Abstract
The aim of this paper is to introduce an adaptive penalized estimator for identifying the true reduced parametric model under the sparsity assumption. In particular, we deal with the framework where the unpenalized estimator of the structural parameters needs simultaneously multiple rates of convergence (i.e., the so-called mixed-rates asymptotic behavior). We introduce a bridge-type estimator by taking into account penalty functions involving \(\ell ^q\) norms (0 < q ≤ 1). We prove that the proposed regularized estimator satisfies the oracle properties. Our approach is useful for the estimation of stochastic differential equations in the parametric sparse setting. More precisely, under the high-frequency observation scheme, we apply our methodology to an ergodic diffusion and introduce a procedure for the selection of the tuning parameters. Furthermore, the paper contains a simulation study as well as a real data prediction in order to assess about the performance of the proposed bridge estimator.
Similar content being viewed by others
References
Antoine, B., Renault, E. (2012). Efficient minimum distance estimation with multiple rates of convergence. Journal of Econometrics, 170(2), 350–367.
Bandi, F., Corradi, V., Moloche, G. (2009). Bandwidth selection for continuous-time Markov processes. Unpublished paper.
Basu, S., Michailidis, G. (2015). Regularized estimation in sparse high-dimensional time series models. The Annals of Statistics, 43(4), 1535–1567.
Caner, M., Knight, K. (2013). An alternative to unit root tests: bridge estimators differentiate between nonstationary versus stationary models and select optimal lag. Journal of Statistical Planning and Inference, 143(4), 691–715.
Clément, E., Gloter, A. (2019). Estimating functions for SDE driven by stable Lévy processes. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 55(3), 1316–1348.
De Gregorio, A., Iacus, S. M. (2012). Adaptive LASSO-type estimation for multivariate diffusion processes. Econometric Theory, 28(4), 838–860.
De Gregorio, A., Iacus, S. M. (2018). On penalized estimation for dynamical systems with small noise. The Electronic Journal Statistics, 12(1), 1614–1630.
Fan, J., Li, R. (2001). Variable selection via nonconcave penalized likelihood and its Oracle properties. Journal of American Statistical Association, 96, 1348–1360.
Fan, J., Li, R. (2006). Statistical Challenges With High Dimensionality: Feature Selection in Knowledge Discovery. In Proceedings of the Madrid international congress of mathematicians, Madrid.
Fan, J., Peng, H. (2004). Nonconcave penalized likelihood with a diverging number of parameters. The Annals of Statistics, 32(3), 928–961.
Frank, L. E., Friedman, J. H., Silverman, B. W. (1993). A statistical view of some chemometrics regression tools. Technometrics, 35(2), 109–135.
Gaïffas, S., Matulewicz, G. (2019). Sparse inference of the drift of a high-dimensional Ornstein-Uhlenbeck process. Journal of Multivariate Analysis, 169, 1–20.
Gloter, A., Sørensen, M. (2009). Estimation for stochastic differential equations with a small diffusion coefficient. Stochastic Processes and their Applications, 119(3), 679–699.
Hastie, T., Tibshirani, R., Friedman, J. (2009). The elements of statistical learning. Data mining, inference, and prediction 2nd ed. Springer Series in Statistics. New York: Springer.
Hastie, T., Tibshirani, R., Wainwright, M. (2015). Statistical learning with sparsity. The LASSO and generalizations. Monographs on Statistics and Applied Probability, 143. Boca Raton: CRC Press.
Iacus, S.M., Yoshida N. (2018). Simulation and inference for stochastic processes with YUIMA. A comprehensive R framework for SDEs and other stochastic processes. Use R!. Cham: Springer.
Kamatani, K., Uchida, M. (2015). Hybrid multi-step estimators for stochastic differential equations based on sampled data. Statistical Inference for Stochastic Processes, 18(2), 177–204.
Kessler, M. (1997). Estimation of an ergodic diffusion from discrete observations. Scandinavian Journal of Statistics., 24(2), 211–229.
