Problem: Maximum partial likelihood estimator for beta (Exam 2022)

Consider a situation where we observe failures occurring \(n\) units of a particular type. Assume we have no censoring and that whenever a unit is failing it is immediately repaired. We assume that a unit that has been repaired may fail again and that the intensity for failure is uninfluenced by previous failures/repairs. We assume there is no limit on how many times a unit can fail, so the number of units at risk is \(n\) at all times.

For each individual we have one (time invariant) binary covariate, \(x_i \in \{0,1\} \) for unit number \(i\). We let the intensity process for failure of unit \(i\) be given by

\[\lambda_i(t) = \alpha_0(t)e^{\beta x_i}\ , \quad \text{for } t\geq 0,\]
where \(\alpha_0(t)\) is an unspecified baseline hazard rate and \(\beta\) is a scalar parameter.

Assuming we observe the process up to a time \(\tau\), find a formula for the partial likelihood function for \(\beta\), and for the corresponding log-partial likelihood function.

Derive a simple expression for the maximum partial likelihood estimator for \(\beta\).