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The Aalen–Johansen estimator is a product of empirical state-transition matrices over event times. In competing risks, the estimated cumulative incidence function for cause $j$ jumps by $\hat{S}(t-)d_j(t)/Y(t)$, while overall survival is updated using all event causes. Right censoring changes subsequent risk sets but does not itself create a probability jump.

Bruce W. Turnbull works on survival analysis and the statistical monitoring of clinical trials. His Cornell profile connects research on statistical methods with the practical problem of deciding what accumulating trial data justify.^[ref]

His 1974 paper studied nonparametric estimation when event times are subject to both left censoring and right censoring. A time known only to precede a threshold and one known only to exceed a threshold convey different information about the same survival function. Retaining those inequalities makes incomplete observations useful without substituting an invented event time.^[ref]

A censored algorithm runtime is an evaluation that ends without observing the algorithm's actual completion time. A timeout or adaptive capping limit can establish that the run had not finished by the cutoff, supplying right censoring rather than an exact completed runtime. The ParamILS capping analysis uses such elapsed-time information to bound comparisons without finishing every run.^[ref]

Write $T$ for completion time and $C$ for a stopping limit. A simple record iswhere $\delta =1$ means completion was observed. The event flag is essential: the same stored duration can mean a completed run or only a lower bound on a still-running computation.

For an original example, two completed runs take $2$ and $5$ seconds and a third is stopped unfinished at $10$ seconds. Averaging the stored durations gives $17/3\approx 5.67$ seconds, but that is only a lower bound on the mean of the actual completion times. If the third would take $40$ seconds, the true mean is $47/3\approx 15.67$ seconds. Discarding it entirely gives $3.5$ seconds and removes the difficult run from the comparison.

Keep exact observations, cutoff information and any timeout penalty separate. A penalized benchmark score may deliberately assign a stated cost to a timeout; that score is not an observation of $T$. Estimating completion-time distributions additionally requires assumptions about how stopping limits were chosen, particularly when the limits depend on earlier performance.

A Cox–Snell residual is the fitted cumulative hazard function evaluated at an observed event or censoring time: $r_i={\hat{H}}_i(Y_i)$. Its reference probability distribution comes from the probability integral transform. If the true event time $T_i$ has a proper continuous survival function $S_i$, then $H_i=-\log S_i$ satisfiesFor example, an exponential distribution with rate $\lambda$ has $H(t)=\lambda t$, so rescaling its event time by $\lambda$ produces a unit-rate exponential distribution.

With right censoring, $Y_i=\min (T_i,C_i)$ and the transformed observation is $H_i(Y_i)$ together with its original event indicator ${\delta }_i$. Under independent censoring, a Kaplan–Meier estimator applied to these transformed observations should approximately follow $e^{-r}$; its logarithm should approximately follow a straight line of slope minus one. Censored residuals must remain censored in this comparison.

The exact exponential identity uses the true model, whereas ${\hat{H}}_i$ is estimated from data: fitted residuals are approximate diagnostics and need not be independent exact exponentials. Atoms in a discrete event-time probability distribution, or a cure fraction with positive probability of no finite event, invalidate the unmodified proper-continuous reference law. Such models require diagnostics respecting their jumps or surviving mass.

Let eventual outcome $I=j$ have probability ${\pi }_j$, with $\log T\mid I=j\sim N({\mu }_j,{\sigma }_j^2)$. An observed outcome at time $t$ contributes ${\pi }_j{f}_j(t)$ to the likelihood, while right censoring at $c$ contributes ${\sum }_j{\pi }_j{S}_j(c)$. The expectation-maximization algorithm imputes both the censored outcome and its latent log event time. Its E-step uses the displayed conditional weights and the first two moments of a truncated normal distribution above $\log c$. With $z=(\log c-{\mu }_j)/{\sigma }_j$ and inverse Mills ratio $\psi (z)$, these moments are $m={\mu }_j+{\sigma }_j\psi (z)$ and $s=m^2+{\sigma }_j^2[1-\psi (z)(\psi (z)-z)]$. Observed outcomes use deterministic class weights and moments $\log t,(\log t{)}^2$. The M-step updates each class probability to its average weight, its mean to the weighted first moment, and its variance to the weighted second moment minus the squared new mean. Outcome-specific cumulative probabilities are cumulative incidence functions, not the conditional distributions $F(t\mid I=j)$ themselves.

