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For competing risks, a cumulative incidence function jumps by the displayed Aalen–Johansen estimator increment. A person right-censored at exactly the event time remains in the immediately preceding risk set under the usual events-before-censoring tie convention. Censoring changes the next risk set, not the current event numerator or survival jump.

For the numerical competing risks update, the estimated probability of remaining event-free just before the tied time isThere were no intervening events, so the cumulative incidence functions did not jump between the two stated times. The Aalen–Johansen estimator uses the risk set immediately before the event time, including the subject censored at that same time under the usual events-before-censoring tie convention. Therefore the cause-$A$ increment is $0.474(2/20)$, andFor a consistency check, ${\hat{F}}_B(7.8)=0.241+0.474/20=0.2647$ and $\hat{S}(7.8)=0.474(1-3/20)=0.4029$; these sum to one. The censoring does not create an incidence jump and leaves $16$ subjects for subsequent follow-up. Dividing by $19$ would instead assume that the censored subject had already left before the event time, contrary to the stated immediately preceding risk set. This is a cumulative incidence update with tied censoring.

The last lung-cancer failure before five years is at 4.85, where the displayed net survival estimate is 0.392. The other-death net survival estimate remains 0.498 from 4.096 until 5.065. Thus the two supplied curves imply net five-year death estimates of $1-0.392=0.608$ for lung cancer and $1-0.498=0.502$ for other causes. Their sum is 1.110, already showing that they cannot be probabilities of two disjoint actual causes of death.

In competing risks, let ${\lambda }_1(t)$ and ${\lambda }_2(t)$ be the cause-specific hazards while alive, with cumulative hazards ${\Lambda }_k(t)={\int }_0^t{\lambda }_k(u)du$. Treating the other cause as censored estimates the net survival $S_k^{\mathrm{net}}(t)=e^{-{\Lambda }_k(t)}$. The actual probability of death from cause $k$ by time $t$ is the cumulative incidence functionSince $S(u)\le e^{-{\Lambda }_k(u)}$, the net probability $1-S_k^{\mathrm{net}}(t)$ generally overstates the cumulative incidence function: death from the other cause prevents a later death of cause $k$. This discrepancy exists even for independent latent cause-specific failure times. Moreover, Kaplan–Meier estimates are not generally exactly unbiased in finite samples even for their proper target. The plotted complements are not unbiased estimators of $\mathbb{P}(T\le 5,D=k)$, and are not even consistent for that target in general.

For the desired probabilities, use the Aalen–Johansen estimator ${\hat{F}}_k(t)={\sum }_{u\le t}\hat{S}(u-)d_k(u)/n(u)$, with all-cause survival $\hat{S}(u)={\prod }_{v\le u}[1-(d_1(v)+d_2(v))/n(v)]$ and independent study censoring. Removing competing deaths from subsequent risk sets is correct for estimating cause-specific hazards; the error is interpreting the separate net survival complements as cumulative incidence functions. A hypothetical elimination-of-a-cause interpretation of net survival would require additional causal assumptions.

Use the Aalen–Johansen estimator of the cumulative incidence function, updating the overall Kaplan–Meier estimator for either event:Right censoring changes subsequent risk sets, not the survival function by itself. Starting with the given estimates at $a_k$, the complete calculation isThe middle event is of the competing type: it decreases overall survival while leaving the disease cumulative incidence function unchanged at that instant. The event-free intervals leave every estimate and risk set unchanged.

At the final time, both the event subject and the subject censored at that time are in the just-before risk set, so its denominator is eight. This is the usual event-before-censoring convention for recorded ties. After the event and right censoring, six subjects remain at risk. ConsequentlyThe nine numbered source items are the inputs to this single calculation, not nine further questions.