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Insert $f_i=h_i{F}_i$ and $F_i=e^{-H_i}$ into the survival likelihood. Each individual's log contribution simplifies toThereforeFailures contribute a log hazard function term as well as exposure, whereas both failures and censored subjects contribute the negative cumulative hazard function over their observed follow-up.

Assume independent individuals and independent censoring conditional on their explanatory variables, with the censoring mechanism carrying no parameter $\theta$. A failure contributes its failure-time probability density function $f_i(x_i;\theta )$, while a censored observation contributes the survival function $F_i(x_i;\theta )$ because its failure time exceeds $x_i$. Terms from the censoring distribution then factor out of the survival-parameter likelihood. Thus the survival likelihood and its logarithm areThe individual subscripts allow the same parameter vector to act through different covariate values.

For individual $i$, let ${\pi }_i={\pi }_i(\psi )$, $q_i(t)$ be the known baseline hazard function, and $r_i(t;\varphi )$ the extra hazard. PutThe class-specific survival functions are $e^{-Q_i(t)}$ and $e^{-Q_i(t)-R_i(t)}$. Consequently the unconditional survivor and failure density areMixture probabilities must be summed before taking a logarithm; a weighted average of class log-likelihoods would be a different, complete-data calculation. Substitution into the right-censored survival likelihood givesEquivalently, this additive excess-hazard mixture likelihood has contributionHere $q_i$ and $q_i+r_i$ must be nonnegative hazards. The factor $-Q_i(x_i)$ can be dropped when maximizing over $\varphi ,\psi$ because $q_i$ is known. Factoring out censoring presumes a common parameter-free censoring mechanism independent of failure and latent class, conditional on the observed covariates; unmodelled class-specific censoring would change these mixture contributions.

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.

For independent individuals under fixed administrative censoring, an observed failure contributes the probability density function $\theta e^{-\theta x_i}$, while a censored observation contributes the survivor function $e^{-\theta x_i}$. Hence the survival likelihood and log-likelihood areFor $d>0$, the score function ${\ell }^{\prime }(\theta )=d/\theta -E$ vanishes at $d/E$, and ${\ell }^{\prime \prime }(\theta )=-d/{\theta }^2<0$. This is the unique maximum over $\theta >0$:Thus maximum likelihood estimation agrees with the observed-exposure estimating equation. If there are no failures, the likelihood decreases with $\theta$: its supremum is approached as $\theta \downarrow 0$, or attained at zero if that boundary is admitted.

Assume independent individuals and independent censoring: conditional on any modelled covariates, the censoring mechanism gives no additional information about the latent event time and has no parameter shared with its survival distribution. An observed event at $x_i$ contributes the probability density function $f(x_i)$. A right-censored observation contributes the survivor function $F(x_i)$, since it establishes only $T_i>x_i$.

More formally, let $C_i$ be an independent censoring time with density $c_i$ and survivor function $G_i$. The observed pair $X_i=\min (T_i,C_i)$, $v_i=1_{\{T_i\le C_i\}}$ has a parameter-dependent factor $f(x_i)G_i(x_i)$ if $v_i=1$ and $F(x_i)c_i(x_i)$ if $v_i=0$. The factors involving only the censoring distribution can be omitted from the event-time likelihood. ThusUsing $f=hF$ also gives the useful survival likelihood formEvery individual therefore contributes survival exposure, whether or not an event is observed. These formulas do not justify ignoring informative censoring; that situation requires a model for the observation mechanism as well.

Write $X={\sum }_i{x}_i>0$ and $d={\sum }_i{v}_i$. The exponential survival likelihood has $\ell (\lambda )=d\log \lambda -\lambda X$. Its score is $d/\lambda -X$, and for $d>0$ its second derivative is $-d/{\lambda }^2<0$. Thus the exponential-rate estimation from censored exposure givesAll observation times, including censoring times, belong in $X$. If $d=0$, the likelihood decreases with positive $\lambda$ and has its supremum as $\lambda \downarrow 0$; under the stated strict positivity constraint there is no interior maximum.

Under independent censoring conditional on the modelled covariates, an observed event contributes $f_i(x_i)=h_i(x_i)e^{-H_i(x_i)}$, and a right-censored observation contributes $S_i(x_i)=e^{-H_i(x_i)}$. Factors from the censoring distribution may be omitted when they contain no survival-model parameters. The survival likelihood and log-likelihood are thereforeCensored individuals still contribute their observed exposure through $H_i(x_i)$; they are not deleted from the analysis.

Assume independent individuals and independent censoring, with the censoring law not involving the event-rate parameters. If $T_j$ is the survival time and $C_j$ its censoring time, we observe $x_j=\min (T_j,C_j)$ and $v_j=1_{\{T_j\le C_j\}}$. The exponential distribution givesAn observed event contributes $f_j(x_j)$, while a right-censored observation contributes $S_j(x_j)$, because it only says the event has not occurred by $x_j$. Multiplying these survival likelihood contributions and omitting censoring factors independent of the parameters givesWithout independent censoring the censoring factors cannot generally be discarded in this way.

