Suppose I have the following accelerated failure time model:
$t = \exp(-\beta t)u$
where the c.d.f of $u$, $F_u(x) = \dfrac{x}{1+x}$.
I tried the following way:
$u = t\exp(\beta t) = \Psi(t)$, or $t = \Psi^{-1}(u)$
$F_t(x)=Pr(t \leq x) = Pr(\Psi^{-1}(u) \leq x) = Pr(u \leq \Psi(x))$
$F_t(x) = F_u(\Psi(x)) = \dfrac{\Psi(x)}{1+\Psi(x)}=\dfrac{x\exp(\beta x)}{1+x\exp(\beta x)}$
The p.d.f of $t$ is: $f_t(x)=\dfrac{(1+\beta x)\exp(\beta x)}{(1+x\exp(\beta x))^2}$
The problem is, when $\beta$ is a negative parameter, $1+\beta x <0$ is possible, and we may end up with a negative p.d.f!
I know there must be something wrong with my procedure, just don't know where.