zbMATH — the first resource for mathematics

Survival analysis under cross-sectional sampling: length bias and multiplicative censoring. (English) Zbl 0969.62062
Summary: Consider a parametric, nonparametric or semiparametric model for survival times. Interest is in estimation of Euclidean and Banach parameters for these models. However, not the survival times themselves will be observed, since this might be quite time consuming. Instead, cross-sectional sampling is applied: at some point in time one identifies a random sample from the population under study and one registers the survival time up to this time-point. Typically, the resulting reduced survival times do not have the same distributions as the true survival times.
On the one hand, longer survival times have a higher probability to be sampled than smaller ones. On the other hand, the observed survival times have been censored multiplicatively. The length bias and multiplicative censoring properties of cross-sectional sampling will be discussed and reviewed as well as estimation in the resulting parametric, nonparametric, and semiparametric models.

62N02 Estimation in survival analysis and censored data
62G05 Nonparametric estimation
62F12 Asymptotic properties of parametric estimators
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
46N30 Applications of functional analysis in probability theory and statistics
62M09 Non-Markovian processes: estimation
Full Text: DOI
[1] Bickel, P.J.; Klaassen, C.A.J.; Ritov, Y.; Wellner, J.A., Efficient and adaptive estimation for semiparametric models, johns hopkins university press, baltimore, MD. paperbound edition 1998, (1993), Springer New York · Zbl 0786.62001
[2] Birgé, L., The grenander estimator: a nonasymptotic approach, Ann. statist., 17, 1532-1549, (1989) · Zbl 0703.62042
[3] Brillinger, D.R., The natural variability of vital rates and associated statistics (with discussion), Biometrics, 42, 693-734, (1986) · Zbl 0611.62136
[4] Cox, D.R., Some sampling problems in technology., (), 506-527
[5] Cox, D.R., Regression models and life-tables, J. roy. statist. soc. ser. B, 34, 187-220, (1972) · Zbl 0243.62041
[6] Cox, D.R., Partial likelihood, Biometrika, 62, 269-276, (1975) · Zbl 0312.62002
[7] Kalbfleisch, J.D., Prentice, R.L., 1980. The Statistical Analysis of Failure Time Data. Wiley Series in Probability and Mathematical Statistics. Wiley, New York. · Zbl 0504.62096
[8] Keiding, N., Statistical inference in the lexis diagram, Philos. trans. roy. soc. London A, 322, 487-509, (1990) · Zbl 0714.62102
[9] Keiding, N., Age specific incidence and prevalence: a statistical perspective, J. roy. statist. soc. ser. A, 154, 371-412, (1991) · Zbl 1002.62504
[10] Kingman, J.F.C., Poisson processes., Oxford studies in probability, Vol. 3, (1993), Clarendon Press Oxford · Zbl 0771.60001
[11] Klaassen, C.A.J., Efficient estimation in the Cox model for survival data., (), 313-319
[12] Lexis, W., Einleitung in die theorie der bevölkerungsstatistik, (1875), Tübner Strassburg
[13] Lund, J., Sampling bias in population studies – how to use the lexis diagram, Scand. J. statist., 27, 589-604, (2000) · Zbl 0962.62109
[14] Simon, R., Length biased sampling in etiological studies, Amer. J. epidemiol., 111, 444-452, (1980)
[15] Sun, J.; Woodroofe, M., Adaptive smoothing for a penalized NPMLE of a non-increasing density, J. statist. planning inference, 52, 143-159, (1996) · Zbl 0848.62020
[16] Vardi, Y., Nonparametric estimation in the presence of length bias, Ann. statist., 10, 616-620, (1982) · Zbl 0491.62034
[17] Vardi, Y., Empirical distributions in selection bias models, Ann. statist., 13, 178-203, (1985) · Zbl 0578.62047
[18] Vardi, Y., Statistical models for intercepted data, J. amer. statist. assoc., 83, 183-197, (1988) · Zbl 0693.62081
[19] Vardi, Y., Multiplicative censoring, renewal theory, deconvolution and decreasing density: nonparametric estimation, Biometrika, 76, 751-761, (1989) · Zbl 0678.62051
[20] Vardi, Y.; Zhang, C.-H., Large sample study of empirical distributions in a random-multiplicative censoring model, Ann. statist., 20, 1022-1039, (1992) · Zbl 0761.62056
[21] Weldon, K.L.; Potvin, D., Nonparametric recovery of duration distributions from cross-sectional sample surveys, Comm. statist. theory methods, 20, 3943-3973, (1991) · Zbl 0800.62206
[22] Wang, M.-C., Nonparametric estimation from cross-sectional survival data, J. amer. statist. assoc., 86, 130-143, (1991) · Zbl 0739.62026
[23] Woodroofe, M.; Sun, J., A penalized maximum likelihood estimate of f(0+) when f is nonincreasing, Statist. sinica, 3, 501-515, (1993) · Zbl 0822.62020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.