×

Weighted likelihood for semiparametric models and two-phase stratified samples, with application to Cox regression. (English) Zbl 1142.62014

In the two-phase stratified sampling at the first phase a large sample is drawn and measurements are made on an auxiliary variable \(U\) that is correlated with the target variable. At the second phase measurements on the target variable \(X\) are made for a subsample drawn from within strata defined by the auxiliary variable. The authors consider estimates based on Horvitz-Thompson (inverse probability) weighted likelihood (WL) equations for the distribution of \((X,U)\). The asymptotic normality of parametric and nonparametric WL estimates normed by \(\sqrt{n}\) is demonstrated. The results are applied to Bernoulli sampling with estimated weights and to the Cox proportional hazards model.

MSC:

62G05 Nonparametric estimation
62D05 Sampling theory, sample surveys
62G20 Asymptotic properties of nonparametric inference
62N02 Estimation in survival analysis and censored data
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Andersen P. K., Ann. Statist. 10 pp 1100– (1982)
[2] Begun J. M., Ann. Statist. 11 pp 432– (1983)
[3] DOI: 10.2307/2337154
[4] DOI: 10.1023/A:1009661900674 · Zbl 0948.62069
[5] DOI: 10.2307/2529620
[6] DOI: 10.1111/1467-9868.00078
[7] DOI: 10.1214/aos/1059655907 · Zbl 1105.62335
[8] DOI: 10.1111/1467-9868.00313 · Zbl 0988.62063
[9] Cox D. R., J. Roy. Statist. Soc. Ser. B 34 pp 187– (1972)
[10] DOI: 10.1093/biomet/91.4.929 · Zbl 1064.62002
[11] DOI: 10.2307/2280784 · Zbl 0047.38301
[12] DOI: 10.1002/sim.4780070116
[13] DOI: 10.1093/biomet/87.1.73 · Zbl 0974.62104
[14] DOI: 10.1198/016214504000000584 · Zbl 1117.62373
[15] DOI: 10.1111/1467-9868.00185 · Zbl 0915.62030
[16] DOI: 10.1093/biomet/87.1.37 · Zbl 0974.62008
[17] DOI: 10.2307/2337050
[18] DOI: 10.2307/1914121 · Zbl 0372.62094
[19] Nan B., Can. J. Statist. 32 pp 403– (2004)
[20] DOI: 10.1214/009053604000000157 · Zbl 1097.62097
[21] DOI: 10.2307/2279117 · Zbl 0018.22603
[22] DOI: 10.2307/2684196 · Zbl 04527611
[23] Pierce D. A., Ann. Statist. 10 pp 475– (1982)
[24] Praestgaard J., Ann. Probab. 21 pp 2053– (1993)
[25] Pugh M., Inference in the Cox proportional hazards model with missing covariates (1994)
[26] DOI: 10.2307/2290910 · Zbl 0815.62043
[27] DOI: 10.1111/j.1467-9469.2004.02-064.x · Zbl 1060.62111
[28] DOI: 10.1111/1467-9868.00333 · Zbl 1059.62010
[29] Scott A. J., J. Roy. Statist. Soc. Ser. B 48 pp 170– (1986)
[30] DOI: 10.1093/biomet/84.1.57 · Zbl 1058.62505
[31] Self S. G., Ann. Statist. 16 pp 64– (1988)
[32] Skinner C. J., Analysis of complex surveys. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics (1989) · Zbl 0701.62018
[33] Therneau T. M., Modeling survival data: extending the Cox model. Statistics for Biology and Health (2000) · Zbl 0958.62094
[34] van der Vaart A. W., Asymptotic statistics, vol. 3 of Cambridge series in statistical and probabilistic mathematics (1998)
[35] van der Vaart A. W., Weak convergence and empirical processes (1996) · Zbl 0862.60002
[36] DOI: 10.1177/0049124189017003003
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.