For independent exponentially distributed failure times subject to independent right censoring, let $d$ be observed failures and $E={\sum }_i{x}_i$ total follow-up exposure. The parameter-dependent likelihood is ${\theta }^d{e}^{-\theta E}$, giving maximum-likelihood estimator $\hat{\theta }=d/E$. The fitted individual cumulative hazards sum to $d$, but this estimating identity alone does not establish that a constant hazard model fits.

Feeding latency is the elapsed time from a defined starting event to a defined feeding event. An experiment might measure from food presentation to an individual's first peck or to the first bird feeding in a social group. Those endpoints answer different questions. The start, the feeding criterion and the observation limit must all be specified.

In an original example, an animal feeds after $20\mathrm{s}$ without a novel object and $80\mathrm{s}$ with one nearby: the novelty-associated increase is $60\mathrm{s}$. Another changes from $100$ to $130\mathrm{s}$, an increase of $30\mathrm{s}$. Comparing only the two novel-condition times reverses the ordering obtained by comparing their changes.

A group-level first-feeding time is $T_{\mathrm{group}}=\min (T_1,\dots ,T_n)$, if $T_i$ denotes each member's time under that same group trial. This identity requires no independence assumption. It also shows why a group endpoint cannot be treated as though it were a measurement of every member: one early feeder determines the recorded minimum.

If recording ends after $180\mathrm{s}$ before feeding occurs, the observation is subject to right censoring: the time exceeds the recorded limit. Replacing it by an observed time of exactly $180\mathrm{s}$, or by infinity, changes what was measured.

A finch study combined whether groups fed at all with feeding latency comparisons in familiar and novel-object conditions. Keeping those outcomes separate preserves information that an average of observed feeding times alone could lose.^[ref]

Germán Rodríguez works on statistical methods for population research. His Princeton teaching and research site connects generalized linear models, survival analysis and demographic questions about event timing. The site's biographical note identifies his training in biostatistics and his work in population research.^[ref]

His Survival Models by Germán Rodríguez explains how survival functions and hazard functions encode the same event-time distribution in different ways. With right censoring, an event observed at $t$ contributes density $f(t)$, whereas an observation ending without an event contributes $S(t)$. Writing the contribution as $S(t)h(t{)}^{\delta }$, with event indicator $\delta \in \{0,1\}$, makes that distinction explicit. These likelihood conventions let incomplete follow-up inform estimation without treating a censored time as an observed failure.^[ref]

Graft patency is the openness of a graft’s lumen to blood flow. For a bypass graft, an open route is a necessary anatomical observation, but it does not by itself establish adequate tissue perfusion or a favorable overall clinical outcome. A conduit can remain open while narrowing progressively.

Reporting must also specify the history of that openness. In the 1997 vascular reporting standard, primary graft patency describes uninterrupted patency without a qualifying intervention on the graft or its anastomosis. Assisted primary graft patency allows intervention before an occlusion; secondary graft patency allows restoration after occlusion under the stated reconstruction rules. Procedures confined to disease beyond the graft and its anastomoses have specific exceptions. The exact endpoint definition must accompany the result.^[ref]

An original complete-follow-up example shows why one percentage is insufficient. Ten grafts are followed for the same interval: six need no procedure, two remain open after preventive graft interventions, one occludes and is successfully reopened, and one remains occluded. The corresponding endpoint proportions are $6/10$ primary, $8/10$ assisted primary and $9/10$ secondary patency. They describe the same cohort, not three competing estimates of one quantity. The last figure includes grafts that never required rescue.

With unequal observation times, the percentage of grafts open at their last visit is not a fixed-time patency estimate. Survival analysis, including the Kaplan–Meier estimator, tracks time to the chosen endpoint and handles right censoring under stated assumptions. Loss to follow-up and death need explicit treatment; simply deleting them can bias a comparison.

Observation is also method-dependent. Ultrasound follow-up can document graft status and detect developing abnormalities. Measurements of narrowing or flow velocity, the occurrence of occlusion and the need for intervention are related but different records. Specifying which was measured makes graft patency useful for comparing graft materials, surveillance strategies and vascular surgery outcomes.^[ref]

At distinct observed event times $t_1<t_2<\cdots$, let $n_j$ be the risk set size immediately before $t_j$, and let $d_j$ count events at $t_j$. The Kaplan–Meier estimator of the survival function isThe empty product is one. Each factor estimates a conditional survival probability among those still observed and event-free; their product joins the successive steps. This is nonparametric statistics: no specified shape for the survival distribution is imposed. Use independent censoring and the appropriate risk sets to interpret the result as an estimate for the intended population.