An exact observed event contributes its event-time density:For exponential-rate estimation from censored exposure, the event-model survival likelihood under independent censoring is ${\prod }_{i}f(x_i{)}^{v_i}F(x_i{)}^{1-v_i}$. For the exponential distribution, put $d={\sum }_i{v}_i$ and $X={\sum }_i{x}_i$. ThenIf $d>0$ and $X>0$, ${\ell }^{\prime }(\theta )=d/\theta -X$ vanishes atThe second derivative $-d/{\theta }^2$ is strictly negative, so this is the unique global maximum-likelihood estimate. The likelihood also tends to zero at both ends of the positive parameter range. If $d=0$ and $X>0$, the likelihood decreases on $\theta >0$; its supremum is at $\theta \downarrow 0$, with no attained positive estimate. Thus every censored time belongs in the exposure denominator.

A right-censored subject contributes survival up to the recorded censoring time, not no information. Under independent censoring, write $x_i=\min (T_i,C_i)$ and $v_i=1_{\{T_i\le C_i\}}$. For exponential death rate $\lambda$, the event-model survival likelihood isDiscarding censored subjects drops observed exposure and selects people whose death occurred before censoring. Their recorded event times are not an unselected sample of all death times. The death-only conditional distribution can differ from the marginal death-time distribution even when censoring is independent. A correct maximum-likelihood estimate uses the total observed person-time, including censored follow-up.

Let $d_j$ be the number of events at potential time $a_j$ and let $r_j$ be the size of the risk set immediately before that time, including observations censored at $a_j$. Define the discrete conditional event probabilityUnder independent censoring, and ignoring the separate censoring distribution, the survival likelihood factorizes asThis follows by multiplying a survival factor for each individual at risk who does not fail at $a_j$, and an event factor for each failure. The log factor $d_j\log q_j+(r_j-d_j)\log (1-q_j)$ is maximized at ${\hat{q}}_j=d_j/r_j$, with the corresponding boundary values when $d_j=0$ or $d_j=r_j$.

The survival function is the product of successive conditional survival probabilities. ThereforeThis is the Kaplan–Meier estimator. A product over no event times is 1. If everyone remaining experiences an event, the product becomes zero and stays zero; subsequent empty risk sets supply no additional factor.

The set must contain every observed event time. Otherwise a discrete survival likelihood would assign probability zero to at least one exact event observation.

The set can include additional times without observed events. Include time zero when events at zero are present, and allow remaining tail probability beyond the fitted range. Estimation of a conditional event mass requires a nonempty risk set; once follow-up ends, later probability masses are not identified by the data. The usual Kaplan–Meier estimator retains the final estimated survival probability as its tail convention.

Choose a parametric baseline hazard $h_0(t;\eta )$ and fitUnder independent right censoring, the full survival likelihood isEstimate $(\eta ,\beta )$ by maximum likelihood estimation and test $H_0:\beta =0$ with a likelihood-ratio test, Wald test, or score test. Equality of the two event-time distributions is exactly $\beta =0$ within this model.

Suppose the recorded durations retain an exponential distribution of rate $\theta$, but an independent fraction $q$ are relabelled as failures and the rest as right-censored. Applying the ordinary survival likelihood gives $\hat{\theta }=d/{\sum }_i{y}_i\to q\theta$, rather than $\theta$. A censoring flag chosen independently of the recorded duration does not establish independent censoring of a latent failure time. Genuine simulation instead draws an event time and an independent censoring time and records their minimum.

An unobserved survival time is right-censored when it is known to exceed the last event-free follow-up time. For a potential survival time $T$ and potential observation-ending time $C$, recordWhen $\delta =1$, the occurrence was observed at $Y$; when $\delta =0$, only $T>Y=C$ is known. An intact machine at the study's closing date and an event-free subject who stops attending follow-up can both yield right censoring, but the reasons for stopping matter for independent censoring. A subject right-censored at year two has contributed two years to person-time at risk and belongs to earlier risk sets. Neither recording a failure at year two nor discarding that subject preserves this information.

Under independent censoring and separate parameters for the observation mechanism, the part of the survival likelihood involving an absolutely continuous probability distribution for $T$ is $f(Y{)}^{\delta }S(Y{)}^{1-\delta }$. Thus a right-censored observation contributes the survival function $S(Y)$, rather than the event probability density function $f(Y)$. See Lu Tian's Stanford notes, Section 2.1, equations (2.1)–(2.2). The Kaplan–Meier six-subject cohort shows exactly how this distinction changes the calculation.