For each of the following situations develop an appropriate expression for the joint likelihood of $N$ observations in terms of the duration density $f(t\mid \mathbf{x},\mathit{\theta })$ and survivor function $S(t\mid \mathbf{x},\mathit{\theta })$.

(a) A sample of independent completed durations, $t_i,i=1,\dots ,N$, is available.

(b) The sample is generated as follows. Initially, individuals are selected from a pool of unemployed and interviewed. Subsequently, they are reinterviewed after $h$ periods. Selected individuals have been unemployed for $t$ weeks on selection. Between selection and interview some find jobs, and others do not. For those who have jobs the time of termination of unemployment spells is known.

(c) The situation is the same as in (b) except that it is not known when the unemployment spell ended.