  In this paper we consider the statistical inference of the unknown parameter
of an exponential distribution based on the time truncated data. The time
truncated data occurs quite often in the reliability analysis for type-I or
hybrid censoring cases. All the results available today are based on the
conditional argument that at least one failure occurs during the experiment. In
this paper we provide some inferential results based on the unconditional
argument. We extend the results for some two-parameter distributions also.
