A clinical study follows a number of patients with an illness. Let $Y_{i} \in[0, \infty)$ be the length of time that patient $i$ lives and $x_{i} \in \mathbb{R}^{p}$ a vector of predictors, for $i \in\{1, \ldots, n\}$. We shall assume that $Y_{1}, \ldots, Y_{n}$ are independent. Let $f_{i}$ and $F_{i}$ be the probability density function and cumulative distribution function, respectively, of $Y_{i}$. The hazard function $h_{i}$ is defined as

$$h_{i}(t)=\frac{f_{i}(t)}{1-F_{i}(t)} \quad \text { for } t \geqslant 0 .$$

We shall assume that $h_{i}(t)=\lambda(t) \exp \left(\beta^{\top} x_{i}\right)$, where $\beta \in \mathbb{R}^{p}$ is a vector of coefficients and $\lambda(t)$ is some fixed hazard function.

(a) Prove that $F_{i}(t)=1-\exp \left(-\int_{0}^{t} h_{i}(s) d s\right)$.

(b) Using the equation in part (a), write the log-likelihood function for $\beta$ in terms of $\lambda, \beta, x_{i}$ and $Y_{i}$ only.

(c) Show that the maximum likelihood estimate of $\beta$ can be obtained through a surrogate Poisson generalised linear model with an offset.