Electronic Journal of Statistics

Non-asymptotic approach to varying coefficient model

Olga Klopp and Marianna Pensky

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In the present paper we consider the varying coefficient model which represents a useful tool for exploring dynamic patterns in many applications. Existing methods typically provide asymptotic evaluation of precision of estimation procedures under the assumption that the number of observations tends to infinity. In practical applications, however, only a finite number of measurements are available. In the present paper we focus on a non-asymptotic approach to the problem. We propose a novel estimation procedure which is based on recent developments in matrix estimation. In particular, for our estimator, we obtain upper bounds for the mean squared and the pointwise estimation errors. The obtained oracle inequalities are non-asymptotic and hold for finite sample size.

Article information

Electron. J. Statist., Volume 7 (2013), 454-479.

First available in Project Euclid: 13 February 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J99: None of the above, but in this section 62H12: Estimation
Secondary: 60G57: Random measures

Varying coefficient model low rank matrix estimation statistical learning


Klopp, Olga; Pensky, Marianna. Non-asymptotic approach to varying coefficient model. Electron. J. Statist. 7 (2013), 454--479. doi:10.1214/13-EJS778. https://projecteuclid.org/euclid.ejs/1360764852

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  • [1] Bühlmann, P. and van de Geer, S. (2011)., Statistics for High-Dimensional Data: Methods, Theory and Applications. Springer.
  • [2] Candès, E. J. and Recht, B. (2009). Exact matrix completion via convex optimization., Foundations of Computational Mathematics, 9(6), 717–772.
  • [3] Chiang, C.-T., Rice, J. A. and Wu, C. O. (2001). Smoothing spline estimation for varying coefficient models with repeatedly measured dependent variables., J. Amer. Statist. Assoc., 96, 605–619.
  • [4] Cleveland, W. S., Grosse, E. and Shyu, W. M. (1991). Local regression models., Statistical Models in S (Chambers, J. M. and Hastie, T. J., eds), 309–376. Wadsworth and Books, Pacific Grove.
  • [5] Fan, J. and Zhang, W. (1999). Statistical estimation in varying coefficient models., Ann. Statist., 27, 1491–1518.
  • [6] Fan, J., and Zhang, W. (2008). Statistical methods with varying coefficient models., Statistics and Its Interface, 1, 179–195.
  • [7] Hastie, T. J. and Tibshirani, R. J. (1993). Varying-coefficient models., J. Roy. Statist. Soc. B. (Chambers, J. M. and Hastie, T. J., eds), 55 757–796.
  • [8] Hoover, D. R., Rice, J. A., Wu, C. O. and Yang, L.-P. (1998). Non-parametric smoothing estimates of time-varying coefficient models with longitudinal data., Biometrika, 85, 809–822.
  • [9] Huang, J. Z., Wu, C. O. and Zhou, L. (2002). Varying-coefficient models and basis function approximations for the analysis of repeated measurements., Biometrika, 89, 111–128.
  • [10] Huang, J. Z. and Shen, H. (2004). Functional coefficient regression models for nonlinear time series: A polynomial spline approach., Scandinavian Journal of Statistics, 31, 515–534.
  • [11] Huang, J. Z., Wu, C. O. and Zhou, L. (2004). Polynomial spline estimation and inference for varying coefficient models with longitudinal data., Statistica Sinica, 14, 763–788.
  • [12] Kauermann, G. and Tutz, G. (1999). On model diagnostics using varying coefficient models., Biometrika, 86, 119–128.
  • [13] Kai, B., Li, R., and Zou, H. (2011). New efficient estimation and variable selection methods for semiparametric varying-coefficient partially linear models., Ann. Stat., 39, 305–332.
  • [14] Keshavan, R. H., Montanari, A. and Oh, S. (2010). Matrix completion from a few entries., IEEE Trans. on Info. Th., 56(6), 2980–2998.
  • [15] Klopp, O. (2011). Matrix completion with unknown variance of the noise., http://arxiv.org/abs/1112.3055
  • [16] Klopp, O. (2012). Noisy low-rank matrix completion with general sampling distribution., Bernoulli, to appear.
  • [17] Klopp, O. (2011). Rank penalized estimators for high-dimensional matrices., Electronic Journal of Statistics, 5, 1161–1183.
  • [18] Koltchinskii, V. (2011). A remark on low rank matrix recovery and noncommutative Bernstein type inequalities., IMS Collections, Festschritt in Honor of J. Wellner.
  • [19] Koltchinskii, V., Lounici, K. and Tsybakov, A. (2011). Nuclear norm penalization and optimal rates for noisy low rank matrix completion., Annals of Statistics, 39(5), 2302–2329.
  • [20] Lian, H. (2012). Spline Estimator for Simultaneous Variable Selection and Constant Coefficient Identification in High-dimensional Generalized Varying-Coefficient Models., Manuscript.
  • [21] Ledoux, M. and Talagrand, M. (1991)., Probability in Banach Spaces: Isoperimetry and Processes. Springer-Verlag, New York, NY.
  • [22] Mallat, S. (2009)., A Wavelet Tour of Signal Processing, Third Ed., Elsevier, New York.
  • [23] Negahban, S. and Wainwright, M. J. (2010). Restricted strong convexity and weighted matrix completion: Optimal bounds with noise., Journal of Machine Learning Research, 13, 1665–1697.
  • [24] Senturk, D. and Mueller, H. G. (2010). Functional varying coefficient models for longitudinal data., J. Amer. Statist. Assoc., 105, 1256–1264.
  • [25] Tropp, J. A. (2011). User-friendly tail bounds for sums of random matrices., Found. Comput. Math., 11(4).
  • [26] Tsybakov, A. (2010)., Introduction to Nonparametric Estimation, Springer Series in Statistics.
  • [27] Wang, L., Kai, B., and Li, R. (2009). Local Rank Inference for Varying Coefficient Models., J. Amer. Statist. Assoc., 104, 1631–1645.
  • [28] Wu, C. O., Chiang, C. T. and Hoover, D. R. (1998). Asymptotic confi- dence regions for kernel smoothing of a varying-coefficient model with longitudinal data., J. Amer. Statist. Assoc., 93, 1388–1402.
  • [29] Yang, L., Park, B.U., Xue, L. and Hardle, W. (2006). Estimation and Testing for Varying Coefficients in Additive Models With Marginal Integration., J. Amer. Statist. Assoc., 101, 1212–1227.