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

Figure 1. xn <sup>1</sup>þ<sup>n</sup> SF SF<sup>1</sup> SF<sup>2</sup> SF3 2½ � plot of exponential distribution <sup>e</sup>� nx� . The horizontal axis is the order of the exponential and the vertical axis is shape factors values.

An intuitive reason for why using shape factor SF in favor of skewness and kurtosis alone is provided by studying the simple example power distribution fam-<sup>1</sup> ily with PDF <sup>n</sup><sup>þ</sup> xn 1 , x∈ ½0; 1�, n , � 1∥n > 0 (or BetaDistribution½1=n þ 1; 1�). This <sup>n</sup> distribution family has the largest value of skewness and kurtosis, and at the same time the smallest shape factor SF when n turns to �1, where the PDF is the steepest, but the skewness and kurtosis take the indistinguishable value of infinity. In comparison, the shape factor SF takes the finite and distribution family specific value of 1.125. The shape factor SF thus makes meaning out of the meaningless infinities.

## 2.2 Alternative way of defining shape factor for symmetric distribution

For symmetric distribution, CM½ �¼ 3 0, our SF will be indiscriminately infinity. We can now employ SF1 in place of SF. Other measures from ACM such as SF2 and SF3 may also be candidates. From the SF3 plot Figure 2 of NormalDistribution½μ; σ� we see that min SF3½�¼ r 0:919824. The lower the value of SF3½ � 2 , the higher the 0 , r , 1 min SF3½ �<sup>r</sup> . We can use either SF3 2½ � or min SF3½ �<sup>r</sup> as <sup>a</sup> shape factor for symmet- <sup>0</sup> , <sup>r</sup> , <sup>1</sup> <sup>0</sup> , <sup>r</sup> , <sup>1</sup> ric distribution to describe the convexity of the ACM curve. The second measure

Figure 2. SF3 plot of Normal distribution. The horizontal axis is the order r of the power and the vertical axis is SF3½ �r .

has the merit of independence to the power order r, by additional efforts of numerical minimization. For our power distribution family, the maximum of the minimum is: max min SF3½r� ¼ <sup>0</sup>:942085, higher than the Normal distribution family. <sup>n</sup> > 0 <sup>0</sup> , <sup>r</sup> , <sup>1</sup>

When all SF, SF1, and SF2 are available, however, we will prefer SF to SF1 and SF2 since its dependency on parameters show simpler patterns than the other two; this can be shown from their contour plots for Beta distribution Figures 3–5, where SF contours are almost lines.
