5. BetaPrime distribution

Beta distribution and Kumaraswamy distribution are a few exceptions which have analytical formulas for the shape factor bounds; for other distributions to be studied, numerical optimization and empirical plot or formula will be the only feasible approach.

� � Transformation of Beta Distribution by x/(1-x) is the GB2([15]), or BetaPrimeDistribution½ ¼<sup>p</sup> <sup>α</sup>; <sup>q</sup> <sup>¼</sup> <sup>β</sup>; <sup>α</sup> <sup>¼</sup> <sup>1</sup>; <sup>β</sup> <sup>¼</sup> <sup>1</sup>� ([11]): TransformedDistribution <sup>x</sup> ; <sup>x</sup>≈BetaDistribution½α; <sup>β</sup>� <sup>≈</sup>BetaPrimeDistribution½α; <sup>β</sup>�. The minimum shape <sup>1</sup>�<sup>x</sup> factor of Beta Distribution is 1, but that of the transformed is 1.5:

$$\begin{aligned} \text{NMimize}\left[\left\{\frac{3(-3+\beta)\left(2(-1+\beta)^2 + a^2(5+\beta) + a(-1+\beta)(5+\beta)\right)}{4(-4+\beta)(-1+2a+\beta)^2}, a>0, \beta>4\right\}, \\ \{a,\beta\} &= \left\{1.5000000239052607, \left\{a \to 0, \beta \to 6.274769836372949 \times 10^7\right\}\right\}.\end{aligned}$$

Empirically, the larger the third parameter α, the smaller the minimum shape factor. The smallest shape factor we get of the BetaPrimeDistribution is 1.125, when α = 446.49537:

$$\begin{aligned} \text{Find Minimum} \left[ \left\{ \begin{aligned} \text{Kurtosis} [\text{BetaTimeDistribution} [p, q, \alpha, \beta]] \\ \text{Skewness} [\text{BetaTimeDistribution} [p, q, \alpha, \beta]] \end{aligned} \right. \end{aligned} \right. \left. \begin{aligned} \text{ $p \to $ \frac{1}{x} $} / \text{$ q \to  $4 \text{ x}$ } \\ \text{ $x \to -10^{\circ}$ , } \text{ $x \to 10^{\circ}$ , } \text{ $p > 0, \,\_{y}$ } \text{ $1, } -4. < z < -1 \text{.} \text{} \text{.} \text{} \text{ ($ p, 6.384125235007732 \times 10^{-10} $)} \text{.} \\ \text{$ (p, 1.0032844709998097) $, } \text{$ (z, -2.157370895027263) $} \text{), } \text{MaxIterations } \rightarrow 5000 \text{ } \text{$  } \\ \text{ $(-1.1250258984236121, } \text{$ (p \to 2.083731454230264 \times 10^{-8} $, } \\ \text{$ y \to 42816363091057056, } \text{ $z \to -2.6498169598310573$ } \text{)} \text{.} \end{aligned} \right. \end{aligned}$$

This is the same value as the minimum shape factor for

GammaDistribution½α; β; γ; μ� (in Section 6). When α > 10,000, the Gamma function involved will not calculate or will calculate incorrectly.

With the transformation of p-> 10^w, α-> 10^-z, q-> 4\*10^z + y, we can study the GB2 shape factor change tendency with respect to α, Figure 19, and shape factor change tendency with respect to p, Figure 20.

The GB2 shape factor is mainly determined by α and p, only slightly changing with respect to q when q is smaller than 5. The change with respect to α and p is similar, having two peaks, or three peaks if we regard the two sides of the infinity as

Figure 19. GB2 distribution shape factor vs. α for fixed p = 10^-3.312 = 0.000487528.

Figure 20. GB2 distribution shape factor vs. p for fixed α = 10^2.6498169598310573 = 446.495.

two branches since that border is not easily crossable for searching or optimization algorithms.

GB2 shape factor's dependency with p and α, or w and z through transformation p = 10^w, α = 10^-z, is mostly unaffected by q except for right-most values of z. They are μ-shaped (Figure 21), this is different from Hyperbolic Distribution (in Section 8), whose shape factor dependency with λ is V-shaped. We guess V-shaped curves have unique global minimums, but μ-shaped curves will show bifurcation behavior: the converged solution in optimization will be very different when the initial point or interval is slightly different.

The knowledge that the shape factor curve attained extreme values in ˜3.3,-1.25 and 1 with respect to z, and attained extreme values in ˜2.65, ˜1.11 and 1 with respect to w, can be used to set the initial interval, the paramount factor determining the quality of the numerical optimization solution, for solving the GB2 fitting problem.
