Estimation of Mean Residual Life

  • W. J. Hall
  • Jon A. WellnerEmail author


Yang (Ann Stat, 6:112–116, 1978) considered an empirical estimate of the mean residual life function on a fixed finite interval. She proved it to be strongly uniformly consistent and (when appropriately standardized) weakly convergent to a Gaussian process. These results are extended to the whole half line, and the variance of the limiting process is studied. Also, nonparametric simultaneous confidence bands for the mean residual life function are obtained by transforming the limiting process to Brownian motion.


Life expectancy Consistency Limiting Gaussian process Confidence bands 


  1. 1.
    Balkema, A. A., & de Haan, L. (1974). Residual life time at great age. Annals of Probability, 2, 792–804.CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Barlow, R. E., & Campo, R. (1975). Total time on test processes and applications to failure data analysis. In Reliability and fault tree analysis (Conference, University of California, Berkeley, 1974); Conference on Reliability and Fault Tree Analysis (pp. 451–481). Society for Industrial and Applied Mathematics; Philadelphia.Google Scholar
  3. 3.
    Berger, R. L., Boos, D. D., & Guess, F. M. (1988). Tests and confidence sets for comparing two mean residual life functions. Biometrics, 44, 103–115.CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Billingsley, P. (1968). Convergence of probability measures. New York: Wiley.zbMATHGoogle Scholar
  5. 5.
    Bjerkedal, T. (1960). Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli. American Journal of Hygiene, 72, 130–148.Google Scholar
  6. 6.
    Bryson, M. C., & Siddiqui, M. M. (1969). Some criteria for aging. Journal of the American Statistical Association, 64, 1472–1483.CrossRefMathSciNetGoogle Scholar
  7. 7.
    Chaubey, Y. P., & Sen, A. (2008). Smooth estimation of mean residual life under random censoring. In Beyond parametrics in interdisciplinary research: Festschrift in honor of professor Pranab K. Sen. Institute of Mathematical Statistics Collection (Vol. 1, pp. 35–49). Beachwood: Institute of Mathematical Statistics.Google Scholar
  8. 8.
    Chaubey, Y. P., & Sen, P. K. (1998). On smooth estimation of hazard and cumulative hazard functions. In Frontiers in probability and statistics (Calcutta, 1994/1995) (pp.91–99). New Delhi: Narosa.Google Scholar
  9. 9.
    Chaubey, Y. P., & Sen, P. K. (1999). On smooth estimation of mean residual life. Journal of Statistical Planning and Inference, 75, 223–236.CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Chen, Y. Q., & Cheng, S. (2005). Semiparametric regression analysis of mean residual life with censored survival data. Biometrika, 92, 19–29.CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Chen, Y. Q., & Cheng, S. (2006). Linear life expectancy regression with censored data. Biometrika, 93, 303–313.CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Chen, Y. Q., Jewell, N. P., Lei, X., & Cheng, S. C. (2005). Semiparametric estimation of proportional mean residual life model in presence of censoring. Biometrics, 61, 170–178.CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Chiang, C. L. (1960). A stochastic study of the life table and its applications: I. Probability distributions of the biometric functions. Biometrics, 16, 618–635.CrossRefzbMATHGoogle Scholar
  14. 14.
    Chiang, C. L. (1968). Introduction to stochastic processes in biostatistics. New York: Wiley.zbMATHGoogle Scholar
  15. 15.
    Cox, D. R. (1962). Renewal theory. London: Methuen.zbMATHGoogle Scholar
  16. 16.
    Csörgő, M., Csörgő, S., & Horváth, L. (1986). An asymptotic theory for empirical reliability and concentration processes. Lecture Notes in Statistics (Vol. 33). Berlin: Springer.Google Scholar
  17. 17.
    Csörgő, M., & Zitikis, R. (1996). Mean residual life processes. Annals of Statistics, 24, 1717–1739.Google Scholar
  18. 18.
    Ebrahimi, N. (1993). Estimation of two ordered mean residual lifetime functions. Biometrics, 49, 409–417.CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Gelfand, A. E., & Kottas, A. (2003). Bayesian semiparametric regression for median residual life. Scandinavian Journal of Statistics, 30, 651–665.CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Gross, A. J., Clark, V. A. (1975). Survival distributions: Reliability applications in the biomedical sciences. New York: Wiley.zbMATHGoogle Scholar
  21. 21.
    Guess, F., & Proschan, F. (1988). Mean residual life: Theory and application. In Handbook of statistics: Quality control and reliability (Vol. 7, pp. 215–224). Amsterdam: North-Holland.Google Scholar
  22. 22.
    Gupta, R. C., & Langford, E. S. (1984). On the determination of a distribution by its median residual life function: A functional equation. Journal of Applied Probability, 21, 120–128.CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Hall, W. J., & Wellner, J. A. (1979). Estimation of mean residual life. In Technical Report, Department of Statistics, University of Rochester.Google Scholar
  24. 24.
    Hall, W. J., & Wellner, J. (1981). Mean residual life. In Statistics and related topics (Ottawa, Ont., 1980) (pp. 169–184). Amsterdam: North-Holland.Google Scholar
  25. 25.