The Kaplan–Meier estimator is a right-continuous function with downward jumps at event times. Right censoring changes later denominators without causing a jump. A tick on the curve marks right censoring, rather than a failure. For a tied group of $d_j$ events, use the one factor $(n_j-d_j)/n_j$; subjects right-censored at that recorded time remain in $n_j$ for this factor. The fraction $d_j/n_j$ estimates a discrete hazard, not a continuous hazard function in inverse-time units. With no censoring, the factors telescope to the fraction of the original cohort whose observed survival times exceed $t$, the complement of its empirical distribution function.

Edward L. Kaplan and Paul Meier published ^[ref]. Their paper addresses lifetime estimation when observation ends before failure and explicitly calls attention to the assumption of independence between lifetime and potential observation loss. The date 1958 is the original publication year; the publisher's later online-posting date is not the discovery date.

In his ^[ref], Kaplan traces Meier's interest to cancer-duration data in 1952 and his own to the lifetimes of vacuum tubes in undersea telephone-cable equipment a year later. The journal encouraged them to combine their manuscripts. The shared mathematical question was how to estimate a survival function when some lifetimes are still running or disappear from observation.

Consider six subjects observed from time zero. Recorded survival data are $A:(1,1)$, $B:(2,0)$, $C:(3,1)$, $D:(3,1)$, $E:(3,0)$, and $F:(5,0)$, with each pair giving $(Y,\delta )$ in years. Thus $A,C,D$ have observed events, while $B,E,F$ are right-censored. The letters label subjects, not treatment groups. Here are all exit times, including rows without events:

At year one, $\hat{S}(1)=1(1-1/6)=5/6$. Removing $B$ at year two leaves the Kaplan–Meier estimator unchanged. Immediately before year three, $C,D,E,F$ are still in the risk set: the two events give $\hat{S}(3)=(5/6)(1-2/4)=5/12\approx 0.416667$. Only afterwards do $C,D,E$ leave, leaving $F$ alone. Removing $E$ before counting the tied events would incorrectly use $n=3$ and give $(5/6)(1-2/3)=5/18$. The event-first convention is part of the recorded-time interpretation; coarsely rounded timestamps do not reveal a hidden ordering within that time bin.

The exact Kaplan–Meier estimator is $1$ on $[0,1)$, $5/6$ on $[1,3)$, and $5/12$ on $[3,5]$. The three right censoring ticks are $(2,5/6)$, $(3,5/12)$, and $(5,5/12)$; at a shared event time the tick is on the post-event step. Beyond year five the risk set is empty, so the horizontal endpoint does not determine later survival. The Kaplan–Meier median survival time is three years, since this is the first step at or below $1/2$.

Now separate the tied events by changing only $D$'s recorded event time from year three to year four. At year three the risk set still has four subjects, but $d=1$, so $\hat{S}(3)=(5/6)(3/4)=5/8$. After the event and $E$'s right censoring, only $D,F$ remain. At year four $n=2,d=1$, giving $\hat{S}(4)=(5/8)(1/2)=5/16=0.3125$. The intervening right censoring prevents the factors from telescoping to the original tied-event factor. If $E$ instead remained observed past year four, the factors would be $(3/4)(2/3)=1/2$, giving the original $5/12$ after both events. This is Kaplan–Meier telescoping across a censor-free interval in numbers.

Incorrectly counting every right-censored exit as an event estimates the distribution of observation-ending times $Y$, rather than survival times $T$. For the original tied cohort, the spurious curve is $5/6$ at year one, $(5/6)(4/5)=2/3$ at year two, $(2/3)(1-3/4)=1/6$ at year three, and zero at year five. That zero asserts a failure for $F$, whose only known information is $T_F>5$.

The computation uses the risk set and event-before-censoring convention in Lu Tian's STAT331 notes, Unit 3, pages 2–6; this six-subject example supplies its own exact counts.

Life testing observes failure times under a specified test protocol to estimate a lifetime probability distribution or check a reliability theory model. A test can end before every unit fails. Under right censoring, one records $Y_i=\min (T_i,C_i)$ and ${\delta }_i=1_{\{T_i\le C_i\}}$, where $T_i$ is the failure time and $C_i$ is the observation limit.

