period of time $[T_j^L,T_j^R]$.

(iii) Right-censoring: when the phenomenon of interest is not observed during the duration of the study. A right-censored observation is an interval of the type $(T_j,\infty]$, where $T_j$ represents the duration of the study for subject $j$.

Ignoring censoring induces bias in the estimation of the parameters. Different types of censoring imply different contributions of the observations to the likelihood function. The contribution of a left-censored observation to the likelihood is $S_l(T_j;\mu,\sigma,\gamma,\delta)$; while the contribution of an interval-censored observation to the likelihood is $S_l(T_j^R;\mu,\sigma,\gamma,\delta)-S_l(T_j^L;\mu,\sigma,\gamma,\delta)$; and the contribution of a right-censored observation to the likelihood is $1-S_l(T_j;\mu,\sigma,\gamma,\delta)$. If we define the sets $\text{Left}=\{j: T_j \text{ is left-censored}\}$, $\text{Int}=\{j: T_j \text{ is interval-censored}\}$, $\text{Right}=\{j: T_j \text{ is right-censored}\}$, and $\text{Obs}=\{j: T_j \text{ in uncensored}\}$, then we can write the likelihood function as follows: $$\begin{aligned}
L(\mu,\sigma,\gamma,\delta) = &\prod_{j\in\text{Obs}} s_l(T_j;\mu,\sigma,\gamma,\delta) \times \prod_{j\in\text{Left}} S_l(T_j;\mu,\sigma,\gamma,\delta) \\&\times  \prod_{j\in\text{Right}}\left[1-S_l(T_j;\mu,\sigma,\gamma,\delta)\right]
\times  \prod_{j\in\text{Int}} \left[ S_l(T_j^R;\mu,\sigma,\gamma,\delta)-S_l(T_j^L;\mu,\sigma,\gamma,\delta)\right].\end{aligned}$$

The latter expression emphasises the practical importance of using distributions with a tractable distribution function.

Accelerated failure time models
-------------------------------

AFT models are a useful tool for modelling the set of survival times ${\bf T}=(T_1,\dots,T_n)$ in terms of a set of covariates $\bm{\beta} = (\beta_1,\dots,\beta_p)$ through the model equation: $$\begin{aligned}
\label{AFTModel}
