# Nonparametric Estimation of Mean Residual Life Function Using Scale Mixtures

## Abstract

It is often of interest in clinical trials and reliability studies to estimate the remaining lifetime of a subject or a device given that it survived up to a given period of time, that is commonly known as the so-called *mean residual life function (mrlf)*. There have been several attempts in literature to estimate the mrlf nonparametrically ranging from empirical estimates to more sophisticated smooth estimation. Given the well known one-to-one relation between survival function and mrlf, one can plug-in any known estimates of the survival function (e.g., Kaplan–Meier estimate) into the functional form of mrlf to obtain an estimate of mrlf. In this chapter, we present a scale mixture representation of mrlf and use it to obtain a smooth estimate of the mrlf under right censoring. Asymptotic properties of the proposed estimator are also presented. Several simulation studies and a real data set are used for investigating the empirical performance of the proposed method relative to other well-known estimates of mrlf. A comparative analysis shows computational advantages of the proposed estimator in addition to somewhat superior statistical properties in terms of bias and efficiency.

## References

- 1.Abdous, B., and A. Berred. 2005. Mean residual life estimation.
*Journal of Statistical Planning and Inference*132: 3–19.MathSciNetCrossRefMATHGoogle Scholar - 2.Agarwal, S.L., and S.L. Kalla. 1996. A generalized gamma distribution and its application in reliability.
*Communications in Statistics—Theory and Methods*25: 201–210.MathSciNetCrossRefMATHGoogle Scholar - 3.Bhattacharjee, M.C. 1982. The class of mean residual lives and some consequences.
*SIAM Journal of Algebraic Discrete Methods*3: 56–65.MathSciNetCrossRefMATHGoogle Scholar - 4.Chaubey, Y.P., and P.K. Sen. 1999. On smooth estimation of mean residual life.
*Journal of Statistical Planning and Inference*75: 223–236.MathSciNetCrossRefMATHGoogle Scholar - 5.Chaubey, Y.P., and A. Sen. 2008. Smooth estimation of mean residual life under random censoring. In Ims collections—beyond parametrics in interdisciplinary research: Festschrift in honor of professor pranab k.
*Sen*1: 35–49.Google Scholar - 6.Elandt-Johnson, R.C., and N.L. Johnson. 1980.
*Survival models and data analysis*. New York: Wiley.MATHGoogle Scholar - 7.Feller. 1968.
*An introduction to probability theory and its applications*. vol. I, 3rd ed. New York: Wiley.Google Scholar - 8.Ghorai, J., A. Susarla., V, Susarla., and Van-Ryzin, J. 1982. Nonparametric Estimation of Mean Residual Life Time with Censored Data. In
*Nonparametric statistical inference. vol. I*, Colloquia Mathematica Societatis, 32. North-Holland, Amsterdam-New York, 269-291.Google Scholar - 9.Guess, F, and Proschan, F. (1988). Mean residual life theory and applications. In
*Handbook of statistics 7, reliability and quality control*, ed. P.R. Krishnaiah., and Rao, C.R. 215–224.Google Scholar - 10.Gupta, R.C., and D.M. Bradley. 2003. Representing the mean residual life in terms of the failure rate.
*Mathematical and Computer Modelling*37: 1271–1280.MathSciNetCrossRefMATHGoogle Scholar - 11.Gupta, R.C., and S. Lvin. 2005. Monotonicity of failure rate and mean residual life function of a gamma-type model.
*Applied Mathematics and Computation*165: 623–633.MathSciNetCrossRefMATHGoogle Scholar - 12.Hall, W.J., and J.A. Wellner. 1981. Mean residual life. In
*Statistics and related topics*, ed. M. Csorgo, D.A. Dawson, J.N.K. Rao, and AKMdE Saleh, 169–184. Amsterdam, North-Holland.Google Scholar - 13.Hille, E. (1948).
*Functional analysis and semigruops*. American Mathematical Society vol.31.Google Scholar - 14.Kalla, S.L., H.G. Al-Saqabi, and H.G. Khajah. 2001. A unified form of gamma-type distributions.
*Applied Mathematics and Computation*118: 175–187.MathSciNetCrossRefMATHGoogle Scholar - 15.Kaplan, E.L., and P. Meier. 1958. Nonparametric estimation from incomplete observations.
*Journal of American Statistical Association*53: 457–481.MathSciNetCrossRefMATHGoogle Scholar - 16.Kopperschmidt, K., and U. Potter. 2003. A non-parametric mean residual life estiamtor: An example from market research.
*Developmetns in Applied Statistics*19: 99–113.Google Scholar - 17.Kulasekera, K.B. 1991. Smooth Nonparametric Estimation of Mean Residual Life.
*Microelectronics Reliability*31 (1): 97–108.MathSciNetCrossRefGoogle Scholar - 18.Lai, C.D., L. Zhang, and M. Xie. 2004. Mean residual life and other properties of weibull related bathtub shape failure rate distributions.
*International Journal of Reliability, Quality and Safety Engineering*11: 113–132.CrossRefGoogle Scholar - 19.Liu, S. (2007).
*Modeling mean residual life function using scale mixtures*NC State University Dissertation. http://www.lib.ncsu.edu/resolver/1840.16/3045. - 20.McLain, A., and S.K. Ghosh. 2011. Nonparametric estimation of the conditional mean residual life function with censored data.
*Lifetime Data Analysis*17: 514–532.MathSciNetCrossRefMATHGoogle Scholar - 21.Morrison, D.G. 1978. On linearly increasing mean residual lifetimes.
*Journal of Applied Probability*15: 617–620.MathSciNetCrossRefMATHGoogle Scholar - 22.Petrone, S., and P. Veronese. 2002. Non parametric mixture priors based on an exponential random scheme.
*Statistical Methods and Applications*11 (1): 1–20.CrossRefMATHGoogle Scholar - 23.Ruiz, J.M., and A. Guillamon. 1996. nonparametric recursive estimator of residual life and vitality funcitons under mixing dependence condtions.
*Communcation in Statistics–Theory and Methods*4: 1999–2011.Google Scholar - 24.Swanepoel, J.W.H., and F.C. Van Graan. 2005. A new kernel distribution function estimator based on a non-parametric transformation of the data.
*The Scadinavian Journal of Statistics*32: 551–562.MathSciNetCrossRefMATHGoogle Scholar - 25.Tsang, A.H.C., and A.K.S. Jardine. 1993. Estimation of 2-parameter weibull distribution from incomplete data with residual lifetimes.
*IEEE Transactions on Reliability*42: 291–298.CrossRefMATHGoogle Scholar - 26.Yang, G.L. 1978. Estimation of a biometric function.
*Annals of Statistics*6: 112–116.MathSciNetCrossRefMATHGoogle Scholar