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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.

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.

The model assumes a time-homogeneous Markov property: transition intensities depend only on the current state, not on calendar time, infection age, or earlier history. A Semi-Markov multi-state model could let recovery and death hazards depend on time since infection or entry into the current state. Weekly tests reveal recovery only by interval censoring, however, so the relevant entry and transition times are latent. Fitting the less restrictive model then requires integrating over unobserved paths, and the available data may contain too little information to identify a flexible duration-dependent hazard.

Acquired immunity means that the reinfection rate is no longer the common constant $a$: it should depend on infection history, usually decreasing immediately after recovery and perhaps rising as immunity wanes. The two observed states are then not Markov unless the state is enlarged to record time since infection, number of prior infections, or latent immune class. Monthly binary tests identify infection status only at visits, so infection and recovery times are interval-censored and past short episodes may be missed. The sparse histories contain little information for estimating several history-dependent transition intensities.

Monthly assessment gives interval censoring, although recording failure at the visit treats it as exact. It also creates many tied times. Ordinary continuous-time log-rank calculations then require a tie convention or grouped-time method, and unequal or missed visits can induce informative observation.

The crude incidence-rate estimator isper person-month. This treats all 3000 person-months as susceptible time and assumes infection times are observed exactly, follow-up is complete, and each first infection is detected. In practice tests occur intermittently, so infection is interval-censored; latent, brief, or asymptomatic infections may also be missed.