For independent censoring, the part of the likelihood function involving a continuous lifetime model with probability density function $f$ and survival function $R$ isA unit that fails at 45 hours contributes $f(45)$; one still working when observation ends at 60 hours contributes $R(60)$. Treating the latter as a failure at 60 hours invents an event that was not observed. The test protocol and any dependence between failure and observation limits therefore affect what can be inferred.^[ref]

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.

A censoring observation means an event time is not observed exactly: the information supplies a bound or interval instead. Under right censoring, one observes $X=\min (T,C)$ and the event indicator $\delta =1_{\{T\le C\}}$; if $\delta =0$, one knows only $T>C$. Left censoring gives an upper bound, while interval censoring locates the event between two observations.

Uninformative censoring means that, conditional on the covariates used in the analysis, the censoring mechanism supplies no additional information about the event time. Independence $T\perp C$ within those covariate strata is a sufficient formulation: subjects remaining in the observed risk set have the same future hazard as the corresponding event-free population. A fixed study cutoff unrelated to prognosis is an example of administrative censoring.

Informative censoring occurs when that condition fails. For example, students at high risk of leaving may also become harder to contact before their departure is formally recorded; censoring at loss of contact selectively removes high-risk students. Ordinary Kaplan–Meier estimator and Nelson–Aalen estimator calculations then use an unrepresentative risk set. The distinction determines whether ordinary survival estimates can be interpreted as population survival. Merely knowing that an observation is censored does not establish independence.

Differentiating the log-likelihood again givesThus the observed Fisher information for the rate is proportional to the total number of observed events. At a positive interior maximum-likelihood estimator, the large-sample variance estimate and standard error arePrecision is therefore driven by the number of observed events rather than by the number recruited alone. Heavy right censoring can leave a large study with little information about its hazard function. Under the regularity conditions for the Fisher information, the expected information is $\mathbb{E}[D]/{\theta }^2$.

A right-censored observation is not wholly uninformative: its exposure contributes to $E$, changes the maximum-likelihood estimator, and affects the expected number of events under a sampling design. Its contribution $-\theta x_i$ happens to have zero second derivative in this rate parameterization. For example, if no events occur, the likelihood still declines as $e^{-\theta E}$ and disfavors large rates, despite its zero curvature. The usual interior variance approximation is then invalid.

Use $F(t)=\mathrm{Pr}(T>t)$, a nonincreasing right-continuous survivor function. The empirical likelihood treats an exact event as a probability mass rather than maximizing an unrestricted probability density function at isolated data points. Under a noninformative observation mechanism, the event-time likelihood contributions areThe second, third and fourth rows account for right censoring, left censoring and interval censoring, respectively. A censoring mechanism with unknown dependence on $T$ cannot simply be dropped; these are the event-time factors under independent censoring or another ignorable mechanism.

There is a useful finite-dimensional reduction before optimization. Let $A_i$ be the set in which observation $i$ tells us its event time lies. Partition the time axis using all observed endpoints, keeping endpoint singletons when required. Within each cell, membership in every $A_i$ is constant. Assign cell masses $w_j\ge 0$, with ${\sum }_j{w}_j=1$, including a tail cell when necessary, and let $A_{ij}$ be the cell-membership indicator. ThenMass locations within a cell are not identifiable and need not be separate parameters. Repeated observation sets can be grouped and their factors raised to their counts. Cells with the same membership column can also be combined. If one column is componentwise dominated by another, transferring its mass to the dominating column weakly increases every factor; hence an unconstrained maximizing distribution can omit the dominated cell. This proves likelihood support reduction under censoring directly. The remaining log-likelihood is a concave function of the probability masses on a simplex, often far smaller than the original unrestricted distribution problem. Added constraints can restrict which transfers remain permissible.

For right censoring alone, take a grid of all distinct observation times $s_j$ and initially allow a discrete hazardLet $d_j$ be the number of events at $s_j$ and $n_j=\#\{i:x_i\ge s_j\}$ the risk set immediately before that time. For tied recorded times, use the convention that events precede censoring, so those censored at $s_j$ remain in this risk set. The mass at $s_j$ is $q_j{\prod }_{k<j}(1-q_k)$, and the probability of surviving past $s_j$ is ${\prod }_{k\le j}(1-q_k)$. Substitution into the exact-event and right-censoring factors yieldsThe exponent $n_j-d_j$ counts all observed individuals whose information requires survival beyond that jump. In particular, censoring observations affect the risk set even though they give no event factor.

