In survival analysis and in the analysis of the life tables an important biometric function of interest is the mean residual life function (MRLF) M whose value at age t is the average future life time given that a subject has survived till time t. Two important classes of the MRLFs are the New Better than Used in Expectation (NBUE) class' and Decreasing Mean Residual Life (DMRL) class'. These two classes are defined as {M(t):M(t)≤M(0), t≥0} and {M(t):M(t)≤M(s), if t≥s} respectively. In this dissertation we consider the problem of estimation and testing for the distributions in the NBUE and DMRL classes under random censoring. Our order restricted estimators of M for the NBUE and DMRL classes are respectively M*n(t) = Mn(t)٨Mn(0) and Mn**(t) = infy≤t Mn(y), where Mn is the Kaplan-Meier estimator of M. We have proven that both these estimators are uniformly strongly consistent, and converge weakly to a Gaussian process. By simulation, we have also shown that our estimators are better than Mn in terms of asymptotic mean sums of squares. Several applications of our estimators have been provided. Tests that identify the NBUE or DMRL behavior have been developed. Both tests are shown to be consistent in their classes. We have derived the asymptotic distributions of our test statistics.