Abstract
In carcinogenicity experiments with animals where the tumor is not palpable it is common to observe only the time of death of the animal, the cause of death (the tumor or another independent cause, as sacrifice) and whether the tumor was present at the time of death. These last two indicator variables are evaluated after an autopsy. A weighted least squares estimator for the distribution function of the disease onset was proposed. Asymptotic properties of that estimator are established here. We demonstrate its strong uniform consistency. A minimax lower bound for the estimation of the disease onset distribution is obtained, as well as the local asymptotic distribution for their estimator.
Similar content being viewed by others
References
Barlow, R. E., Bartholomew, D. J., Bremner, J. M. and Brunk, H. D. (1972).Statistical Inference Under Order Restrictions, Wiley, New York.
Dudley, R. M. (1989).Real Analysis and Probability, Wadsworth and Brooks/Cole, Pacific Gorve, Calfornia.
Gomes, A. E., Groeneboom, P. and Wellner, J. A. (2001). Nonparametric estimation of the lifetime and disease onset distributions for a survival-sacrifice model, (pre-print), http://ssor.twi.tudelft.nl/pietg/surv.pdf
Groeneboom, P. (1987).Asymptotics for the Interval Censored Observations, Tech. Report, No. 87-18, Department of Mathematics, University of Amsterdam.
Groeneboom, P. (1989). Brownian motion with a parabolic drift and Airy functions,Probability Theory and Related Fields,81, 79–109.
Groeneboom, P. (1996).Lectures on Inverse Problems, Lecture Notes in Mathematics,1648, 67–164, Springer, Berlin.
Groeneboom, P. and Wellner, J. A. (1992).Information Bounds and Nonparametric Maximum Likelihood Estimation, Birkhauser, Boston.
Groeneboom, P. and Wellner, J. A. (2001). Computing Chernoff's distribution,Journal of Computational and Graphical Statistics,10, 388–400.
Holland, J. M., Mitchell, T. J. and Walburg, H. E. (1977). Effects of prepubertal ovariectomy on survival and specific diseases in female RFM mice given 300 R of X rays,Radiation Research,69, 317–327.
Jewell, N. P. (1982). Mixtures of exponential distribution,The Annals of Statistics,10, 479–484.
Kodell, R., Shaw, G. and Johnson, A. (1982). Nonparametric joint estimators for disease resistance and survival functions in survival/sacrifice experiments,Biometrics,38, 43–58.
Robertson, T., Wright, F. T. and Dykstra, R. L. (1988).Order Restricted Statistical Inference, Wiley, New York.
Turnbull, B. W. and Mitchell, T. J. (1984). Nonparametric estimation of the distribution of time to onset for specific diseases in survival/sacrifice experiments,Biometrics,40, 41–50.
van der Laan, M. J., Jewell, N. P. and Peterson, D. (1997). Efficient estimation of the lifetime and disease onset distribution,Biometrika,84, 539–554.
van der Vaart, A. W. and Wellner, J. A. (1996).Weak Convergence and Empirical Processes, Springer, New York.
Wang, J.-G. (1987). A note on the uniform consistency of the Kaplan-Meier estimator,The Annals of Statistics,15, 1313–1316.
Author information
Authors and Affiliations
About this article
Cite this article
Gomes, A.E. Asymptotics for a weighted least squares estimator of the disease onset distribution function for a survival-sacrifice model. Ann Inst Stat Math 56, 683–700 (2004). https://doi.org/10.1007/BF02506483
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02506483