A consistent estimator in the accelerated failure time model with censored observations and measurement errors. (English. Ukrainian original) Zbl 1232.62129

Usol’tseva, O. S.

Theory Probab. Math. Stat. 82, 161-169 (2011); translation from Teor. Jmovirn. Mat. Stat. No. 82, 156-162.

Summary: We consider the following accelerated failure time model used in the statistical analysis of the survival data: \[ T_i=\exp\left\{\beta_0+\beta_X^T X_i+\varepsilon_i\right\}, \quad i\geq 1. \] The life times \( T_i\) are observed under censoring. We also observe the vectors \(W_i=X_i+U_i\) instead of the regressors \(X_i\), where the \(U_i\) are measurement errors. The vector of the regression parameters \(\beta=\bigl(\beta_0,\beta_X^T\bigr)^T\) is estimated from the observations. We construct an estimator as a solution of the corresponding unbiased estimating equation and show that this estimator is consistent if the censoring distribution is known. We also prove the consistency of the estimators for the case of an unknown censoring distribution if the regressors \(X_i\) are bounded and the errors \(\varepsilon_i\) are bounded from above. For the latter case, we estimate the censoring distribution by the Kaplan-Meier method.

MSC:

62N02	Estimation in survival analysis and censored data
62N05	Reliability and life testing
62N01	Censored data models
62G05	Nonparametric estimation
62F12	Asymptotic properties of parametric estimators

References:

[1]	T. Augustin, Survival Analysis under Measurement Error, Habilitationsschrift, Universität München, 2002.
[2]	Antónia Földes and Lidia Rejtö, Strong uniform consistency for nonparametric survival curve estimators from randomly censored data, Ann. Statist. 9 (1981), no. 1, 122 – 129. · Zbl 0453.62034
[3]	Christopher C. Heyde, Quasi-likelihood and its application, Springer Series in Statistics, Springer-Verlag, New York, 1997. A general approach to optimal parameter estimation. · Zbl 0879.62076
[4]	A. Kukush and Y. Chernikov, Goodness-of-fit tests in Nevzorov’s model, Theory Stoch. Process. 7 (2001), no. 1-2, 215-230. · Zbl 0974.62097
[5]	A. Kukush and A. Malenko, Goodness-of-fit test in a structural errors-in-variables model based on a score function, Australian J. Statist. 37 (2008), no. 1, 71-79.
[6]	G. Pfanzagl, On the measurability and consistency of minimum contrast estimates, Metrika 14 (1969), 249-273. · Zbl 0181.45501
[7]	O. S. Usoltseva, A consistent estimator in a survival model under measurement errors, Visnyk Kiev National University 22 (2009), 45-50. (Ukrainian)