In classical statistics, the observations are frequently assumed to include *independent* random variables *X*_{1}, …, *X*_{n}, with *X*_{i} having the density function

where α_{i}^{θ}(*x*) is the hazard function, *S*_{i}^{θ}(*x*) is the survival function, and θ is a vector of unknown parameters. Then inference on θ may be based on the **likelihood** function,

in the usual way. In survival analysis, however, one can rarely avoid various kinds of incomplete observation. The most common form of this is *right-censoring* where the observations are

(1)

where *D*_{i} is the indicator *I*{_{i} = *X*_{i}}, and _{i} = *X*_{i}, the true survival time, if the observation of the lifetime of *i* is uncensored and _{i} = *U*_{i}, the time of right-censoring, otherwise. Thus, *D*_{i} = 1 indicates an uncensored observation, *D*_{i} = 0 corresponds to a right-censored observation. Other kinds of incomplete observation will ...

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