Abstract
We consider a Cox-type regression model with change-points in the covariates. A change-point specifies the unknown threshold at which the influence of a covariate shifts smoothly, i.e., the regression parameter may change over the range of a covariate and the underlying regression function is continuous but not differentiable. The model can be used to describe change-points in different covariates but also to model more than one change-point in a single covariate. Estimates of the change-points and of the regression parameters are derived and their properties are investigated. It is shown that not only the estimates of the regression parameters are \({\sqrt{n}}\) -consistent but also the estimates of the change-points in contrast to the conjecture of other authors. Asymptotic normality is shown by using results developed for M-estimators. At the end of this paper we apply our model to an actuarial dataset, the PBC dataset of Fleming and Harrington (Counting processes and survival analysis, 1991) and to a dataset of electric motors.
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References
Andersen PK, Gill RD (1982) Cox’s regression model for counting processes: a large sample study. Ann Stat 10: 1100–1120
Chappell R (1989) Fitting bent lines to data, with applications to allometry. J Theor Biol 138: 235–256
Fleming TR, Harrington DP (1991) Counting processes and survival analysis. Wiley, New York
Gandy A, Jensen U (2005) On goodness of fit tests for Aalen’s additive risk model. Scand J Stat 32: 425–445
Gandy A, Jensen U (2006) Model checks for Cox-type regression models based on optimally weighted martingale residuals. Preprint Hohenheim
Gandy A, Jensen U, Lütkebohmert C (2005) A Cox model with a change-point applied to an actuarial problem. Braz J Probab Stat 19: 93–109
Ibragimov IA, Has’minskii RZ (1981) Statistical estimation. Asymptotic theory. Springer, New York
Kosorok M, Song R (2007) Inference under right censoring for transformation models with a change-point based on a covariate threshold. Ann Stat 35: 957–989
Liang K, Self S, Liu X (1990) The Cox proportional hazard model with change-point: an epidemiologie application. Biometrics 46: 783–793
Luo X, Boyett J (1997) Estimations of a threshold parameter in Cox regression. Commun Stat Theor Methods 26: 2329–2346
Pons O (2003) Estimation in a Cox regression model with a change-point according to a threshold in a covariate. Ann Stat 31: 442–463
Vander Vaart A, Wellner J (1996) Weak convergence and empirical processes. Springer, New York
Vander Vaart A (1998) Asymptotic statistics. Cambridge University Press, New York
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Jensen, U., Lütkebohmert, C. A Cox-type regression model with change-points in the covariates. Lifetime Data Anal 14, 267–285 (2008). https://doi.org/10.1007/s10985-008-9083-3
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DOI: https://doi.org/10.1007/s10985-008-9083-3