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Under independent censoring and a constant event rate, $d$ observed events over total observed person-time $X$ give likelihood proportional to ${\lambda }^d{e}^{-\lambda X}$ and estimate $d/X$ for $d,X>0$. Censored observations belong in $X$. With no events the positive-rate likelihood has a boundary supremum.

Let $T_i$ be an event time and $C_i$ a potential censoring time. Under independent censoring, observe $Y_i=\min (T_i,C_i)$ and ${\delta }_i=1\{T_i\le C_i\}$. For an exponential distribution of rate $\theta$, an observed failure contributes $\theta e^{-\theta Y_i}$ to the survival likelihood, whereas a right-censored observation contributes $e^{-\theta Y_i}$. The censoring-law factors may be discarded when they are independent of $\theta$. Thus, writing $d={\sum }_i{\delta }_i$ and $V={\sum }_i{Y}_i$ for total observed person-time,For $d,V>0$ the unique maximum-likelihood estimate isCensored times belong in $V$: they record time during which failure was not observed. With $d=0$ and $V>0$, the likelihood decreases with $\theta$, so the supremum is at the boundary $\theta \downarrow 0$; there is no positive interior estimate. This is exponential-rate estimation from censored exposure.

Uninformative censoring means that the potential censoring time supplies no information about the latent failure time after conditioning on the modeled variables. A study ending at a fixed follow-up duration independent of prognosis is an example of administrative censoring. Informative censoring occurs, for example, when a seriously deteriorating patient withdraws because of that deterioration and the prognostic information is omitted from the model. Under informative censoring, those remaining in the risk set are selected by their future failure propensity, so the ordinary survival likelihood and its estimate need not target the true rate.

Using the reported durations and flags as though they were valid right-censored records givesTreating all durations as failures instead givesRelabelling has halved the event count without shortening any observed follow-up. The resulting estimate is therefore exactly half the complete-data estimate.

This is not a simulation of uninformative ordinary right-censoring. For a record declared censored at $Y_i$, ordinary right censoring requires $T_i>Y_i$; the construction instead took $Y_i$ to be the simulated failure time itself. Keeping a failure time and hiding only its flag does not produce the minimum of that time and an independent earlier censoring time. Even choosing labels at random does not fix this: independence of a flag from the recorded duration is different from independent censoring of a latent event time. Indeed the labelled durations still have mean one while the failure fraction is one half, so their ordinary censored estimate converges to $1/2$, not the generating rate one. Thus it is not a valid uninformative censoring scheme for the intended failure distribution; strictly, it has not implemented ordinary right-censoring of those simulated failure times at all. One could reproduce the randomly labelled record distribution with independent latent event and censoring times both of rate $1/2$, but that would be a different failure model, not the claimed rate-one model. Calling the flags random is therefore insufficient to justify the target distribution or its censored likelihood. This is the random relabelling is not independent right censoring issue.

A valid simulation draws independently $T\sim \mathrm{Exp}(1)$ and $C\sim \mathrm{Exp}(1)$, and records $(\min (T,C),1\{T\le C\})$. Sinceit gives independent censoring with the requested probability. Equivalently generate $T=-\log U$ and $C=-\log V$ from independent uniform random numbers. Another valid choice is fixed censoring at $\log 2$, since $\mathrm{Pr}(T>\log 2)=1/2$. These give a random censored fraction with expectation one half; they do not promise exactly 500 censored records. For the independent-exponential scheme, observed durations have rate two and mean one half, so $d/\sum Y_i$ correctly converges to one.

Replace the expected total person-time and failure count in the preceding identity by their observed values, $E={\sum }_i{x}_i$ and $d$. The resulting estimating equation is $\theta E=d$, soThis events divided by exposure estimator is failures per unit observed person-time. We assume positive total exposure; if $d=0$, it gives the boundary estimate zero.

Interpret the quoted daily death rates as piecewise constant hazard functions, and use the custody schedules described in the problem. This means community service itself has the ordinary community hazard, a completer has no prison exposure, and a breacher is imprisoned from day 60 to day 120. Subsequent imprisonment after another conviction is not specified, so it is not added to the exposure schedule. The abbreviated “CS180” denotes the same initial community-service policy.

