Adaptive estimation of the baseline hazard function in the Cox model by model selection, with high-dimensional covariates. (English) Zbl 1334.62160
Summary: The purpose of this article is to provide an adaptive estimator of the baseline function in the Cox model with high-dimensional covariates. We consider a two-step procedure: first, we estimate the regression parameter of the Cox model via a Lasso procedure based on the partial \(\log\)-likelihood, secondly, we plug this Lasso estimator into a least-squares type criterion and then perform a model selection procedure to obtain an adaptive penalized contrast estimator of the baseline function.
Using non-asymptotic estimation results stated for the Lasso estimator of the regression parameter, we establish a non-asymptotic oracle inequality for this penalized contrast estimator of the baseline function, which highlights the discrepancy of the rate of convergence when the dimension of the covariates increases.
Using non-asymptotic estimation results stated for the Lasso estimator of the regression parameter, we establish a non-asymptotic oracle inequality for this penalized contrast estimator of the baseline function, which highlights the discrepancy of the rate of convergence when the dimension of the covariates increases.
MSC:
| 62N02 | Estimation in survival analysis and censored data |
| 62G05 | Nonparametric estimation |
| 62J07 | Ridge regression; shrinkage estimators (Lasso) |
| 62G07 | Density estimation |
| 62G08 | Nonparametric regression and quantile regression |
| 62N01 | Censored data models |
Keywords:
survival analysis; conditional hazard rate function; Cox’s proportional hazards model; right-censored data; semiparametric model; nonparametric model; high-dimensional covariates; model selection; non-asymptotic oracle inequalities; concentration inequalitiesReferences:
| [1] | Aalen, O., A model for nonparametric regression analysis of counting processes, (Mathematical Statistics and Probability Theory (Proc. Sixth Internat. Conf., Wisła, 1978). Mathematical Statistics and Probability Theory (Proc. Sixth Internat. Conf., Wisła, 1978), Lecture Notes in Statist., vol. 2 (1980), Springer: Springer New York), 1-25 · Zbl 0445.62095 |
| [2] | Akaike, H., Information theory and an extension of the maximum likelihood principle, (In Second International Symposium on Information Theory, Tsahkadsor, 1971 (1973), Akadémiai Kiadó: Akadémiai Kiadó Budapest), 267-281 · Zbl 0283.62006 |
| [3] | Andersen, P. K.; Borgan, Ø.; Gill, R. D.; Keiding, N., (Statistical Models Based on Counting Processes. Statistical Models Based on Counting Processes, Springer Series in Statistics (1993), Springer-Verlag: Springer-Verlag New York) · Zbl 0769.62061 |
| [4] | Baraud, Y., A Bernstein-type inequality for suprema of random processes with applications to model selection in non-Gaussian regression, Bernoulli, 16, 4, 1064-1085 (2010) · Zbl 1459.60044 |
| [5] | Barron, A.; Birgé, L.; Massart, P., Risk bounds for model selection via penalization, Probab. Theory Related Fields, 113, 3, 301-413 (1999) · Zbl 0946.62036 |
| [6] | Bertin, K.; Le Pennec, E.; Rivoirard, V., Adaptive Dantzig density estimation, Ann. Inst. H. Poincaré Probab. Statist., 47, 1, 43-74 (2011) · Zbl 1207.62077 |
| [7] | Bickel, P. J.; Ritov, Y.; Tsybakov, A. B., Simultaneous analysis of Lasso and Dantzig selector, Ann. Statist., 37, 4, 1705-1732 (2009) · Zbl 1173.62022 |
