A Journal "Theory of Probability and Mathematical Statistics"
2021
2020
2019
2018
2017
2016
2015
2014
2013
2012
2011
2010
2009
2008
2007
2006
2005
2004
2003
2002
2001
2000
1999
1998
1997
1996
1995
1994
1993
1992
1991
1990
1989
1988
1987
1986
1985
1984
1983
1982
1981
1980
1979
1978
1977
1976
1975
1974
1973
1972
1971
1970

Archive

# Theory of Probability and Mathematical Statistics

## CONSISTENT ESTIMATION IN COX PROPORTIONAL HAZARDS MODEL WITH MEASUREMENT ERRORS AND UNBOUNDED PARAMETER SET

### A. G. KUKUSH, O. O. CHERNOVA

Abstract: Cox proportional hazards model with measurement error is investigated. In \cite{KuBar} and \cite{ChiKu} asymptotic properties were studied of simultaneous estimator $\lambda_n(\cdot)$, $\beta_n$ for baseline hazard rate $\lambda(\cdot)$ and regression parameter $\beta$, at that the parameter set $\Theta=\Theta_{\lambda}\times \Theta_{\beta}$ was assumed bounded. In the present paper, the set $\Theta_{\lambda}$ is unbounded from above and is not separated away from $0$. We construct the estimator in two steps: first we derive a strongly consistent estimator and then modify it to provide its asymptotic normality.

Keywords: Асимптотично нормальна оцінка, консистентна оцінка, модель Кокса з пропорційними ризиками, сумісне оцінювання базової функції ризику та параметра регресії

Bibliography:
1. P. K. Andersen and R. D. Gill, Cox's regression model for counting processes: a large sample study, Ann. Statist. 10 (1982), no. 4, 1100-1120.
2. T. Augustin, An exact corrected log-likelihood function for Cox's proportional hazards model under measurement error and some extensions, Scand. J. Stat. 31 (2004), no. 1, 43-50.
3. C. Chimisov and A. Kukush, Asymptotic normality of corrected estimator in Cox proportional hazards model with measurement error, Mod. Stoch. Theory Appl. 1 (2014), no. 1, 13-32.
4. D. R. Cox, Regression models and life tables, J. R. Stat. Soc. Ser. B. Stat. Methodol. 34 (1972), 187-220.
5. M. Gu and F. H. Kong, Consistent estimation in Cox proportional hazards model with covariate measurement errors, Statist. Sinica 32 (1999), no. 9, 953-969.
6. A. Kukush, S. Baran, I. Fazekas, and E. Usoltseva, Simultaneous estimation of baseline hazard rate and regression parameters in Cox proportional hazards model with measurement error, J. Statist. Res. 45 (2011), no. 2, 77-94.
7. G. Pfanzagl, On the measurability and consistency of minimum contrast estimates, Metrika 14 (1969), 249-273.
8. P. Royston (2011), Estimating a smooth baseline hazard function for the Cox model, Research report no. 314, Department of Statistical Science, University College London.