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Asymptotic results for maximum likelihood estimators in joint analysis of repeated measurements and survival time. (English) Zbl 1086.62034
Summary: Maximum likelihood estimation has been extensively used in the joint analysis of repeated measurements and survival time. However, there is a lack of theoretical justification of the asymptotic properties for the maximum likelihood estimators. This paper intends to fill this gap. Specifically, we prove the consistency of the maximum likelihood estimators and derive their asymptotic distributions. The maximum likelihood estimators are shown to be semiparametrically efficient.

MSC:
62F12 Asymptotic properties of parametric estimators
62N02 Estimation in survival analysis and censored data
62H12 Estimation in multivariate analysis
62E20 Asymptotic distribution theory in statistics
62G20 Asymptotic properties of nonparametric inference
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[1] Andersen, P. K., Borgan, Ø., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes . Springer, New York. · Zbl 0769.62061
[2] Andersen, P. K. and Gill, R. D. (1982). Cox’s regression model for counting processes: A large sample study. Ann. Statist. 10 1100–1120. JSTOR: · Zbl 0526.62026
[3] Bickel, P. J., Klaassen, C. A. J., Ritov, Y. and Wellner, J. A. (1993). Efficient and Adaptive Estimation for Semiparametric Models . Johns Hopkins Univ. Press, Baltimore, · Zbl 0786.62001
[4] Billingsley, P. (1995). Probability and Measure , 3rd ed. Wiley, New York. · Zbl 0822.60002
[5] Chen, H. Y. and Little, R. J. A. (1999). Proportional hazards regression with missing covariates. J. Amer. Statist. Assoc. 94 896–908. · Zbl 0996.62092
[6] Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). J. Roy. Statist. Soc. Ser. B 39 1–38. · Zbl 0364.62022
[7] Diggle, P. J., Liang, K.-Y. and Zeger, S. L. (1994). Analysis of Longitudinal Data . Oxford Univ. Press, New York. · Zbl 1031.62002
[8] Dupuy, J.-F., Grama, I. and Mesbah, M. (2003). Asymptotic normality of semiparametric estimators in the Cox model with nonignorable missing covariate. C. R. Math. Acad. Sci. Paris 336 81–84. (In French.) · Zbl 1032.62093
[9] Evans, M. and Swartz, T. (2000). Approximating Integrals via Monte Carlo and Deterministic Methods . Oxford Univ. Press, New York. · Zbl 0958.65009
[10] Gill, R. D., van der Laan, M. J. and Robins, J. M. (1997). Coarsening at random: Characterizations, conjectures and counterexamples. In Proc. First Seattle Symposium in Biostatistics : Survival Analysis (D. Y. Lin and T. R. Fleming, eds.) 255–294, Springer, New York. · Zbl 0918.62003
[11] Henderson, R., Diggle, P. and Dobson, A. (2000). Joint modelling of longitudinal measurements and event time data. Biostatistics 1 465–480. · Zbl 1089.62519
[12] Hogan, J. W. and Laird, N. M. (1997). Mixture models for the joint distribution of repeated measures and event times. Statistics in Medicine 16 239–257.
[13] Hu, P., Tsiatis, A. A. and Davidian, M. (1998). Estimating the parameters in the Cox model when covariates are measured with error. Biometrics 54 1407–1419. · Zbl 1058.62557
[14] Huang, W., Zeger, S. L., Anthony, J. C. and Garrett, E. (2001). Latent variable model for joint analysis of multiple repeated measures and bivariate event times. J. Amer. Statist. Assoc. 96 906–914. · Zbl 1072.62659
[15] Johansen, S. (1983). An extension of Cox’s regression model. Internat. Statist. Rev. 51 165–174. JSTOR: · Zbl 0526.62081
[16] Kalbfleisch, J. D. and Prentice, R. L. (2002). The Statistical Analysis of Failure Time Data , 2nd ed. Wiley, New York. · Zbl 1012.62104
[17] Li, Y. and Lin, X. (2000). Covariate measurement errors in frailty models for clustered survival data. Biometrika 87 849–866. · Zbl 1028.62078
[18] Murphy, S. A. and van der Vaart, A. W. (2000). On profile likelihood (with discussion). J. Amer. Statist. Assoc. 95 449–485. · Zbl 0995.62033
[19] Parner, E. (1998). Asymptotic theory for the correlated gamma-frailty model. Ann. Statist. 26 183–214. · Zbl 0934.62101
[20] Rudin, W. (1973). Functional Analysis . McGraw–Hill, New York. · Zbl 0253.46001
[21] Rudin, W. (1987). Real and Complex Analysis , 3rd ed. McGraw–Hill, New York. · Zbl 0925.00005
[22] Tsiatis, A. A. and Davidian, M. (2001). A semiparametric estimator for the proportional hazards models with longitudinal covariates measured with error. Biometrika 88 447–458. · Zbl 0984.62078
[23] Tsiatis, A. A., DeGruttola, V. and Wulfsohn, M. S. (1995). Modeling the relationship of survival to longitudinal data measured with error. Applications to survival and CD4 counts in patients with AIDS. J. Amer. Statist. Assoc. 90 27–37. · Zbl 0818.62102
[24] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes . Springer, New York. · Zbl 0862.60002
[25] Wu, M. C. and Bailey, K. R. (1989). Estimation and comparison of changes in the presence of informative right censoring: Conditional linear model. Biometrics 45 939–955. Corrigendum 46 889. · Zbl 0715.62123
[26] Wu, M. C. and Carroll, R. J. (1988). Estimation and comparison of changes in the presence of informative right censoring by modeling the censoring process. Biometrics 44 175–188. · Zbl 0707.62210
[27] Wulfsohn, M. S. and Tsiatis, A. A. (1997). A joint model for survival and longitudinal data measured with error. Biometrics 53 330–339. · Zbl 0874.62140
[28] Xu, J. and Zeger, S. L. (2001). The evaluation of multiple surrogate endpoints. Biometrics 57 81–87. · Zbl 1209.62340
[29] Xu, J. and Zeger, S. L. (2001). Joint analysis of longitudinal data comprising repeated measures and times to events. Appl. Statist. 50 375–387. · Zbl 1112.62312
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