Problem: Maximum likelihood estimators for a shared frailty model (Exam 2022)

Introduction (to part A): Let \(Z\) be exponentially distributed with mean \(1/\lambda\) so that the density function for \(Z\) is

\[ f(z) = \lambda e^{-\lambda z} \quad\text{ for } z \geq 0. \]

Problem A: From the definition of the Laplace transform, show that the Laplace transform of \(Z\), \(\mathscr{L}(c)\), is given by

\[\mathscr{L}(c) = \frac{\lambda}{\lambda + c}.\]

Show by induction that the \(r\)'th derivative of \(\mathscr{L}(c)\) is given by

\[ \mathscr{L}^{(r)}(c) = (-1)^r \frac{\lambda\cdot r!}{(\lambda+c)^{r+1}} \]

Introduction (to part B): Assume we have \(m\) clusters of individuals. Numbering the clusters from \(1\) to \(m\), let \(n_i\) denote the number of individuals in cluster number \(i\). Assume that all the \(n_i\) individuals in cluster number \(i\) share a frailty variable \(Z_i\) and that the hazard rate for each individual in this cluster is given by

\[ \alpha(t\mid Z_i) = Z_i \cdot t^{k-1}, \]
where \(k>0\) is a parameter. Finally we assume \(Z_1, \dots, Z_m\) to be independent and to be exponentially distributed with mean \(1/\lambda\). Note that the values of the two parameters \(\lambda\) and \(k\) are the same for all individuals.

Consider a situation with right censoring, so that for individual number \(j\) in cluster number \(i\) we observe \( (\widetilde{T}_{ij}, D_{ij}) \) where \(D_{ij}\) is an indicator specifying whether or not the lifetime of this individual is censored and \(\widetilde{T}_{ij}\) is the corresponding observed lifetime or censoring time.

Problem B: Find an expression for the log-likelihood function in terms of the quantities specified above and \( D_{i\bullet} = \sum_{j = 1}^{n_i}D_{ij}\).

If possible, find explicit expressions for the maximum likelihood estimators for \(\lambda\) and \(k\). If this is not possible, optimise (if possible) analytically with respect one of the parameters, find the profile likelihood for the other parameter, and explain how the profile likelihood can be used to find the maximum likelihood estimates for \(\lambda\) and \(k\). If it is neither possible to optimise analytically with respect to one of the parameters, discuss briefly how one can then find the maximum likelihood
estimates.