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The Dantzig selector for diffusion processes with covariates. (English) Zbl 1390.62167

Summary: The Dantzig selector for a special parametric model of diffusion processes is studied in this paper. In our model, the diffusion coefficient is given as the exponential of the linear combination of other processes which are regarded as covariates. We propose an estimation procedure which is an adaptation of the Dantzig selector for linear regression models and prove the \(l_q\) consistency of the estimator for all \(q\in[1,\infty]\).

MSC:

62M05 Markov processes: estimation; hidden Markov models
60J60 Diffusion processes
62F12 Asymptotic properties of parametric estimators
62N02 Estimation in survival analysis and censored data
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[1] Antoniadis, A., Fryzlewicz, P. and Letué, F. (2010). The Dantzig selector in Cox’s proportional hazards model, Scand. J. Stat., 37(4), 531-552. · Zbl 1349.62473
[2] Bradic, J., Fan, J. and Jiang, J. (2011). Regularization for Cox’s proportional hazards model with NP-dimensionality, Ann. Statist., 39(6), 3092-3120. · Zbl 1246.62202
[3] Candés, E. and Tao, T. (2007). The Dantzig selector: Statistical estimation when p is much larger than n, Ann. Statist., 35(6), 2313-2351. · Zbl 1139.62019
[4] Fan, J. and Li, R. (2002). Variable selection for Cox’s proportional hazards model and frailty model, Ann. Statist., 30(1), 74-99. · Zbl 1012.62106
[5] Freedman, D. A. (1975). On tail probability for martingales, Ann. Probab., 3(1), 100-118. · Zbl 0313.60037
[6] Fujimori, K. and Nishiyama, Y. (2017). The lq consistencyof the Dantzig selector for Cox’s proportional hazards model, J. Statist. Plann. Inference., 181, 62-70. · Zbl 1356.62183
[7] Genon-Catalot, V. and Jacod, J. (1993). On the estimation of the diffusion coefficient for multidimensional diffusion processes, Ann. Inst. H. Poincaré Probab. Statist., 29(1), 119-151. · Zbl 0770.62070
[8] Gregorio, A. D. and Iacus, S. M. (2012). Adaptive LASSO-type estimation for multivariate diffusion processes, Econometric Theory, 28(4), 838-860. · Zbl 1419.62170
[9] Huang, J., Sun, T., Ying, Z., Yu, Y. and Zhang, C.-H. (2013). Oracle inequalities for the LASSO in the Cox model, Ann. Statist., 41(3), 1142-1165. · Zbl 1292.62135
[10] Kessler, M. (1997). Estimation of an ergodic diffusion from discrete observations, Scand. J. Statist., 24(2), 211-229. · Zbl 0879.60058
[11] Masuda, H. and Shimizu, Y. (2014). Moment convergence in regularized estimation under multiple and mixed-rates asymptotics, arXiv: 1406.6751. · Zbl 1380.62082
[12] Tibshirani, R. (1996). Regression shrinkage and selection via the Lasso, J. Roy. Statist. Soc. Ser. B, 58(1), 267-288. · Zbl 0850.62538
[13] Tibshirani, R. (1997). The lasso method for variable selection in the Cox model, Stat. Med., 16, 385-395.
[14] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. With Applications to Statistics, Springer Series in Statistics, Springer-verlag, New York. · Zbl 0862.60002
[15] Yoshida, N. (1992). Estimation for diffusion processes from discrete observation, J. Multivariate Anal., 41(2), 220-242. · Zbl 0811.62083
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