= Negative log-scale counterexample to proportional hazards
For $\log T_z=bz-X$, with $e^X$ unit <exponential distribution> and $b=\log2$, the <survivor functions> are $S_z(t)=1-e^{-2^z/t}$ for $t>0$. These are time-scaled <Fréchet distributions>, so they form an <accelerated life family>. However their <hazard ratio> is $2/(e^{1/t}+1)$, tending to zero as $t\downarrow0$ and to one as $t\to\infty$. It is not constant. Thus a positive-scale assumption matters when deriving a <proportional hazards family> from the usual log-location-scale model.
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