= Solution
Place discrete hazards $q_j$ at the ordered distinct event times $t_j$. If $d_j$ events occur among $r_j$ individuals at risk immediately before $t_j$, the nonparametric empirical likelihood is
$$
L(q)=\prod_jq_j^{d_j}(1-q_j)^{r_j-d_j},
\qquad0\leq q_j\leq1.
$$
Equivalently, an observed event at $x$ contributes the probability mass at $x$, while a right-censored observation contributes the survivor probability beyond its censoring time. Maximization gives $\widehat q_j=d_j/r_j$ and the <Kaplan–Meier estimator>
$$
\widehat S(t)=\prod_{t_j\leq t}(1-\widehat q_j).
$$
<Left truncation> means an individual is observed only conditional on surviving beyond an entry time $L$. Ignoring it overrepresents long survivors and creates <survivorship bias>. A subject with event or censoring time $X>L$ contributes its usual likelihood divided by $S(L)$; in risk-set form, that individual enters each $r_j$ only for event times satisfying $L<t_j\leq X$.
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