Just wondering why in the literature Weibull distribution is always defined for positive shapes, whereas the extension in the negative direction is possible and has many useful properties.
Suppose $X \propto \mathrm{Weibull}(\theta, \lambda)$, i.e. is Weibull-distributed with shape $\theta > 0$ and scale $\lambda > 0$ with PDF
$$
p_X(x) = \frac{\theta}{\lambda}\left(\frac{x}{\lambda}\right)^{\theta-1}e^{-\left(x/\lambda\right)^\theta}, \quad x > 0.
$$
$Y=1/X$ then follows the inverse Weibull distribution with PDF
$$
p_{Y}(y) = \left|\left(1/y\right)'_y\right| p_X\left(1/y\right) =
\frac{\theta}{\lambda y^2} \left(\frac{1}{\lambda y} \right)^{\theta-1} e^{-\left(1/\lambda y\right)^\theta} =
\frac{\theta}{\lambda^{-1}} \left(\frac{y}{\lambda^{-1}}\right)^{-\theta-1} e^{-\left(y/\lambda^{-1} \right)^{-\theta}}.
$$
I.e. if negative shapes would be allowed, we could say $Y \propto \mathrm{Weibull}(-\theta,\lambda^{-1})$ (just $\theta$ in the front of $p_X(x)$ has to be replaced with $|\theta|$). CDF/mean/mode etc also require very minor adaptation for the negative shapes.
I guess the same trick is possible for the whole generalized Gamma family, and e.g. inverse Gamma would then become its member.
Best Answer
There is no good reason not to do such a generalization, which would unite the Weibull and inverse Weibull distributions. So reasons must be historical or accidental.
Also see the comments thread.