= Weibull accelerated-life and proportional-hazards families
{c}
{title2=$S_z(t)=\exp(-\lambda_z t^k),\quad k>0$}
<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 $e^{-bz/c}$.
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