The Annals of Statistics

Estimation of a monotone mean residual life

Subhash C. Kochar, Hari Mukerjee, and Francisco J. Samaniego

Full-text: Open access

Abstract

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.

Article information

Source
Ann. Statist., Volume 28, Number 3 (2000), 905-921.

Dates
First available in Project Euclid: 12 March 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1015952004

Digital Object Identifier
doi:10.1214/aos/1015952004

Mathematical Reviews number (MathSciNet)
MR1792793

Zentralblatt MATH identifier
1105.62379

Subjects
Primary: 62P10: Applications to biology and medical sciences 62G05: Estimation 62E10: Characterization and structure theory

Keywords
Mean residual life order restricted inference asymptotic theory

Citation

Kochar, Subhash C.; Mukerjee, Hari; Samaniego, Francisco J. Estimation of a monotone mean residual life. Ann. Statist. 28 (2000), no. 3, 905--921. doi:10.1214/aos/1015952004. https://projecteuclid.org/euclid.aos/1015952004


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