= Solution
For subject $i$, write $\delta_i=\mathbf1_{\{T_i\le C_i\}}$, and define the <counting process> and pre-event <at-risk process> by
$$
N(t)=\sum_i\mathbf1_{\{X_i\le t,\ \delta_i=1\}},\qquad R(t)=\sum_i\mathbf1_{\{X_i\ge t\}}.
$$
Under independent <censoring> and a common <hazard>, each observed event-free subject has conditional event <probability> $h(t)dt$ in a short interval. Therefore $\mathbb E(dN(t)\mid\text{past})=R(t)dH(t)$ to first order. Equivalently the <compensator of a counting process> is $\int_0^tR(s)dH(s)$. Dividing the observed increments by current exposure gives the estimating equation $d\widehat H=dN/R$, hence
$$
\boxed{\widehat H(t)=\int_0^t\frac{\mathbf1_{\{R(s)>0\}}}{R(s)}dN(s)=\sum_{t_j\le t}\frac{d_j}{r_j}.}
$$
Here $t_j$ are distinct event times, $d_j$ the number of events at that time, and $r_j=R(t_j)$ the number at risk immediately before it. One can also obtain the same increment by maximizing the local multiplicative-intensity likelihood $a^{d_j}e^{-r_ja}$: differentiation gives $a=d_j/r_j$. This derives the <Nelson–Aalen estimator>, rather than merely naming it.
A censored observation contributes to the denominator while under observation, and then leaves the <risk set>; it contributes no event jump. For tied events, use the single increment $d_j/r_j$, not $d_j$ successive one-event increments with shrinking denominators. If <censoring> and events share a recorded time, a convention is needed; counting the event before removing same-time censored subjects uses the pre-time <risk set> above. Rounded or interval-censored data may require a model suited to their observation mechanism.
For the distinct ordered observations in the question, let $\delta_j$ indicate whether $X_j$ is an event. The <risk set> at $X_j$ contains precisely the $n-j+1$ subjects whose recorded time is at least $X_j$. Thus
$$
\widehat H(X_i)=\sum_{j\le i}\frac{\delta_j}{n-j+1},\qquad
\sum_{i=1}^n\widehat H(X_i)=\sum_{j=1}^n\frac{\delta_j}{n-j+1}\sum_{i=j}^n1
=\sum_{j=1}^n\delta_j.
$$
Therefore \b[the sum of the fitted integrated <hazards> at all observation times equals the observed number of events], the <event-count identity for Nelson–Aalen cumulative hazards>. The <grouped Nelson–Aalen event-count identity> extends the same argument to tied observations with their multiplicities and the stated risk-set convention.
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