Each factor can now be maximized separately. For $0<d_j<n_j$, its derivative is $d_j/q_j-(n_j-d_j)/(1-q_j)$, giving ${\hat{q}}_j=d_j/n_j$; the second derivative is negative. The same result holds at the boundaries: ${\hat{q}}_j=0$ if $d_j=0$, and ${\hat{q}}_j=1$ if $d_j=n_j$. The Kaplan–Meier estimator is thereforeOnly event times contribute nontrivial factors. Residual mass after the last observation may be placed anywhere later without changing the empirical likelihood. A terminal plateau is therefore not evidence of permanent survival; the tail is unobserved.

For a constrained Kaplan–Meier estimator with $F(t)=p$, first split the support cells at $t$. In the probability-mass representation the exact optimization problem isThe last sum excludes a singleton mass at $t$, because $F(t)=\mathrm{Pr}(T>t)$. This is a concave maximization with linear equality constraints, so it directly gives a constrained nonparametric maximum-likelihood estimator. It also handles a time lying between observed events or beyond all follow-up.

An equivalent hazard formulation adds $t$ to the grid, permits zero-event jumps when needed, and imposesFor an interior solution using only positive-event jumps before $t$, add $\lambda \{{\sum }_{s_j\le t}\log (1-q_j)-\log p\}$ to the log-likelihood. Differentiating givesThe multiplier is found by solving the scalar equationwith denominators and fitted hazards in their admissible ranges. On an admissible event-only interior branch, this product is increasing in $\lambda$, making a one-dimensional root search straightforward.

A constraint can require mass at a time with no observed event. The event-only multiplier equation must not be used blindly. To include boundary cases, put $u_j=-\log (1-q_j)\ge 0$. Before $t$, the constraint becomes ${\sum }_j{u}_j=-\log p$, and each contribution isFor $d_j>0$ its multiplier equation again gives $q_j=d_j/(n_j+\lambda )$. For $d_j=0$, its derivative is $-n_j$, so the maximum has $u_j=0$ when $\lambda >-n_j$, and may have $u_j>0$ when $\lambda =-n_j$. Such a jump supplies mass needed by the constraint; an unobserved tail with $n_j=0$ is included by the same rule. For example, if every observation is right-censored after $t$ and there are no events, the ordinary estimate has $F(t)=1$. To impose $p<1$, put mass $1-p$ at $t$ and the remaining $p$ after all censoring times. The maximized likelihood is $p^n$, although there are no event-time factors at all. The mass-constrained optimization above automatically accounts for this possibility and for any remaining tail nonuniqueness.

Use the standard right censoring setup with everyone entering observation at time zero. At each observed time $x_j$, the pre-exit risk set has size $r_j=\#\{i:x_i\ge x_j\}$. The Nelson–Aalen estimator of the cumulative hazard function isCensoring removes an individual from later risk sets but contributes no event increment.

Exchange the two finite sums:Each event increment appears in exactly as many subjects' fitted cumulative hazards as there were subjects at risk for that increment. This proves the event-count identity for Nelson–Aalen cumulative hazards. Common entry is relevant: with delayed entry, a subject does not accumulate the increments before entering, so an entry-time subtraction would be needed.

One method for ties is to group observations at each distinct recorded time. If $d_k$ events occur at ${\tau }_k$, use the pre-time risk set size $R_k$ including observations censored at that same recorded time, and take one jump $d_k/R_k$. This is the usual events-before-censoring convention within a recorded tie. Since $R_k=\#\{i:x_i\ge {\tau }_k\}$,Thus the grouped Nelson–Aalen event-count identity preserves the required identity at the recorded observation times. It is a sum over individuals, not an unweighted sum over distinct times.

A second method, suitable when ties reflect rounded times, is to break ties by small perturbations or an artificial ordering, then apply the untied estimator. The event-count identity holds on that new, untied time grid. It does not generally persist if all estimates are mapped back to the original common time. For example, two tied failures among two individuals give a grouped jump $2/2=1$, so the original-time sum is $2$. Sequentially breaking the failures gives jumps $1/2$ and $1$, whose total is $3/2$; assigning the completed jump to both original tied observations gives sum $3$. On the perturbed grid the two values are $1/2$ and $3/2$, whose sum is again $2$. This distinction states exactly what the second method does and does not preserve.

