3.1 Minimum shape factor for given CV

Using Mathematica, we can solve the parameter α and β by cv and std for BetaDistribution½α; β�:

$$\rho \rightarrow \frac{\mathbf{c}\mathbf{v} - \mathbf{s}\mathbf{t}\mathbf{d} - \mathbf{c}\mathbf{v}^2 \mathbf{s}\mathbf{t}\mathbf{d}}{\mathbf{c}\mathbf{v}^3}, \beta \rightarrow \frac{\mathbf{c}\mathbf{v}^2 - 2\mathbf{c}\mathbf{v}\mathbf{s}\mathbf{t}\mathbf{d} - \mathbf{c}\mathbf{v}^3 \mathbf{s}\mathbf{t}\mathbf{d} + \mathbf{s}\mathbf{t}\mathbf{d}^2 + \mathbf{c}\mathbf{v}^2 \mathbf{s}\mathbf{t}\mathbf{d}^2}{\mathbf{c}\mathbf{v}^3 \mathbf{s}\mathbf{t}\mathbf{d}}.$$

Since α > 0, we must have:

$$\text{std} < \frac{\mathbf{cv}}{1 + \mathbf{cv}^2},$$

or

$$\frac{1-\sqrt{1-4\text{std}^2}}{2\text{std}} < \text{cv} < \frac{1+\sqrt{1-4\text{std}^2}}{2\text{std}}.$$

We also know std must be between 0 and 0.5 for these solutions to exist. By computer-aided exploration through contour plot, we can find the location of the std where SF takes minimum for a given cv.

The overall observation is that when cv < 1, SF approaches infinity in the middle value of std, and decreases when deviating from it. When cv > 1, SF approaches its minimum in the middle value of std and increases when departing. Together with the fact that std has an allowable upper bound of cv/(1 + cv^2) and lower bound of 0, the minimum of SF must be attained either at the global extreme where the derivative of SF with respect to std is zero or at the two boundaries when cv > 1, and attained at the two boundaries when cv < 1.

Using Mathematica to take the derivative of the shape factor with respect to std to find the std where shape factor attained extreme values, and solving it for the intersection with std upper and lower bound, we know the minimal shape factor for Beta distribution for a given CV when CV is below 0.707107 or above 2.48239 cv (intersecting std upper bound) is attained at std upper bound with value: <sup>1</sup>þcv2

$$\min\_{10 \le \varepsilon dt \le \frac{cv}{1 + \sigma^2}} \text{SF} = \frac{1 - cv^2 + cv^4}{\left(-1 + c\nu^2\right)^2}, \\
when \text{ } cv \le 0.707107 \|cv > 2.48239. \tag{3}$$

When CV is between 0.707107 and 1.024766 (intersecting std lower bound) the minimal shape factor is attained at std lower bound 0 with value:

$$\min\_{0 \le std < \frac{cv}{1 + v^2}} \text{SF} = \mathbf{1.5} + \frac{\mathbf{0.75}}{cv^2}, \\ \text{when } \mathbf{0.707107} \le cv \le \mathbf{1.024766}. \tag{4}$$

When CV is between 1.024766 and 2.48239, the minimum SF is attained at std that is the zero derivative points of the shape factor. The piecewise curve plot of the minimum SF for given CV is in Figure 6. The formula for the central piece, minshape, is given in Figure 7 which is too complex for manual derivation without the aid of computer algebra system.