    Hollander, M., & Proschan, F. (1975). Tests for the mean residual life. Biometrika, 62, 585–593.CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Horvitz, D. G., & Thompson, D. J. (1952). A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47, 663–685.CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    Hu, X., Kochar, S. C., Mukerjee, H., & Samaniego, F. J. (2002). Estimation of two ordered mean residual life functions. Journal of Statistical Planning and Inference, 107, 321–341.CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Jeong, J.-H., Jung, S.-H., & Costantino, J. P. (2008). Nonparametric inference on median residual life function. Biometrics, 64, 157–163.CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    Joe, H., & Proschan, F. (1984). Comparison of two life distributions on the basis of their percentile residual life functions. The Canadian Journal of Statistics, 12, 91–97.CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Joe, H., & Proschan, F. (1984). Percentile residual life functions. Operations Research, 32, 668–678.CrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    Jupp, P. E., & Mardia, K. V. (1982). A characterization of the multivariate Pareto distribution. Annals of Statistics, 10, 1021–1024.CrossRefMathSciNetzbMATHGoogle Scholar
  32. 32.
    Kochar, S. C., Mukerjee, H., & Samaniego, F. J. (2000). Estimation of a monotone mean residual life. Annals of Statistics, 28, 905–921.CrossRefMathSciNetzbMATHGoogle Scholar
  33. 33.
    Kulkarni, H. V., & Rattihalli, R. N. (2002). Nonparametric estimation of a bivariate mean residual life function. Journal of the American Statistical Association, 97, 907–917.CrossRefMathSciNetzbMATHGoogle Scholar
  34. 34.
    Lillo, R. E. (2005). On the median residual lifetime and its aging properties: a characterization theorem and applications. Naval Research Logistics, 52, 370–380.CrossRefMathSciNetzbMATHGoogle Scholar
  35. 35.
    Ma, C. (1996). Multivariate survival functions characterized by constant product of mean remaining lives and hazard rates. Metrika, 44, 71–83.CrossRefMathSciNetzbMATHGoogle Scholar
  36. 36.
    Ma, C. (1998). Characteristic properties of multivariate survival functions in terms of residual life distributions. Metrika, 47, 227–240.CrossRefMathSciNetzbMATHGoogle Scholar
  37. 37.
    Ma, Y., & Yin, G. (2010). Semiparametric median residual life model and inference. The Canadian Journal of Statistics, 38, 665–679.CrossRefMathSciNetzbMATHGoogle Scholar
  38. 38.
    Maguluri, G., & Zhang, C.-H. (1994). Estimation in the mean residual life regression model. Journal of the Royal Statistical Society: Series B (Methodological), 56, 477–489.MathSciNetzbMATHGoogle Scholar
  39. 39.
    Oakes, D., & Dasu, T. (1990). A note on residual life. Biometrika, 77, 409–410.CrossRefMathSciNetzbMATHGoogle Scholar
  40. 40.
    Oakes, D., & Dasu, T. (2003). Inference for the proportional mean residual life model. In Crossing boundaries: Statistical essays in honor of Jack Hall. IMS Lecture Notes Monograph Series (Vol. 43, 105–116). Beachwood: Institute of Mathematical Statistics.Google Scholar
  41. 41.
    Qin, G., & Zhao, Y. (2007). Empirical likelihood inference for the mean residual life under random censorship. Statistics & Probability Letters, 77, 549–557.CrossRefMathSciNetzbMATHGoogle Scholar
  42. 42.
    Robins, J. M., & Rotnitzky, A. (1992). Recovery of information and adjustment for dependent censoring using surrogate markers. In AIDS epidemiology, methodological issues (pp. 297–331). Boston: BirkhauserCrossRefGoogle Scholar
  43. 43.
    Schmittlein, D. C., & Morrison, D. G. (1981). The median residual lifetime: A characterization theorem and an application. Operations Research, 29, 392–399.CrossRefMathSciNetzbMATHGoogle Scholar
  44. 44.
    Shorack, G. R. (1972). Functions of order statistics. Annals of Mathematical Statistics, 43, 412–427.CrossRefMathSciNetzbMATHGoogle Scholar
  45. 45.
    Shorack, G. R., & Wellner, J. A. (1986). Empirical processes with applications to statistics. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. New York: Wiley.Google Scholar
  46. 46.
    Wellner, J. A. (1977). A Glivenko-Cantelli theorem and strong laws of large numbers for functions of order statistics. Annals of Statistics, 5, 473–480.CrossRefMathSciNetzbMATHGoogle Scholar
  47. 47.
    Wellner, J. A. (1978). Limit theorems for the ratio of the empirical distribution function to the true distribution function. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 45, 73–88.CrossRefMathSciNetzbMATHGoogle Scholar
  48. 48.
    Wilson, E. B. (1938). The standard deviation of sampling for life expectancy. Journal of the American Statistical Association, 33, 705–708.CrossRefzbMATHGoogle Scholar
  49. 49.
    Yang, G. (1977/1978). Life expectancy under random censorship. Stochastic Processes and Their Applications, 6, 33–39.CrossRefMathSciNetzbMATHGoogle Scholar
  50. 50.
    Yang, G. L. (1978). Estimation of a biometric function. Annals of Statistics, 6, 112–116.CrossRefMathSciNetzbMATHGoogle Scholar
  51. 51.
    Zhao, Y., & Qin, G. (2006). Inference for the mean residual life function via empirical likelihood. Communications in Statistics Theory Methods, 35, 1025–1036.CrossRefMathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Biostatistics and Computational BiologyUniversity of RochesterRochesterUSA
  2. 2.University of WashingtonSeattleUSA

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