Kinoshita Y., Yoshida N. (2019). Penalized quasi likelihood estimation for variable selection. https://arxiv.org/abs/1910.12871.
Knight, K., Fu, W. (2000). Asymptotics for LASSO-type estimators. The Annals of Statistics, 28(5), 1536–1378.
Lee, L.-F. (2004). Asymptotic distributions of quasi-maximum likelihood estimators for spatial autoregressive models. Econometrica, 72(6), 1899–1925.
Masuda, H. (2019). Non-Gaussian quasi-likelihood estimation of SDE driven by locally stable Lévy process. Stochastic Processes and their Applications, 129(3), 1013–1059.
Masuda, H., Shimizu, Y. (2017). Moment convergence in regularized estimation under multiple and mixed-rates asymptotics. Mathematical Methods of Statistics, 26(2), 81–110.
McCrorie, J. R., Chambers, M. J. (2006). Granger causality and the sampling of economic processes. Journal of Econometrics, 132(2), 311–336.
Nardi, Y., Rinaldo, A. (2011). Autoregressive process modeling via the LASSO procedure. Journal of Multivariate Analysis, 102(3), 528–549.
Radchenko, P. (2008). Mixed-rates asymptotics. The Annals of Statistics, 36(1), 287–309.
Shimizu, Y., Yoshida, N. (2006). Estimation of parameters for diffusion processes with jumps from discrete observations. Statistical Inference for Stochastic Processes, 9(3), 227–277.
Sørensen, M., Uchida, M. (2003). Small-diffusion asymptotics for discretely sampled stochastic differential equations. Bernoulli, 9(6), 1051–1069.
Suzuki, T., Yoshida, N. (2019). Penalized least squares approximation methods and their applications to stochastic processes. To appear in Japanese Journal of Statistics and Data Science, https://arxiv.org/abs/1811.09016.
Tibshirani, R. (1996). Regression shrinkage and selection via the LASSO. Journal of the Royal Statistical Society: Series B, 58(1), 267–288.
Uchida, M. (2011). Contrast-based information criterion for ergodic diffusion processes from discrete observations. Annals of the Institute of Statistical Mathematics, 62(1), 161–187.
Uchida, M., Yoshida, N. (2012). Adaptive estimation of an ergodic diffusion process based on sampled data. Stochastic Processes and their Applications, 122(8), 2885–2924.
Uchida, M., Yoshida, N. (2014). Adaptive Bayes type estimators of ergodic diffusion processes from discrete observations. Statistical Inference for Stochastic Processes, 17(2), 181–219.
Wang, H., Leng, C. (2007). Unified LASSO estimation by Least Squares Approximation. Journal of American Statistical Association, 102(479), 1039–1048.
Wang, H., Li, G., Tsai, C.-L. (2007). Regression coefficient and autoregressive order shrinkage and selection via the LASSO. Journal of the Royal Statistical Society Series B, 169(1), 63–78.
Yoshida, N. (1992). Estimation for diffusion processes from discrete observation. Journal of Multivariate Analysis, 41(2), 220–242.
Yoshida, N. (2011). Polynomial type large deviation inequalities and quasi-likelihood analysis for stochastic differential equations. Annals of the Institute of Statistical Mathematics, 63(3), 431–479.
Yu, J., Phillips, P. C. B. (2001). Gaussian estimation of continuous time models of the short term interest rate. The Econometrics Journal, 4(2), 210–224.
Zou, H. (2006). The adaptive LASSO and its Oracle properties. Journal of American Statistical Association, 101(476), 1418–1429.
Acknowledgements
We would like to thank the Associate Editor and the Referees for their insightful remarks which led to a substantial improvement of the first version of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
De Gregorio, A., Iafrate, F. Regularized bridge-type estimation with multiple penalties. Ann Inst Stat Math 73, 921–951 (2021). https://doi.org/10.1007/s10463-020-00769-w
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-020-00769-w