The Annals of Statistics

Minimax adaptive tests for the functional linear model

Nadine Hilgert, André Mas, and Nicolas Verzelen

Full-text: Open access

Abstract

We introduce two novel procedures to test the nullity of the slope function in the functional linear model with real output. The test statistics combine multiple testing ideas and random projections of the input data through functional principal component analysis. Interestingly, the procedures are completely data-driven and do not require any prior knowledge on the smoothness of the slope nor on the smoothness of the covariate functions. The levels and powers against local alternatives are assessed in a nonasymptotic setting. This allows us to prove that these procedures are minimax adaptive (up to an unavoidable $\log\log n$ multiplicative term) to the unknown regularity of the slope. As a side result, the minimax separation distances of the slope are derived for a large range of regularity classes. A numerical study illustrates these theoretical results.

Article information

Source
Ann. Statist. Volume 41, Number 2 (2013), 838-869.

Dates
First available in Project Euclid: 29 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.aos/1369836962

Digital Object Identifier
doi:10.1214/13-AOS1093

Mathematical Reviews number (MathSciNet)
MR3099123

Zentralblatt MATH identifier
1267.62059

Subjects
Primary: 62J05: Linear regression
Secondary: 62G10: Hypothesis testing

Keywords
Functional linear regression eigenfunction principal component analysis adaptive testing minimax hypothesis testing minimax separation rate multiple testing ellipsoid goodness-of-fit

Citation

Hilgert, Nadine; Mas, André; Verzelen, Nicolas. Minimax adaptive tests for the functional linear model. Ann. Statist. 41 (2013), no. 2, 838--869. doi:10.1214/13-AOS1093. https://projecteuclid.org/euclid.aos/1369836962.


Export citation

References

  • [1] Ash, R. B. and Gardner, M. F. (1975). Topics in Stochastic Processes. Probability and Mathematical Statistics 27. Academic Press, New York.
  • [2] Baraud, Y. (2002). Non-asymptotic minimax rates of testing in signal detection. Bernoulli 8 577–606.
  • [3] Baraud, Y., Huet, S. and Laurent, B. (2003). Adaptive tests of linear hypotheses by model selection. Ann. Statist. 31 225–251.
  • [4] Cai, T. T. and Hall, P. (2006). Prediction in functional linear regression. Ann. Statist. 34 2159–2179.
  • [5] Cardot, H., Ferraty, F., Mas, A. and Sarda, P. (2003). Testing hypotheses in the functional linear model. