Properties of non-parametric Stute estimators

  • Didier Alain NJAMEN NJOMEN Department of Mathematics and Computer Science, Faculty of Science, University of Maroua
Keywords: Censored lifetimes, Linear regression, Kaplan-Meier jump, Nonparametric estimator, Asymptotic distribution.


In this paper, we use the linear regression model for survival data, explaining that it corresponds to an accelerated time model of lifetime see as described in Kalbfleisch and Prentice [12] and Koul et al. [15]. In this context, we adapt the jumps of the KM estimator as defined in Lopez [16] to the accelerated lifetime model. The introduction of a more restrictive hypothesis allows us to establish a strong consistency property of the Stute [24] estimator obtained by minimizing the sum of the least squares. Using the asymptotic normality of the bivariate distribution estimator proposed by Stute [26] and the Slutsky theorem, we succeed in establishing the asymptotic distribution of the Stute [24]estimator.


O. O. Aalen, A linear regression model for the analysis of life times, Statistics in medicine, Wiley Online Library, vol. 8, no. 8,pp. 907–925, 1989. DOI: 10.1002/sim.4780080803.

O. O. Aalen, Further results on the non-parametric linear regression model in survival analysis, Statistics in medicine, Wiley Online Library, vol. 12, no. 17, pp. 1569–1588, 1993. DOI: 10.1002/sim.4780121705.

P. K. Andersen, and R. D. Gill, Cox’s regression model for counting processes: a large sample study The annals of statistics, JSTOR,vol. 10, no. 4, pp. 1100–1120, 1982.

V. Bagdonavicius, and M. Nikulin, Accelerated life models: modeling and statistical analysis, CRC Press, 2001

J. Buckley, and I. James,Linear regression with censored data, Biometrika, vol. 66, no. 3, pp. 429–436, 1979. DOI: 10.230/2335161.

D. R. Cox, Regression models and life-tables, Wiley for the Royal Statistical Society, vol. 34, no. 2, pp. 187–220, 1972.

T. Fleming, and D. Harrington, Counting processes and survival Analysis, John Wiley and Sons, Inc., New York. 1991.

R. Gill, Censoring and Stochastic Integrals, Statistica Neerlandica, Wiley Online Library, vol.34, no.2, 1980. DOI:10.1111/j.1467-9574.198.tb00692.x.

R. Gill, Testing with replacement and the product limit estimator, The Annals of Statistics, vol. 9, no. 4, pp. 853–860, 1981.

R. Gill, Large sample behaviour of the product-limit estimator on the whole line Ann. Statist. vol. 11, no. 1, pp. 49–58, 1983.

B. Jørgensen, The theory of Linear Models, New York: Chapman and Hall, vol. 21, 1993.

J. Kalbfleisch, and R. Prentice, The survival analysis of failure time data, John Wiley & Sons, Inc., Hoboken, New Jersey. 2nd edn.,2002.

E.L.Kaplan,andP.Meier, Nonparametricestimationfromincompleteo

bservations, Journal of the American Statistical Association,vol. 53, no. 282, pp. 457–481, 1958.

J. Klein, and M. Moeschberger, Survival analysis: Techniques for censored and truncated regression, Statistics for Biology and Health, New York, NY: Springer-Verlag, 1997.

H. Koul, V. Susarla, and J. V. Ryzin, Regression analysis with randomly right-censored data The Annals of Statistics,vol. 9, no. 6,pp. 1276-1288, 1981.

O. Lopez, Réduction de dimension en présence de données censurées Ph. D. thesis, ENSAE ParisTech,, 2007.

R. G. Miller, Least squares regression with censored data, Biometrika, vol. 63, no. 3, pp. 449-464, 1976. DOI:10.2307/2335722.

D. A. Njamen-Njomen, and J. Ngatchou-Wandji, Nelson-Aalen and Kaplan-Meier Estimators in Competing Risks, Applied Mathematics,vol. 5, no. 4, pp. 765–776. 2014.,

D. A. Njamen Njomen, Convergence of the Nelson-Aalen Estimator in Competing Risks, International Journal of Statistics and Probability, vol. 6, no. 3, pp. 9–23. Canadian Center of Science and Education. 2017. doi:10.5539/ijsp.v6n3p9.

D. A. Njamen Njomen, and J. Ngatchou-Wandji (2018).Consistency of the Kaplan-Meier Estimator of the Survival Function in Competiting Risks,The Open Statistics and Probability Journal, vol. 9, pp. 1–17. DOI:10.2174/1876527001809010001,

C. R. Rao, and H. Toutenburg, Linear models, Least squares and alternatives. Springer Series in Statistics, New York : Springer-Verlag, 1995.

A. C. Rencher, and G. B. Schaalje, Linear models in statistics, New York, John Wiley & Sons, Inc., Hoboken, New Jersey, Second edition, 2008.

S. R. Searle, Linear models, Reprint of the 1971 original, Wiley Classics Library, New York: John Wiley and Sons Inc., 1997.

W. Stute, Consistent estimation under random censorship when covariables are present, Journal of Multivariate Analysis, vol. 45,no. 1, pp. 89–103, 1993. doi¿10.1006/jmva.1993.1028

W. Stute, and J. L. Wang The strong law under randomcensorship, The Annals of Statistics, vol. 21, no. 3, pp. 1591–1607, 1993.

W. Stute, Distributional convergence under random censorship when covariables are present, Scandinavian journal of statistics,vol. 23, no. 4, pp. 461–471, 1996.

V. Susarla, and J. V. Ryzin, Large sample theory for an estimator of the mean survival time from censored samples, The Annals of Statistics, vol. 8, no. 5, pp. 1002–1016, 1980.

W. Weibull, The Phenomenon of Rupture in Solids, Ingeniors Vetenskaps Akademien Handlinga, Stockholm, Sweden, vol. 55,no. 153, 1939.

How to Cite
NJAMEN NJOMEN, D. A. (2019). Properties of non-parametric Stute estimators. Statistics, Optimization & Information Computing, 7(2), 370-382.
Research Articles