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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.

Condition on the bulb being alive at age $\tau$. The next hour's survival likelihood contribution isIt does not depend on the unknown age $\tau$. This is memorylessness of the exponential distribution: the residual lifetime given survival has the same exponential distribution as a new lifetime.

Condition on each patient's initial observed state and assume independent patient trajectories and observation times that do not carry additional information about the unobserved path. The Markov property factors each trajectory's likelihood function into its successive one-year transition probabilities. ThusMore explicitly, with $q=\lambda +\mu >0$ and $E=e^{-q}$,Successive transitions within one patient need not be independent unconditionally: this is a product of conditional factors justified by the Markov property. If initial-state probabilities are modeled rather than conditioned on, their likelihood contributions must also be included. Zero-count terms contribute zero; a positive count attached to a zero model probability makes the log-likelihood equal to $-\infty$.

The four-parameter panel-observed multi-state likelihood requires the following additional assumptions.

If age is instead updated continuously, the model is time-inhomogeneous and the displayed homogeneous transition formula is not exact. For example, setThen the exact irreversible likelihood contribution isThus the simple closed form uses an explicit frozen-age approximation; adjacency alone and continuously evolving age do not determine that form.

Use an irreversible continuous-time multi-state model $1\to 2\to 3$, with state 3 absorbing. Let $a=q_{12}(50)$ and $b=q_{23}(50)$, and define log-linear age effects byFor a closed-form panel-observed multi-state likelihood, freeze the age covariate at the start of each observation interval. The transition intensities are then $(a,b)$ over the first interval and $(\lambda ,\mu )=(a{e}^{5{\beta }_{12}},b{e}^{5{\beta }_{23}})$ over the second. Conditional on the initial state, the likelihood contribution isIndeed, $p_{11}(5)=e^{-5a}$ and $p_{13}(1)=1-p_{11}(1)-p_{12}(1)$. In this irreversible continuous-time Markov chain, the intermediate state has probabilityThe second expression is the continuous limit of the first and handles equal rates. The observed two-state jump across an interval is compatible with adjacent instantaneous transitions: an unobserved visit to state 2 occurs between the examinations.

Assume a negative test identifies the recovered state. Person 1 remains infected through day 7 and recovers during $(7,14]$, so their interval-censored contribution isPerson 2 remains infected through day 7 and makes an exact $I\to D$ transition at day 10. A transition at an exact time contributes a transition probability density, givingAssuming independent individuals, their joint likelihood contribution is