Smooth estimation of mean residual life under random censoring
Yogendra P. Chaubey, Arusharka Sen
Abstract
We propose here a smooth estimator of the mean residual life function based on randomly censored data. This is derived by smoothing the product-limit estimator using the Chaubey-Sen technique (Chaubey and Sen (1998)). The resulting estimator does not suffer from boundary bias as is the case with standard kernel smoothing. The asymptotic properties of the estimator are investigated. We establish strong uniform consistency and asymptotic normality. This complements the work of Chaubey and Sen (1999) which considered a similar estimation procedure in the case of complete data. It is seen that the properties are similar, though technically more difficult to prove, to those in the complete data case with appropriate modifications due to censoring.
First Page:
Show
Hide
Primary Subjects: 62G05, 62G20
Secondary Subjects: 62G07
Keywords: asymptotics; Hille’s theorem; mean residual life; random censoring; smoothing; survival function
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.imsc/1207058262
Digital Object Identifier: doi:10.1214/193940307000000031
References
[1] Abdous, B. and Berred, A. (2005). Mean residual life estimation. J. Statist. Plann. Inference 132 3–19.
Mathematical Reviews (MathSciNet):
MR2163677
Zentralblatt MATH:
1075.62089
Digital Object Identifier: doi:10.1016/j.jspi.2004.06.012
[2] Chaubey, Y. P. and Sen, P. K. (1996). On smooth estimation of survival and density functions. Statistics and Decisions 14 1–22.
Mathematical Reviews (MathSciNet):
MR1381202
[3] Chaubey, Y.P. and Sen, P.K. (1997). On smooth estimation of hazard and cumulative hazard functions. In Frontiers of Probability and Statistics (S. P. Mukherjee et al., eds.) 91–99. Narosa Publishing House, New Delhi.
[4] Chaubey, Y. P. and Sen, P. K. (1998). On smooth functional estimation under random censorship. In Frontiers in Reliability 4. Series on Quality, Reliability and Engineering Statistics (A. P. Basu et al., eds.) 83–97. World Scientific, Singapore.
[5] Chaubey, Y. P. and Sen, P. K. (1999). On smooth estimation of mean residual life. J. Statist. Plann. Inference 75 223–236.
Mathematical Reviews (MathSciNet):
MR1678973
Zentralblatt MATH:
0943.62101
Digital Object Identifier: doi:10.1016/S0378-3758(98)00144-X
[6] Csörgo, M. and Zitikis, R. (1996). Mean residual life process. Ann. Statist. 24 1717–1739.
Mathematical Reviews (MathSciNet):
MR1416657
Zentralblatt MATH:
0933.62106
Digital Object Identifier: doi:10.1214/aos/1032298292
Project Euclid: euclid.aos/1032298292
[7] Efron, B. (1967). The power likelihood ratio test. Ann. Math. Statist. 38 802–806.
Mathematical Reviews (MathSciNet):
MR212935
Digital Object Identifier: doi:10.1214/aoms/1177698874
Project Euclid: euclid.aoms/1177698874
[8] Feller, W. (1965). An Introduction to Probability Theory and Its Applications. II. Wiley, New York.
[9] Ghorai, J., Susarla, A., Susarla, V. and van Ryzin, J. (1980). Non-parametric estimation of mean residual life with censored data. In Colloquia Mathematica Societatis Janos Bolyai 32. Nonparametric Statistical Inference (B. V. Gnedenko et al., eds.) 269–291. North-Holland, Amsterdam.
[10] Deheuvels, P. and Einmahl, J. H. J. (2000). Functional limit laws for the increments of Kaplan-Meier product-limit processes and applications. Ann. Probab. 28 1301–1335.
Mathematical Reviews (MathSciNet):
MR1797314
Zentralblatt MATH:
1016.62031
Digital Object Identifier: doi:10.1214/aop/1019160265
Project Euclid: euclid.aop/1019160336
[11] Gill, R. D. (1983). Large sample behaviour of the product-limit estimator on the whole line. Ann. Statist. 19 1457–1470.
Mathematical Reviews (MathSciNet):
MR684862
Zentralblatt MATH:
0518.62039
Digital Object Identifier: doi:10.1214/aos/1176346055
Project Euclid: euclid.aos/1176346055
[12] Hall, W. J. and Wellner, J. A. (1981). Mean residual life. In Statistics and Related Topics (M. Csörgö et al., eds.) 169–184. North-Holland, Amsterdam.
Mathematical Reviews (MathSciNet):
MR665274
Zentralblatt MATH:
0481.62078
[13] Hille, E. (1948). Functional Analysis and Semigroups. Amer. Math. Soc. Colloq. Pub. 31. American Mathematical Society, New York.
Mathematical Reviews (MathSciNet):
MR25077
[14] Kim, J. S. and Proschan, F. (1991). Piecewise exponential estimator of the survivor function. IEEE Trans. Reliability 40 134–139.
[15] Mielniczuk, J. (1986). Some aymptotic properties of kernel estimators of a density function in case of censored data. Ann. Statist. 14 766–773.
Mathematical Reviews (MathSciNet):
MR840530
Zentralblatt MATH:
0603.62047
Digital Object Identifier: doi:10.1214/aos/1176349954
Project Euclid: euclid.aos/1176349954
[16] Padgett, W. J. and McNichols, D. T. (1984). Nonparametric density estimation from censored data. Comm. Statist. Theory Methods A13 1581–1611.
Mathematical Reviews (MathSciNet):
MR752184
Digital Object Identifier: doi:10.1080/03610928408828780
[17] Ruiz, J. M. and Guillamòn, A. (1996). Nonparametric recursive estimator of mean residual life and vitality parameter under mixing dependent conditions. Comm. Statist. Theory Methods 4 1999–2011.
[18] Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
Mathematical Reviews (MathSciNet):
MR838963
[19] Stute, W. (1995). The central limit theorem under random censorship. Ann. Statist. 23 422–439.
Mathematical Reviews (MathSciNet):
MR1332574
Zentralblatt MATH:
0829.62055
Digital Object Identifier: doi:10.1214/aos/1176324528
Project Euclid: euclid.aos/1176324528
[20] Stute, W. and Wang, J. L. (1993). The strong law under random censorship. Ann. Statist. 21 1591–1607.
Mathematical Reviews (MathSciNet):
MR1241280
Zentralblatt MATH:
0785.60020
Digital Object Identifier: doi:10.1214/aos/1176349273
Project Euclid: euclid.aos/1176349273
[21] Yang, G. L. (1977). Life expectancy under random censorship. Stochastic Process. Appl. 6 33–39.
Mathematical Reviews (MathSciNet):
MR458678
Zentralblatt MATH:
0372.62075
Digital Object Identifier: doi:10.1016/0304-4149(77)90015-1
[22] Yang, G. L. (1978). Estimation of a biometric function. Ann. Statist. 6 1–116.
Mathematical Reviews (MathSciNet):
MR471233
Zentralblatt MATH:
0371.62055
Digital Object Identifier: doi:10.1214/aos/1176344070
Project Euclid: euclid.aos/1176344070
Institute of Mathematical Statistics Collections
