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.
"Estimation of a monotone mean residual life." Ann. Statist. 28 (3) 905 - 921, June 2000. https://doi.org/10.1214/aos/1015952004