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Use the usual semiparametric Cox proportional-hazards model with observed covariates $z_i$: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 $R_i=\{j:x_j\ge x_i\}$ and $w_j(\beta )=e^{{\beta }^T{z}_j}$. Conditional on a single event at $x_i$ in its risk set, the chance that individual $i$ supplies it isThe baseline hazard cancels. Consequently estimate $\beta$ by maximizing Cox partial likelihoodFor example, the score and information of Cox partial likelihood follow by differentiation. Define $S_i^{(0)}={\sum }_{j\in R_i}{w}_j$, $S_i^{(1)}={\sum }_{j\in R_i}{w}_j{z}_j$ and $S_i^{(2)}={\sum }_{j\in R_i}{w}_j{z}_j{z}_j^T$. ThenWhen 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 $\hat{\beta }$, estimate the cumulative hazard of the reference subject by the Breslow estimatorIts 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 ${\hat{H}}_i(t)=e^{{\hat{\beta }}^T{z}_i}{\hat{H}}_0(t)$ 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.

The Cox proportional-hazards model specifieswhere $h_0(t)$ is an unspecified common baseline hazard and $z_i$ is the individual's covariate vector. The hazard ratio for two fixed covariate vectors is $\exp [{\beta }^T(z_i-z_k)]$, constant in time. Assume independent subjects and independent censoring conditional on their covariates, and use the question's absence of tied events.

Let $t_j$ be an event time, $i_j$ its failing subject, and $R_j$ the risk set immediately before that time. In a short interval of length $dt$, the conditional probability that subject $i$ fails, given one failure among those at risk, isThe baseline hazard cancels. Multiplying the successive conditional event-identity factors gives the Cox partial likelihood for $\beta$:This is a partial likelihood, not the full survival likelihood for the unspecified $h_0$. Risk sets evolve with the observed history; the derivation uses their successive conditional hazards, rather than assuming the risk sets themselves are independent.

Define $w_{jk}(\beta )=e^{{\beta }^T{z}_k}/{\sum }_{\ell \in R_j}{e}^{{\beta }^T{z}_{\ell }}$ and ${\bar{z}}_j(\beta )={\sum }_{k\in R_j}{w}_{jk}(\beta )z_k$. For component $r$,Differentiating again givesThus the score and information of Cox partial likelihood are the sum of event-minus-risk-mean contrasts and the sum of risk-weighted covariance matrices, respectively. The Hessian is negative semidefinite; the likelihood is concave, with strict identifiability requiring variation in the relevant risk-set covariates.