In survival analysis and in the analysis of life tables an
important biometric function of interest is the life expectancy at age $x,
M(x)$, defined by
$$M(x) = E[X - x|X > x],$$
¶ where $X$ is a lifetime. $M$ is called the mean residual life
function.In many applications it is reasonable to assume that $M$ is decreasing
(DMRL) or increasing (IMRL); we write decreasing (increasing) for nonincreasing
(non-decreasing). There is some literature on empirical estimators of $M$ and
their properties. Although tests for a monotone $M$ are discussed in the
literature, we are not aware of any estimators of $M$ under these order
restrictions. In this paper we initiate a study of such estimation. Our
projection type estimators are shown to be strongly uniformly consistent on
compact intervals, and they are shown to be asymptotically
“root-$n$” equivalent in probability to the (unrestricted)
empirical estimator when $M$ is strictly monotone. Thus the monotonicity is
obtained “free of charge”, at least in the aymptotic sense. We
also consider the nonparametric maximum likelihood estimators. They do not
exist for the IMRL case. They do exist for the DMRL case, but we have found the
solutions to be too complex to be evaluated efficiently.