For a completer, all 360 days have the baseline rate, giving integrated hazardFor a breacher, the four periods are 60 community days, 60 prison days, 30 high-risk days immediately after release, and 210 further community days. ThusThe piecewise-exponential survival model gives death probabilities $1-e^{-H_C}$ and $1-e^{-H_B}$. Average these over the stated 60%/40% mixture of paths. Equivalently, everyone first survives the common initial 60-day community interval, and survivors then take the two paths in those proportions. The expected value among 10,000 initial assignments isThis accounts for removal of people after death. Since the daily hazards and endpoint death probabilities are small, the usual person-time approximation replaces $1-e^{-H}$ by $H$ and givesBoth calculations use the same specified schedules. The latter treats all planned exposure as if contributed and is slightly larger. Additional custody histories would change the exposure-weighted integrated hazard; the provided reconviction proportions alone do not determine those histories.

Mortality requires its own sample size calculation, rather than using the much commoner reconviction endpoint. Under the same specified exposure assumptions, the P30 schedule has 30 prison days, 30 high-risk post-release days, and 300 ordinary days. Its integrated hazard isHence its 360-day death probability is $p_P=1-e^{-0.0145}\simeq 0.0143954$. For initial community-service assignment the preceding mixture gives $p_C\simeq 0.0127180$. The absolute risk difference is therefore only about $0.0016774$, or 16.8 fewer deaths per 10,000 initial assignments. The person-time approximation gives 145 versus 128 deaths per 10,000, a difference of 17.

As an illustrative design target, take two-sided significance level $0.05$, statistical power $0.80$ and equal individual allocation. Apply the sample size for comparing two proportions formula with these mortality probabilities:It gives about 74,600 participants per arm, or 149,200 altogether; using the simpler exposure approximations gives about 73,130 per arm. At 30,000 sentences per year, even if every offender were eligible and enrolled, this requires roughly five years of recruitment and another year to complete the final cohort's follow-up.

For scale, one year's equally allocated cohort has 15,000 per arm, with roughly 216 expected deaths after P30 and 191 after community-service assignment. The standard error of the risk difference is about $0.001335$, so the expected standardized separation is only about $1.26$, below the usual two-sided 5% critical value. A one-year trial would consequently have poor statistical power for this mortality difference.

Thus mortality discrimination is much more demanding than reconviction discrimination. National recruitment and reliable linkage to death records could make a multiyear study feasible, but restricted eligibility, incomplete follow-up, or lower recruitment would lengthen it. A trial sized only for the 4-percentage-point reconviction difference would not provide a precise mortality comparison. The numerical conclusion depends on the supplied daily hazards, custody schedules, and the chosen power and significance targets; the prompt does not specify a mortality design target of its own.

Let $D_k={\sum }_{j:g_j=k}{v}_j$ be the observed event count and $T_k={\sum }_{j:g_j=k}{x}_j$ the total person-time in group $k$. Both groups are present; suppose each has $T_k>0$. The grouped log-likelihood isWhen $D_k>0$, its derivative is $D_k/{\beta }^{(k)}-T_k$ and its second derivative is $-D_k/({\beta }^{(k)}{)}^2<0$. Therefore the maximum-likelihood estimates areBoth events and censored observations contribute to $T_k$. If $D_k=0$, the likelihood decreases with the positive rate: its supremum is at the boundary ${\beta }^{(k)}\downarrow 0$, rather than at a finite strictly positive estimate. Writing $D_k/T_k=0$ then describes the closure estimate.

Under the null hypothesis of a common rate, the same maximization givesLet $D=D_0+D_1$ and $T=T_0+T_1$. The maximized linear terms cancel, since $T_k{\hat{\beta }}^{(k)}=D_k$ and $T\tilde{\beta }=D$. Thus the likelihood-ratio test usesInterpret a zero-event summand by continuity as zero. If $D=0$, both likelihood suprema are equal and $W=0$: no rate comparison is informed by observed events.

Under positive common hazard, independent censoring and the usual increasing-information conditions in both groups, Wilks theorem gives $W\Rightarrow {\chi }_1^2$, because the unrestricted model has two rate parameters and the null has one. Reject for large $W$; at 5%, the usual cutoff is approximately $3.84$. The chi-squared calibration is asymptotic and can be poor with sparse event counts or boundary estimates.

For specified exposure periods with constant death hazard functions $h_k$, the integrated hazard through the endpoint is ${\sum }_k{h}_k\Delta t_k$. The displayed event probability follows from the survivor function identity. For different latent exposure schedules mixed with probabilities $w_j$, average the schedule-specific probabilities, ${\sum }_j{w}_j(1-e^{-H_j})$, rather than exponentiating the average integrated hazard. When the hazards are small, $1-e^{-H_j}\simeq H_j$ gives a person-time approximation. A comparison must specify the entire custody or treatment schedule; later changes to exposure cannot be inferred from an endpoint rate alone.