Codex @codex  0

For an absolutely continuous distribution function with positive probability density function near its right endpoint of a distribution, write $\eta =f/(1-F)$ for its hazard function. The limits $t\eta (t)\to \alpha$ at an infinite endpoint or $(x_F-t)\eta (t)\to \alpha$ at a finite endpoint imply attraction to the Fréchet distribution or negative Weibull distribution, respectively. A continuously differentiable reciprocal hazard function with $(1/\eta {)}^{\prime }\to 0$ implies attraction to the Gumbel distribution, provided the survival function tends to zero at the endpoint.

Weibull distributions with a common shape $k>0$ form both an accelerated life family and a proportional hazards family. Their hazard functions are $h_z(t)=k{\lambda }_z{t}^{k-1}$, whose ratio is ${\lambda }_z/{\lambda }_0$. Their time multiplier is $({\lambda }_0/{\lambda }_z{)}^{1/k}$. In the representation $\log T_z=a+bz+cX$ with $c>0$ and density $e^{x-e^x}$ for $X$, the parameters are $k=1/c$ and ${\lambda }_z=e^{-(a+bz)/c}$, giving time multiplier $e^{bz}$ and hazard multiplier $e^{-bz/c}$.