup> <sup>¼</sup> <sup>10</sup>�10:<sup>5</sup> , β ¼ 0:149286 through list calculation; this is higher than 1 + 1/S^2 = 1 + 1/5.99378199789956^2 = 1.02784, the lower boundary of Beta distribution.

The Mathematica contour plot works for large α, and we see the shape factor is increasing along the contour of skewness, which attains its maximum when β ! ∞. For example, for NATH skewness 5.99378199789956, the maximum shape factor is 1.97131, arriving at α = 0.5239510562868946. The maximum shape factor of Kumaraswamy distribution for given skewness is in Figure 18, which is algebraically represented by the parametric curve of Eq. (6) and Eq. (8).

So the permissible shape factor range of the Kumaraswamy distribution still spans the lower end of the whole allowable range of (1,∞), but higher than that of the Beta distribution. Affine transformed Kumaraswamy distribution can fit all the first four moments of NATH, with the fitted distribution TVaR close to NATH TVaR in the error range of 5–6%, while the best effort affine transformed Beta distribution is in the error range of 9–10%.

To further improve the fit, we need additional freedom in parameters, such as the GB1 distribution [15], since

KumaraswamyDistribution½α; β�≈GeneralizedBetaDistributionI½1; β; α; 1�, and the maximum shape factor plot in Figure 18 is lower than that of LogNormalDistribution, the upper bound of GB1. The following section will study a sibling distribution to GB1, fitted as good as GB1, but is more widely known.
