Journal of Statistical Planning and Inference 75 (1999) 223–236 On smooth estimation of mean residual life Yogendra P. Chaubey a; ∗ , Pranab K. Sen b a Department of Mathematics & Statistics, Concordia University, Montreal, Quebec, Canada b University of North Carolina, Chapel Hill, NC, USA Abstract The methodology developed in Chaubey and Sen (1996), (Statistics and Decision, 14, 1–22) is adopted here for smooth estimation of mean residual life. It is seen that Hille (1948), (Functional Analysis and Semigroups, AMS, New York) theorem, which has been vital in the development of smooth estimators of the distribution, density, hazard and cumulative hazard functions, does not work well in the current context. For this reason a modied weighting scheme is proposed for estimation of the mean residual life. Asymptotic properties of the resulting estimator is investigated along with its aging aspects. c  1999 Elsevier Science B.V. All rights reserved. AMS classications: 62G07; 62G20 Keywords: Exponential smoothing; Hille’s theorem; Strong consistency; Survival function; Tail-behavior 1. Introduction Let T be a non-negative random variable with distribution function (d.f.) F (·), and density function f(·). Denote the associated survival function (s.f.) and hazard function by S (t )=1 − F (t ) and h(t )= f(t )=S (t )= −(d= dt ) log S (t ), respectively. Fur- ther, H (t ), the cumulative hazard function (c.h.f.), is dened by H (t )=  t 0 h(y)d y = − log S (t ); t ¿0. All these concepts have been extensively used in reliability and survival analysis. Another function, known as life expectancy at age t in life table analysis, and mean residual life (MRL) in survival analysis is dened as m(t )= E(T − t |T ¿t ) = 1 S (t )  ∞ t S (y)d y: (1.1) The reader may be referred to Guess and Proschan (1988) for an extensive review of this function. In this paper we are concerned with estimation of m(t ) when f(t ) is * Correspondence address: Department of Mathematics & Statistics, Concordia University, Montreal, Quebec, Canada. E-mail: chaubey@vax2.concordia.ca. 0378-3758/99/$ – see front matter c  1999 Elsevier Science B.V. All rights reserved. PII: S0378-3758(98)00144-X