| [8] | Birgé, L.; Massart, P., From Model Selection to Adaptive Estimation (1997), Springer · Zbl 0920.62042 |
| [9] | Bradic, J.; Fan, J.; Jiang, J., Regularization for Cox’s proportional hazards model with NP-dimensionality, Ann. Statist., 39, 6, 3092-3120 (2012) · Zbl 1246.62202 |
| [11] | Brunel, E.; Comte, F., Penalized contrast estimation of density and hazard rate with censored data, Sankhyā, 67, 3, 441-475 (2005) · Zbl 1192.62102 |
| [12] | Brunel, E.; Comte, F.; Guilloux, A., Nonparametric density estimation in presence of bias and censoring, TEST, 18, 1, 166-194 (2009) · Zbl 1203.62052 |
| [13] | Brunel, E.; Comte, F.; Lacour, C., Minimax estimation of the conditional cumulative distribution function, Sankhyā Ser. A, 72, 2, 293-330 (2010) · Zbl 1213.62063 |
| [14] | Bühlmann, P.; van de Geer, S., On the conditions used to prove oracle results for the Lasso, Electron. J. Stat., 3, 1360-1392 (2009) · Zbl 1327.62425 |
| [15] | Bunea, F.; Tsybakov, A. B.; Wegkamp, M. H., Aggregation and sparsity via l1 penalized least squares, (Proceedings of the 19th Annual Conference on Learning Theory. Proceedings of the 19th Annual Conference on Learning Theory, COLT’06 (2006), Heidelberg, Springer-Verlag: Heidelberg, Springer-Verlag Berlin), 379-391 · Zbl 1143.62319 |
| [16] | Bunea, F.; Tsybakov, A.; Wegkamp, M., Aggregation for gaussian regression, Ann. Statist., 35, 4, 1674-1697 (2007) · Zbl 1209.62065 |
| [17] | Bunea, F.; Tsybakov, A.; Wegkamp, M., Sparse density estimation with l1 penalties, (Learning Theory (2007), Springer), 530-543 · Zbl 1203.62053 |
| [18] | Bunea, F.; Tsybakov, A. B.; Wegkamp, M., Sparsity oracle inequalities for the Lasso, Electron. J. Stat., 1, 169-194 (2007) · Zbl 1146.62028 |
| [19] | Comte, F.; Gaïffas, S.; Guilloux, A., Adaptive estimation of the conditional intensity of marker-dependent counting processes, Ann. Inst. H. Poincaré Probab. Statist., 47, 4, 1171-1196 (2011) · Zbl 1271.62222 |
| [20] | Cox, D. R., Regression models and life-tables, J. Roy. Statist. Soc. Ser. B, 34, 187-220 (1972) · Zbl 0243.62041 |
| [21] | Donoho, D.; Elad, M.; Temlyakov, V., Stable recovery of sparse overcomplete representations in the presence of noise, IEEE Trans. Inform. Theory, 52, 1, 6-18 (2006) · Zbl 1288.94017 |
| [22] | Efron, B.; Hastie, T.; Johnstone, I.; Tibshirani, R., Least angle regression, Ann. Statist., 32, 2, 407-499 (2004) · Zbl 1091.62054 |
| [23] | Fleming, T.; Harrington, D., Counting Processes and Survival Analysis, Vol. 169 (2011), John Wiley & Sons |
| [24] | Greenshtein, E.; Ritov, Y., Persistence in high-dimensional linear predictor selection and the virtue of overparametrization, Bernoulli, 10, 6, 971-988 (2004) · Zbl 1055.62078 |
| [25] | Grégoire, G., Least squares cross-validation for counting process intensities, Scand. J. Statist., 20, 4, 343-360 (1993) · Zbl 0795.62031 |
| [26] | Huang, J.; Sun, T.; Ying, Z.; Yu, Y.; Zhang, C., Oracle inequalities for the Lasso in the Cox model, Ann. Statist., 41, 3, 1142-1165 (2013) · Zbl 1292.62135 |
| [27] | Juditsky, A.; Nemirovski, A., Functional aggregation for nonparametric regression, Ann. Statist., 28, 3, 681-712 (2000) · Zbl 1105.62338 |
| [28] | Knight, K.; Fu, W., Asymptotics for lasso-type estimators, Ann. Statist., 28, 5, 1356-1378 (2000) · Zbl 1105.62357 |
| [30] | Letué, F., Modèle de Cox: Estimation par Sélection de Modèle et Modèle de Chocs bivarié (2000), Université de Paris Sud, UFR Scientifique d’Orsay, (Ph. D. thesis) |
