The Annals of Statistics

Estimations in homoscedastic linear regression models with censored data: an empirical process approach

Fushing Hsieh

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Pertaining to the estimating equations proposed by Tsiatis, based on the linear rank test, we show the existence of local confounding between the baseline hazard function and the covariates. Due to the local confounding, an estimating equation in Tsiatis' family with a larger time point of truncation could contain less information about the regression parameter than the estimating equation with a smaller time point of truncation. This phenomenon further indicates significant loss of efficiency of Tsiatis' estimating equations as well as the power loss of log-rank type tests when the baseline hazard function goes up and down along the time scale. To take care of this local confounding without using nonparametric estimates of the derivative of the baseline hazard function, we propose the empirical process approach (EPA) based on an empirical process constructed from Tsiatis' log-rank estimating equation by varying its truncating time point. The EPA will provide very tractable estimations of the regression parameters as well as Pearson's chi-squared statistics for testing the model's assumptions. Specifically, the performance of the EPA estimator is shown to be very close to the best estimator in Tsiatis' family.

Article information

Ann. Statist., Volume 25, Number 6 (1997), 2665-2681.

First available in Project Euclid: 30 August 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62G20: Asymptotic properties

Accelerated failure time model log-rank test local confounding martingale central limit theorem time-dependent covariates


Hsieh, Fushing. Estimations in homoscedastic linear regression models with censored data: an empirical process approach. Ann. Statist. 25 (1997), no. 6, 2665--2681. doi:10.1214/aos/1030741090.

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  • ANDERSEN, P. K., BORGAN, O., GILL, R. D. and KEIDING, N. 1992. Statistical Models Based on Counting Processes. Springer, New York. Z.
  • ANDERSEN, P. K. and GILL, R. D. 1982. Cox's regression model for counting processes: a large sample study. Ann. Statist. 10 1100 1120. Z.
  • BERKSON, J. 1980. Minimum chi-square, not maximum likelihood! Ann. Statist. 8 547 469. Z.
  • BICKEL, P. J. 1978. Using residuals robustly. I: tests for heteroscedasticity, nonlinearity. Ann. Statist. 6 266 291. Z.
  • BUCKLEY, P. J. and JAMES, I. 1979. Linear regression with censored data. Biometrika 66 429 436. Z.
  • BURKE, M. D., CSORGO, S. and HORVATH, L. 1988. A correction to and improvement of strong ¨ ´ approximations of some biometric estimates under random censorship. Probab. Theory Related Fields 79 51 57. Z.
  • CARROLL, R. J. and RUPPERT, D. 1988. Transformation and Weighting in Regression. Chapman and Hall, London. Z.
  • COX, D. R. and OAKES, D. 1984. Analy sis of Survival Data. Chapman and Hall, London. Z.
  • Fy GENSON, F. and RITOV, Y. 1994. Monotone estimating equations for censored data. Ann. Statist. 22 732 746. Z.
  • HSIEH, F. 1996a. Empirical process approach in two-sample location-scale model with censored data. Ann. Statist. 24 2705 2719. Z. Z.
  • HSIEH, F. 1996b. Empirical process approach EPA in heteroscedastic regression models with right-censored data. Unpublished manuscript. Z.
  • HSIEH, F. and HSU, C. 1996. Empirical process approach in heteroscedastic linear model with sy mmetric error. Unpublished manuscript. Z.
  • HSIEH, F. and TURNBULL, B. W. 1996. The empirical process approach for general semiparametric regression models: theory and applications. Unpublished lecture notes. Z.
  • KALBFLEISCH, J. D. and PRENTICE, R. L. 1980. The Statistical Analy sis of Failure Time Data. Wiley, New York. Z.
  • KOLMOS, J., MAJOR, P. and TUSNADY, G. 1975. An approximation of partial sum of independent ´ ´ R.V.'s and sample D.F. I. Z. Wahrsch. Verw. Gebiete 32 111 131. Z.
  • KONING, A. 1994. Approximation of the basic martingale. Ann. Statist. 22 565 579. Z.
  • KOUL, H., SUSARLA, V. and VAN RAy ZIN, J. 1981. Regression analysis with randomly right censored data. Ann. Statist. 9 1276 1288. Z.
  • LAI, T. L. and YING, Z. 1992. Linear rank statistics in regression analysis with censored or truncated data. J. Multivariate Anal. 40 13 45. Z.
  • LOUIS, T. A. 1981. Nonparametric analysis of an accelerated failure time model. Biometrika 68 381 390. Z.
  • MILLER, R. 1976. Least squares regression with censored data. Biometrika 63 449 464. Z.
  • POLLARD, D. 1990. Empirical Processes: Theory and Applications. IMS, Hay ward, CA. Z.
  • PORTNOY, S. 1988. Asy mptotic behavior of likelihood methods for exponential family when the number of parameters tends to infinity. Ann. Statist. 16 356 366. Z. RAMLAU-HANSEN, H. 1983. Smoothing counting process intensities by means of kernel functions. Ann. Statist. 11 453 466. Z.
  • RITOV, Y. 1990. Estimation in a linear regression model with censored data. Ann. Statist. 18 303 328. Z.
  • RITOV, Y. and WELLNER, J. A. 1988. Censoring, martingales and the Cox model. Contemp. Math. 80 191 220. Z.
  • ROBINS, J. and TSIATIS, A. A. 1992. Semiparametric estimation of an accelerated failure time model with time-dependent covariates. Biometrika 79 311 319. Z.
  • SLUD, E. V. 1992. Partial likelihood of continuous-time stochastic processes. Scand. J. Statist. 19 97 109. Z.
  • TSIATIS, A. A. 1990. Estimating regression parameters using linear rank tests for censored data. Ann. Statist. 18 354 372.
  • WEI, L. J. and GAIL, M. H. 1983. Nonparametric estimation for a scale-change with censored observations. J. Amer. Statist. Assoc. 78 382 388. Z.
  • YING, Z. 1993. A large sample study of rank estimation for censored regression data. Ann. Statist. 21 76 99. DEPARTMENT OF MATHEMATICS NATIONAL TAIWAN UNIVERSITY TAIPEI TAIWAN E-MAIL: