= Solution
For a proper continuous event time, the <survival function> is $S(t)=e^{-H(t)}$. The assumed invertibility of the <cumulative hazard function> gives, for $u\ge0$,
$$
P\{H(T)>u\}=P\{T>H^{-1}(u)\}=e^{-H(H^{-1}(u))}=e^{-u}.
$$
Therefore
$$
\boxed{U=H(T)\sim\operatorname{Exp}(1).}
$$
This is the <cumulative hazard probability transformation>; it applies also conditionally on a subject's covariates, using that subject's correctly specified <cumulative hazard function>.
The fitted transformed times are <Cox–Snell residuals>. They retain their event/<right censoring> indicators, so a <right-censored> residual represents an exponential observation known only to exceed its displayed value. Under the fitted model and independent <right censoring>, calculate the <Kaplan–Meier estimator> of residual survival and compare it with $e^{-y}$, or calculate the residual <Nelson–Aalen estimator> and compare its <cumulative hazard> with the diagonal $H(y)=y$. Systematic departures reveal model inadequacy; sparse extreme residual <risk sets> and parameter estimation require caution. Treating all censored residuals as observed event times would invalidate this diagnostic.
Without <right censoring>, the residual mean should be approximately one. With <right censoring>, use the <modified Cox–Snell residual>
$$
\boxed{y_i^*=y_i+(1-v_i).}
$$
For a true unit-rate <exponential distribution>, the <memoryless property> gives $E(U\mid U>c)=c+1$. Thus an event keeps its known transformed time, while a censored observation is replaced by the conditional expected event time. Under independent <right censoring>, iterated <expectation> makes the mean of these adjusted values one when the true hazards are used, and approximately one when fitted hazards are used. These mean-imputed values do not themselves have an exponential distribution, so the survival-curve diagnostic should still use the original censored residual dataset. Moreover fitting equations can force the adjusted sample mean to one, making its mean alone a weak diagnostic.
For the proposed mixture of a finite <right censoring> time $c\ge0$ and no <right censoring>, $P(C<U)=\pi e^{-c}$ and
$$
E\{\min(U,C)\}=(1-\pi)E(U)+\pi\int_0^cP(U>u)\,du=1-\pi e^{-c}.
$$
Consequently
$$
\boxed{E(U^*)=1+(k-1)\pi e^{-c},\qquad k=1.}
$$
When <right censoring> has positive <probability> this choice is unique; if $\pi e^{-c}=0$, no correction is needed and any $k$ has the same effect. Its independence from $c$ is the content of exponential memorylessness: the expected extra lifetime after any <right censoring> time is one. Conditioning on an arbitrary independent <right censoring> time proves the same correction beyond this special two-point mixture.
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