= Solution
Use the product-limit estimate in (b). Between consecutive event times there are no mortality jumps, so
$$
\frac{d\widehat F_E(t)}{dt}=\overline h_B(t)\widehat F_E(t).
$$
Thus \b[the estimated relative survivor curve rises between events] when background mortality is positive, and is constant if it is zero. <Censoring> changes the <risk set> and can change the slope, but supplies no downward event jump. At the next event time the curve is multiplied by $1-d_k/r_k$.
Provided the curve is still positive, the ratio of the post-event estimates at the two consecutive event times is
$$
\frac{\widehat F_E(t_k)}{\widehat F_E(t_j)}
=\exp\left[\int_{t_j}^{t_k}\overline h_B(u)\,du\right]
\left(1-\frac{d_k}{r_k}\right).
$$
For $d_k<r_k$, the later value is larger precisely when
$$
\boxed{\int_{t_j}^{t_k}\overline h_B(u)\,du>
-\log\left(1-\frac{d_k}{r_k}\right).}
$$
For example, suppose there are 100 individuals in the <risk set> after the earlier event, no intervening <censoring>, a 20-day gap, and one death at its end. If every <background hazard> is $0.001$ per day, then the integrated background contribution is $0.02$ and
$$
\frac{\widehat F_E(t_k)}{\widehat F_E(t_j)}=e^{0.02}(1-1/100)\simeq1.010>1.
$$
The individual <background hazards> are small, but the interval contains less observed mortality than expected from that background. This can occur by chance in a small event sample or when the cohort is healthier than the life-table reference on other dimensions.
A nonnegative true <excess hazard> makes $F_E$ nonincreasing, but this unconstrained <estimator> need not inherit that restriction. An increase is therefore not proof of a clinical survival benefit. If the exponential cumulative-hazard version is chosen instead, its corresponding condition is $\int_{t_j}^{t_k}\overline h_B>d_k/r_k$; the qualitative explanation remains the same. If the <risk set> is exhausted by deaths, the product-limit factor is zero and no later increase is possible.
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