6. Generalized gamma distribution

The generalized gamma distribution in Mathematica is the Amoroso distribution [16], with the parameter correspondence: α \$ α, β \$ θ, γ \$ β, μ \$ a:

For generalized gamma distribution GammaDistribution½α; β; γ; μ�, the shape factor depends only on α and γ. It seems the smaller the α, and the bigger the γ, the smaller the <sup>S</sup> K 2. When <sup>α</sup> <sup>¼</sup> <sup>3</sup>:<sup>318512677036329</sup> � <sup>10</sup>�<sup>12</sup>, <sup>γ</sup> <sup>¼</sup> <sup>8811</sup>:572418686921, <sup>K</sup> <sup>¼</sup> <sup>1</sup>:125, close to the global minimum <sup>1</sup> of K/S^2. <sup>S</sup><sup>2</sup>

So there arises the question: the generalized gamma and GB2 can match smaller shape factors than Hyperbolic Distribution (Section 8), why they cannot fit as good as the latter for NATH with shape factor 1.83409?

One explanation is that the numerical solution for GB2 or generalized Gamma distribution is trapped in the shape factor curve right branch by the combined constraints of skewness and kurtosis, which is not the branch that can attain 1.125, unlike the generalized hyperbolic distribution whose shape factor has a global minimum in λ = 0.
