5.2.2. Weibull

It can be verified that the ccdf *FX*(*x*) = *e*<sup>−</sup>(*u*/*a*)*<sup>c</sup>* with fixed shape parameter *c* = 1/2 arises as a c.m. distribution (Jewell 1982), where the mixing measure (measure of the spectral function) *S* is given by

$$dS(y) = \frac{e^{-\frac{1}{4\alpha y}}}{2\sqrt{a\pi y^3}}dy.$$

Similarly, we can find using Proposition 1 that

$$dS\_H(y) = \frac{1}{\phi c^2} \left( \frac{\nu\_1 \nu\_2 - c\rho\_1(\theta \nu\_1 + (1-\theta)\nu\_2)}{-(y+\rho\_1)\rho\_1} + \frac{\nu\_1 \nu\_2}{y\rho\_1} \right) \frac{e^{-\frac{1}{4\pi y}}}{2\sqrt{a\pi y^3}} dy.$$
