= Solution
For <administrative censoring with uniform entry>, let entry time $A\sim\operatorname{Uniform}(\tau_a,\tau_b)$. Then $C=\tau_c-A$ has the <uniform distribution> on $[\ell,r]$, where $\ell=\tau_c-\tau_b$, $r=\tau_c-\tau_a$, and $d=r-\ell=\tau_b-\tau_a$. \b[Its density and survivor function] are
$$
\boxed{g_C(t)=\begin{cases}1/d,&\ell<t<r,\\0,&\text{otherwise},\end{cases}\qquad G(t)=\begin{cases}1,&0\le t<\ell,\\(r-t)/d,&\ell\le t<r,\\0,&t\ge r.\end{cases}}
$$
Dividing density by survival and integrating gives \b[the censoring hazard and integrated hazard]:
$$
\boxed{h_C(t)=\begin{cases}0,&0\le t<\ell,\\1/(r-t),&\ell\le t<r,\end{cases}\qquad H_C(t)=\begin{cases}0,&0\le t<\ell,\\\log\frac{d}{r-t},&\ell\le t<r,\\+\infty,&t\ge r.\end{cases}}
$$
The <hazard function> is undefined once no uncensored individuals remain. As $t\uparrow r$, it diverges like $1/(r-t)$ and the <cumulative hazard function> diverges logarithmically. This reflects the certainty of administrative censoring by the longest possible follow-up time, not dependence of censoring on the event process.
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