Another common grouped survival approach is the Kaplan–Meier estimator, from which one may estimate cumulative hazard by $-\log \hat{S}(t)$. Its jump is $-\log (1-d_k/R_k)$, not $d_k/R_k$, and it likewise does not have the Nelson–Aalen event-count identity. In particular it becomes infinite at a time when all remaining individuals fail.

Use the question's decreasing survivor function convention $F(t)=\mathbb{P}(T>t)$, rather than interpreting $F$ as a cumulative distribution function. An exact failure at $t$ contributes the mass $F(t-)-F(t)$. The right censoring contribution is $F(3)$, and the interval censoring contribution is $F(2)-F(5)$. Under independent observations and a noninformative observation mechanism, the empirical likelihood with mixed censoring, omitting observation-mechanism factors, isThis is a likelihood over probability distributions with possible atoms; replacing the exact-event masses by a smooth density would be a different model.

Set $u=F(2)$ and $v=F(5)$. Monotonicity and $0\le F\le 1$ giveFor instance, $F(4-)\le F(2)=u$ while $F(4)\ge F(5)=v$, proving the second inequality. The interval-censoring factor already equals $u-v$. Multiplying gives the monotonicity bound for a mixed-censoring likelihoodFor any $0<v<u<1$, this bound is attained by a distribution with masses $1-u$, $u-v$ and $v$ at $1$, $4$ and $6$ respectively. Its survivor function is one before the first mass, $u$ between the first and second, $v$ between the second and third, and zero thereafter.

The maximum likelihood is positive, because such a three-mass distribution has positive likelihood. At any maximizer all the above inequalities must therefore be equalities; otherwise replacing it by that three-mass distribution at the same $u,v$ would strictly improve its likelihood. Equality, together with monotonicity, yieldsThus these relations are consequences of maximum likelihood, not assumptions about where an arbitrary distribution must put its mass.

The reduced log-likelihood isIts stationary equations areThe second gives $u=3v$. Substituting in the first gives $-1/(1-u)+4/u=0$, henceThe reduced log-likelihood is strictly concave: its second directional derivative in a nonzero direction $(a,b)$ isThe likelihood vanishes on the boundary of the admissible triangle, so this stationary point is the unique maximum. The corresponding nonparametric maximum-likelihood estimator puts masses $1/5$, $8/15$ and $4/15$ at the three exact failure times.

With right censoring, an event has not occurred by the last observation time $C$, so the data establish only $T>C$. One records $Y=\min (T,C)$ and $\delta =1_{\{T\le C\}}$, distinguishing an observed event from a censored exit. A business still operating at the follow-up deadline is administratively right-censored. Its observed age is a lower bound on its lifetime, not its eventual closure age. Standard survival analysis requires suitable independent censoring, possibly conditional on the fitted covariates.

Statistician B uses the more appropriate analysis. Its start–stop representation includes left truncation as well as right censoring. Both entry and exit are ages since opening, so a business belongs to the risk set at age $t$ only when $\mathrm{ageatentry}<t\le \mathrm{obsage}$. The event indicator distinguishes closure from a censored end of observation. The counting-process representation does not mean that multiple closures per business have been observed.

Statistician A records only exit and status and treats every sampled business as at risk from opening. That incorrectly adds already-established businesses to risk sets at ages before they entered observation, ignoring their necessary survival to entry. Its first risk count is $60$, whereas B's is $25$. The extra non-events dilute early estimated closure hazards and here bias the estimated survival upward. Correcting left truncation makes later risk sets capable of increasing as businesses enter at their respective ages.

For the Kaplan–Meier estimator with delayed entry, at event ages $t_j$ with $d_j$ closures and $r_j$ businesses at risk,The standard error follows from the Greenwood formula. Between event times this survival function estimate is constant. At age five the last preceding event is at $4.52$, and at age ten it is at $9.90$. Thus the requested estimates, including their precision, areandThese correspond to estimated proportions 46.2% and 28.0%, not to the probability of surviving five or ten additional years conditional on being sampled at an arbitrary age. A's corresponding estimates $0.650$ and $0.407$ use the inappropriate risk sets.

This interpretation assumes appropriately independent entry and censoring and a suitable common lifetime distribution across entry cohorts. A delayed-entry correction does not remove arbitrary informative sampling or calendar effects. More generally, if no observation covers the earliest ages, absolute survival from opening requires additional information about survival before the first observable age; the reported product-limit normalization is the one used in the supplied analysis.