  In survival analysis, the lifetime under study is not always observed. In
certain applications, for some individuals, the value of the lifetime is only
known to be smaller or larger than some random duration. This framework
represent an extension of standard situations where the lifetime is only left
or only right randomly censored. We consider the case where the independent
observation units include also some covariates, and we propose two
semiparametric regression models. The new models extend the standard Cox
proportional hazard model to the situation of a more complex censoring
mechanism. However, like in Cox's model, in both models the nonparametric
baseline hazard function still could be expressed as an explicit functional of
the distribution of the observations. This allows to define the estimator of
the finite-dimensional parameters as the maximum of a likelihood-type criterion
which is an explicit function of the data. Given an estimate of the
finite-dimensional parameter, the estimation of the baseline cumulative hazard
function is straightforward.
