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Presmoothing the Aalen-Johansen estimator in the illness-death model. (English) Zbl 1337.62205
Summary: One major goal in clinical applications of multi-state models is the estimation of transition probabilities. The usual nonparametric estimator of the transition matrix for non-homogeneous Markov processes is the Aalen-Johansen estimator [O. O. Aalen and S. Johansen, Scand. J. Stat. 5, 141–150 (1978; Zbl 0383.62058)]. In this paper we propose a modification of the Aalen-Johansen estimator in the illness-death model based on presmoothing. The consistency of the proposed estimators is formally established. Simulations show that the presmoothed estimators may be much more efficient than the Aalen-Johansen estimator. A real data illustration is included.

MSC:
62M05 Markov processes: estimation; hidden Markov models
62G05 Nonparametric estimation
62N02 Estimation in survival analysis and censored data
62G20 Asymptotic properties of nonparametric inference
62N05 Reliability and life testing
62P10 Applications of statistics to biology and medical sciences; meta analysis
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References:
[1] Aalen, O. and Johansen, S. (1978). An Empirical Transition Matrix for Non-homogeneous Markov Chains Based on Censored Observations, Scandinavian Journal of Statistics 5 , 141-150. · Zbl 0383.62058
[2] Amorim, A., de Uña- Álvarez, J. and Meira-Machado, L. (2011). Presmoothing the transition probabilities in the illness-death model, Statistics & Probability Letters 81(7) , 797-806. · Zbl 1218.62111
[3] Andersen, P. K., Borgan, Ø., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes , Springer-Verlag, New York. · Zbl 0769.62061
[4] Andersen, P., Esbjerj, S. and Sorensen, T. (2000). Multi-state models for bleeding episodes and mortality in liver cirrhosis, Statistics in Medicine 19 , 587-599.
[5] Borgan, Ø. (1998). Aalen-Johansen Estimator, Encyclopedia of Biostatistics 1 , 5-10.
[6] Cao, R. and Jácome, M. (2004). Presmoothed kernel density estimator for censored data, Journal of Nonparametric Statistics 16(1-2) , 289-309. · Zbl 1049.62035
[7] Cox, D. R. (1972). Regression Models and Life-Tables, Journal of the Royal Statistical Society Series B 34(2) , 187-220. · Zbl 0243.62041
[8] Crowley, J. and Hu, M. (1977). Covariance Analysis of Heart Transplant Survival Data, Journal of the American Statistical Association 72(357) , 27-36.
[9] Datta, S. and Satten, G. (2001). Validity of the Aalen-Johansen estimators of stage occupation probabilities and Nelson-Aalen estimators of integrated transition hazards for non-markov models, Statistics & Probability Letters 55 , 403-411. · Zbl 0998.62072
[10] de la Peña, V. and Giné, E. (1999). Decoupling: from Dependence to Independence , Springer, New York. · Zbl 0918.60021
[11] de Uña- Álvarez, J. and Amorim, A. P. (2011). A semiparametric estimator of the bivariate distribution function for censored gap times, Biometrical Journal 53(1) , 113-127. · Zbl 1207.62076
[12] de Uña-Álvarez, J. and Rodríguez-Campos, C. (2004). Strong consistency of presmoothed Kaplan-Meier integrals when covariables are present, Statistics 38(6) , 483-496. · Zbl 1055.62052
[13] Devroye, L. (1978a). The uniform convergence of nearest neighbor regression function estimators and their application in optimization, IEEE Transactions on Information Theory 24(2) , 142-151. · Zbl 0375.62083
[14] Devroye, L. (1978b). The uniform convergence of the nadaraya-watson regression function estimate, Canadian Journal of Statistics 6(2) , 179-191. · Zbl 0405.62033
[15] Dikta, G. (1998). On semiparametric random censorship models, Journal of Statistical Planning and Inference 66(2) , 253-279. · Zbl 0927.62101
[16] Dikta, G. (2000). The strong law under semiparametric random censorship models, Journal of Statistical Planning and Inference 83(1) , 1-10. · Zbl 0941.62055
[17] Dikta, G., Ghorai, J. and Schmidt, C. (2005). The central limit theorem under semiparametric random censorship models, Journal of Statistical Planning and Inference 127(1-2) , 23-51. · Zbl 1056.60021
[18] Dikta, G., Kvesic, M. ad Schmidt, C. (2006). Bootstrap Approximations in Model Checks for Binary Data, Journal of the American Statistical Association 101 (474, 521-530.) · Zbl 1119.62332
[19] Härdle, W. and Luckhaus, S. (1984). Uniform Consistency of a Class of Regression Function Estimators, Annals of Statistics 12(2) , 612-623. · Zbl 0544.62037
[20] Hosmer, D. W. and Lemeshow, S. (1989). Applied Logistic Regression , Wiley. · Zbl 0715.62125
[21] Hougaard, P. (1999). Multi-state Models: A Review, Lifetime Data Analysis 5 , 239-264. · Zbl 0934.62112
[22] Kaplan, E. and Meier, P. (1958). Nonparametric Estimation from Incomplete Observations, Journal of the American Statistical Association 53(282) , 457-481. · Zbl 0089.14801
[23] Mack, Y. P. and Silverman, B. W. (1982). Weak and Strong Uniform Consistency of Kernel Regression Estimates, Probability Theory and Related Fields 61 , 405-415. · Zbl 0495.62046
[24] Meira-Machado, L., de Uña-Álvarez, J. and Cadarso-Suárez, C. (2006). Nonparametric estimation of transition probabilities in a non-Markov illness-death model, Lifetime Data Analysis 12 , 325-344. · Zbl 1356.62127
[25] Meira-Machado, L., de Uña-Álvarez, J., Cadarso-Suárez, C. and Andersen, P. (2009). Multi-state models for the analysis of time to event data, Statistical Methods in Medical Research 18 , 195-222.
[26] Miller, R. G. (1983). What Price Kaplan-Meier?, Biometrics 39 , 1077-1081. · Zbl 0532.62027
[27] Yuan, M. (2005). Semiparametric Censorship Model with Covariates, Test 14(2) , 489-514. · Zbl 1087.62054
[28] Zhang, H. (2004). Asymptotic effciency of estimation in the partial Koziol-Green model, Journal of Statistical Planning and Inference 124(2) , 399-408. · Zbl 1047.62093
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