I want to fit a survival model using a proportional hazards assumption
$$h(t) = h_0(t)\exp(x^T\beta),$$
where $$h_0(t) = \dfrac{\frac{1}{\sigma t} \phi \left(\frac{\log(t) - \mu}{\sigma}\right)}{\Phi \left(-\frac{\log(t) - \mu}{\sigma}\right)},$$ is the baseline hazard associated to the lognormal distribution. The likelihood function is:
$$L(\beta,\mu,\sigma) = \prod_{i=1}^n h(t_i)^{d_i}\exp\left(-H(t_i)\right) ,$$
The corresponding log-likelihood function associated to a sample $t_1,\dots,t_n$ with censoring indicators $d_1,\dots,d_n$ is
$$l(\beta,\mu,\sigma) = \sum_{i=1}^n d_i \log h_0(t_i) + \sum_{i=1}^n d_i x_i^T\beta - \sum_{i=1}^n H_0(t_i)\exp(x_i^T\beta),$$ where $H_0(t) = - \log\Phi \left(-\frac{\log(t) - \mu}{\sigma}\right)$ is the cumulative hazard function.
Question. How can I prove that this log likelihood function is concave?
