In this paper we consider the problems of testing 
: a mean residual life (MRL) is exponential
versus 
: it is new better than used in expectation (NBUE), but not exponential
first and then versus 
: it is decreasing MRL (DMRL), but not exponential. The
paper gives a nice chronological summary of the tests developed for these two classes both
for the complete data and incomplete data cases. We have developed a new test for each
class under censoring. Both tests are shown to be consistent in their classes. We have
derived the asymptotic distributions of our test statistics.