= Solution
Use the usual semiparametric <Cox proportional-hazards model> with observed <covariates> $z_i$:
$$
h_i(t)=h_0(t)e^{\beta^Tz_i}.
$$
The <covariates> are implicit in the question's regression fit, even though the displayed data notation only lists exit times and event indicators. Assume entry at a common origin, time-constant <covariates>, distinct observed times and <independent censoring>. Write $R_i=\{j:x_j\geq x_i\}$ and $w_j(\beta)=e^{\beta^Tz_j}$. Conditional on a single event at $x_i$ in its <risk set>, the chance that individual $i$ supplies it is
$$
\frac{h_0(x_i)w_i(\beta)}{\sum_{j\in R_i}h_0(x_i)w_j(\beta)}
=\frac{w_i(\beta)}{\sum_{j\in R_i}w_j(\beta)}.
$$
The <baseline hazard> cancels. Consequently estimate $\beta$ by maximizing <Cox partial likelihood>
$$
L_C(\beta)=\prod_{i=1}^m\left[\frac{e^{\beta^Tz_i}}{\sum_{j\in R_i}e^{\beta^Tz_j}}\right]^{v_i},
\qquad
\ell_C(\beta)=\sum_i v_i\left[\beta^Tz_i-\log\sum_{j\in R_i}e^{\beta^Tz_j}\right].
$$
For example, the <score and information of Cox partial likelihood> follow by differentiation. Define $S_i^{(0)}=\sum_{j\in R_i}w_j$, $S_i^{(1)}=\sum_{j\in R_i}w_jz_j$ and $S_i^{(2)}=\sum_{j\in R_i}w_jz_jz_j^T$. Then
$$
U(\beta)=\sum_i v_i\left(z_i-\frac{S_i^{(1)}}{S_i^{(0)}}\right),
\qquad
I(\beta)=\sum_i v_i\left[\frac{S_i^{(2)}}{S_i^{(0)}}-
\frac{S_i^{(1)}S_i^{(1)T}}{(S_i^{(0)})^2}\right].
$$
When a finite identifiable maximum exists, iterate $\beta\leftarrow\beta+I(\beta)^{-1}U(\beta)$ until convergence. The <Observed Fisher information> also gives the usual asymptotic coefficient uncertainty. Censored subjects enter the relevant <risk sets>, even though they supply no numerator event factor.
After fitting $\widehat\beta$, estimate the <cumulative hazard> of the reference subject by the <Breslow estimator>
$$
\boxed{\widehat H_0(t)=\sum_{i:x_i\leq t}\frac{v_i}{\sum_{j\in R_i}e^{\widehat\beta^Tz_j}}.}
$$
Its jump at an event is the observed event count divided by the total fitted <hazard multiplier> at risk. The estimate is an integrated <baseline hazard>, with hazard represented by a measure placing these jumps at observed event times. If a smooth ordinary hazard function is wanted, smooth these increments with a chosen kernel and bandwidth; differentiating the step function as an ordinary function would incorrectly give zero away from the jumps. The fitted individual <cumulative hazard> is $\widehat H_i(t)=e^{\widehat\beta^Tz_i}\widehat H_0(t)$ over that individual's follow-up. The invariances below concern this <Cox partial likelihood> fit and this <Breslow estimator>; they need not hold for a fully parametric proportional-hazards fit or an arbitrarily smoothed hazard estimator.
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