| [31] | Liptser, R. S.; Shiryayev, A. N., (Theory of Martingales. Theory of Martingales, Mathematics and its Applications (Soviet Series), vol. 49 (1989), Kluwer Academic Publishers Group: Kluwer Academic Publishers Group Dordrecht), Translated from the Russian by K. Dzjaparidze [Kacha Dzhaparidze] · Zbl 0369.60001 |
| [32] | Mallows, C., Some comments on c p, Technometrics, 15, 4, 661-675 (1973) · Zbl 0269.62061 |
| [33] | Massart, P., (Concentration Inequalities and Model Selection. Concentration Inequalities and Model Selection, Lecture Notes in Mathematics, vol. 1896 (2007), Springer: Springer Berlin), Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour, July 6-23, 2003, With a foreword by Jean Picard · Zbl 1170.60006 |
| [34] | Meinshausen, N.; Bühlmann, P., High-dimensional graphs and variable selection with the Lasso, Ann. Statist., 34, 3, 1436-1462 (2006) · Zbl 1113.62082 |
| [35] | Meinshausen, N.; Yu, B., Lasso-type recovery of sparse representations for high-dimensional data, Ann. Statist., 246-270 (2009) · Zbl 1155.62050 |
| [36] | Nemirovski, A., (Topics in Nonparametric Statistics. Ecole d’Ete de Probabilites de Saint-Flour XXVIII, 1998, Vol. 28 (2000)), 85 · Zbl 0998.62033 |
| [37] | Ramlau-Hansen, H., The choice of a kernel function in the graduation of counting process intensities, Scand. Actuar. J., 1983, 3, 165-182 (1983) · Zbl 0523.62035 |
| [38] | Ramlau-Hansen, H., Smoothing counting process intensities by means of kernel functions, Ann. Statist., 11, 2, 453-466 (1983) · Zbl 0514.62050 |
| [39] | Reynaud-Bouret, P., Penalized projection estimators of the aalen multiplicative intensity, Bernoulli, 12, 4, 633-661 (2006) · Zbl 1125.62027 |
| [40] | Talagrand, M., New concentration inequalities in product spaces, Invent. Math., 126, 3, 505-563 (1996) · Zbl 0893.60001 |
| [41] | Talagrand, M., (The Generic Chaining: Upper and Lower Bounds of Stochastic Processes. The Generic Chaining: Upper and Lower Bounds of Stochastic Processes, Springer Monographs in Mathematics (2005), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1075.60001 |
| [42] | Tibshirani, R., Regression shrinkage and selection via the Lasso, J. Roy. Statist. Soc. Ser. B, 58, 1, 267-288 (1996) · Zbl 0850.62538 |
| [43] | Tibshirani, R., The Lasso method for variable selection in the Cox model, Stat. Med., 16, 4, 385-395 (1997) |
| [44] | van de Geer, S., Exponential inequalities for martingales, with application to maximum likelihood estimation for counting processes, Ann. Statist., 23, 5, 1779-1801 (1995) · Zbl 0852.60019 |
| [45] | van de Geer, S., The deterministic lasso. Technical report (2007), ETH Zürich: ETH Zürich Switzerland, Available at: http://stat.ethz.ch/research/publarchive/2007/140 |
| [46] | van de Geer, S., High-dimensional generalized linear models and the Lasso, Ann. Statist., 36, 2, 614-645 (2008) · Zbl 1138.62323 |
| [47] | Verzelen, N., Minimax risks for sparse regressions: Ultra-high dimensional phenomenons, Electron. J. Stat., 6, 38-90 (2012) · Zbl 1334.62120 |
| [48] | Zhang, C.; Huang, J., The sparsity and bias of the Lasso selection in high-dimensional linear regression, Ann. Statist., 36, 4, 1567-1594 (2008) · Zbl 1142.62044 |
| [49] | Zhao, P.; Yu, B., On model selection consistency of Lasso, J. Mach. Learn. Res., 7, 2541-2563 (2006) · Zbl 1222.62008 |
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