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For the Kaplan–Meier estimator, the estimated variance isThe formula follows by summing approximate variances of the logarithms of the conditional survival factors and applying the delta method to their product. Its square root is the estimated standard error. The same expression uses the actual delayed-entry risk sets for the Kaplan–Meier estimator with delayed entry. When $0<\hat{S}(t)<1$ and every contributing $r_j>d_j$, an approximate log-scale confidence interval is $\exp \{\log \hat{S}(t)\pm z_{1-\alpha /2}\widehat{\mathrm{se}}(\hat{S}(t))/\hat{S}(t)\}$, with the upper endpoint capped at one. This interval can be inaccurate with very small risk sets.

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.

The risk set immediately before an event time contains the individuals who remain observed, event-free, and eligible to experience the event. For a cohort observed from time zero, with exit times $Y_i=\min (T_i,C_i)$ under right censoring, write $R_j=\{i:Y_i\ge t_j\}$ and $n_j=|R_j|$. Individuals with an event or right censoring exactly at $t_j$ are included in this denominator; process events before removing subjects right-censored at that same recorded time. Earlier right censoring removes subjects from later risk sets. With delayed entry, use the entry times too, as in the Kaplan–Meier estimator with delayed entry.