5.1 Minimum shape factor for GB2

The skewness and kurtosis matching problem for GB2 is very sensitive to the initial parameter ranges given. A study of the minimum shape factor of GB2 with

Figure 21. GB2 distribution skewness kurtosis and shape factor vs. α or z, vs. p or w plots for fixed y = q-4/α.

Figure 22. The numerical minimum GB2 shape factor for given p in horizontal axis.

respect to each parameter will give us permissible ranges for those parameters. Direct work with shape factor encounters problems from Mathematica's numerical optimization function NMinimize, minimizing the log shape factor instead can overcome this difficulty. The plot is in Figure 22.

In the range (0.0001, 5.0) of p, the numerical minimum shape factor plot of GB2 is a very smooth curve. The fitted formula of GB2 min SF for given p by Mathematica's machine learning function FindFormula is Eq. (9).

$$\min \frac{K}{S^2} = 1.1593871374775397 + 1.4702458297305288 \ast 0.51484991588003610^{\frac{1}{0.0295050250000}}\tag{9}$$

As a test, for NATH the log shape factor is Log½1:83408� ¼ 0:60654412, the solution of Eq. (9) for p with NATH SF is p ¼ 0:608342; the minimum log shape factor of GB2 for this p is 0.60603997, only 0.08% smaller than input.

From the contour plot Figure 20 we know for given α, the shape factor of GB2 has two singular points with p or 10<sup>w</sup>. The minimization for given α needs to carry out in each of the three regions cut by these two singular points. The plot is in Figure <sup>23</sup>. With <sup>a</sup> new parameterization, <sup>p</sup> <sup>¼</sup> <sup>λ</sup> , q <sup>¼</sup> <sup>4</sup>þ<sup>ν</sup> , the minimization of shape α α

#### Figure 23.

The numerical minimum GB2 shape factor for given α or given pα in horizontal axis.

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

factor for GB2, for given λ ¼ pα, is easier to perform. The plot is included in Figure 23 as well.

Figures 22 and 23 show that the permissible parameters for NATH are p , 0:63, α > 0:5, pα , 0:5: This is confirmed by GB2 fit practice. The best fit by GB2 for NATH is at w ¼ �0:329075005, p ¼ 0:468732, with about 5% error from input TVaR. The discontinuity of fitted GB2 TVaR with respect to parameter change is also observed, this w value is such a critical point.
