Biometrika Advance Access originally published online on November 19, 2007
Biometrika 2007 94(4):873-892; doi:10.1093/biomet/asm062
Articles |
A General Approach to the Predictability Issue in Survival Analysis with Applications
Department of Economics, University of Mannheim, L7, 3-5, 68131 Mannheim, Germany emammen{at}rumms.uni-mannheim.de
Cass Business School, City University, 106 Bunhill Row, London EC1Y 8TZ, U.K. festinalente{at}nielsen.mail.dk
Very often in survival analysis one has to study martingale integrals where the integrand is not predictable and where the counting process theory of martingales is not directly applicable, as for example in nonparametric and semiparametric applications where the integrand is based on a pilot estimate. We call this the predictability issue in survival analysis. The problem has been resolved by approximations of the integrand by predictable functions which have been justified by ad hoc procedures. We present a general approach to the solution of this problem. The usefulness of the approach is shown in three applications. In particular, we argue that earlier ad hoc procedures do not work in higher-dimensional smoothing problems in survival analysis.
Key Words: Bias-correction • Kernel hazard estimation • Martingale central limit theorem • Nonparametric smoothing • Predictability • Survival analysis
References
-
Aalen O. O. Nonparametric inference for a family of counting processes. Ann. Statist. (1978) 6:701–26.[CrossRef]
Aalen O. O., Johansen S. An empirical transition matrix for nonhomogeneous Markov chains based on censored observations. Scand. J. Statist. (1978) 5:141–50.
Andersen P. K., Borgan O., Gill R. D., Keiding N. Censoring, truncation and filtering in statistical models based on counting process theory. Contemp. Math. (1988) 80:19–59.
Andersen P. K., Borgan O., Gill R. D., Keiding N. Statistical Models Based on Counting Processes (1993) New York: Springer-Verlag.
Andersen P. K., Gill R. D. Cox's regression model for counting processes. A large sample study. Ann. Statist. (1982) 10:1100–20.[CrossRef]
Berman S. M., Frydman H. Parametric estimation of hazard functions with stochastic covariate processes. Sankhya (1999) 61:174–88.
Bosq D. Nonparametric Statistics for Stochastic Processes. Estimation and Prediction (1996) New York: Springer-Verlag.
Cox D. R. Regression models and life tables (with Discussion). J. R. Statist. Soc. B (1972) 34:187–220.
Cramer H., Wold H. Mortality variations in Sweden: a study in graduation and forecasting. Scand. Actuar. J. (1935) 18:161–241.
Fledelius P., Guillen M., Nielsen J. P., Petersen K. A comparative study of parametric and nonparametric estimators of old-age mortality in Sweden. J. Actuar. Prac. (2004) 11:103–46.
Gill R. D. Censoring and Stochastic Integrals (1980) Mathematical Centre Tracts 124: Mathematisch Centrum, Amsterdam.
Gill R. D. Large sample behavior of the product-limit estimator. Ann. Statist. (1983) 11:49–58.[CrossRef]
Gompertz B. On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Phil. Trans. R. Soc. Lond. (1825) 115:513–85.[CrossRef]
Jewell N. P., Kalbfleisch J. D. Marker models in survival analysis and applications to issues associated with aids. In: AIDS Epidemiology: Methodological Issues—Jewell N., Dietz K., Farewell V., eds. (1992) Boston: Birkhäuser. 211–30.
Jewell N. P., Nielsen J. P. A framework for consistent prediction rules based on markers. Biometrika (1993) 80:153–64.
Jones M. C., Linton O. B., Nielsen J. P. A simple bias reduction method for density estimation. Biometrika (1995) 82:327–38.
Kaplan E. L., Meier P. Non-parametric estimation from incomplete observations. J. Am. Statist. Assoc. (1958) 53:457–81.[CrossRef][ISI]
Kutoyants Y. A. Statistical Inference for Ergodic Diffusion Processes (2004) London: Springer-Verlag.
Linton O. B., Nielsen J. P. A multiplicative bias reduction method for non-parametric regression. Statist. Prob. Lett. (1994) 19:181–7.[CrossRef]
Linton O. B., Nielsen J. P., Van de Geer S. Estimating multiplicative and additive marker dependent hazard functions. Ann. Statist. (2003) 31:464–92.[CrossRef]
Nielsen J. P. Multiplicative bias correction in kernel hazard estimation. Scand. J. Statist. (1998) 11:453–66.
Nielsen J. P. Marker dependent hazard estimation from local linear estimation. Scand. Actuar. J. (1998) 2:113–24.
Nielsen J. P. Super-efficient hazard estimation based on high-quality marker information. Biometrika (1999) 86:227–32.
Nielsen J. P. Super-efficient prediction based on high-quality marker information. Astin Bull. (2000) 30:295–304.[CrossRef]
Nielsen J. P. Smoothing and prediction with a view to actuarial science, biostatistics and finance. Scand. Actuar. J. (2003) 1:51–74.
Nielsen J. P. Variable bandwidth kernel hazard estimators. J. Nonparam. Statist. (2003) 15:355–76.[CrossRef]
Nielsen J. P., Linton O. B. Kernel estimation in a nonparametric marker dependent hazard model. Ann. Statist. (1995) 23:1735–48.[CrossRef]
Nielsen J. P., Linton O. B., Bickel P. On a semiparametric survival model with flexible covariate effect. Ann. Statist. (1998) 26:215–41.[CrossRef]
Nielsen J. P., Tanggaard C. Boundary and bias correction in kernel hazard estimation. Scand. J. Statist. (2001) 28:675–98.[CrossRef]
Perozzo L. Statistica grafica. Ann. Statistica. (1880) 12:1–16.
Ramlau-Hansen H. Smoothing counting process intensities by means of kernel functions. Ann. Statist. (1983) 11:453–66.[CrossRef]
Yong F. H. L., Taylor J. M. G., Bryant J. L., Chmiel J. S., Gange S. J., Hoover D. Dependence of the hazard of aids on markers. aids (1997) 11:217–28.[CrossRef][ISI][Medline]
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