**Dr. Hari Mukerjee**

Wichita State University

**"Estimation of a Monotone Mean Residual Life"**

**Abstract:**

In survival analysis and in the analysis of life tables an important biometric function of interest is the life expectancy at age defined by

where is a lifetime. Is called the mean residual life function. In many applications it is reasonable to assume that is decreasing (DMRL)
or increasing (IMRL); we write decreasing (increasing) for nonincreasing (nondecreasing). There is some literature on empirical estimators of
and their properties. Although tests for a monotone are discussed in the literature, we are not aware of any estimators of 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 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.

#### Friday, February 12, 1999

3:00 PM in 335 Jabara Hall

Please join us for refreshments before the lecture at 2:30p.m. in room 353 Jabara Hall.

[ Spring 1999]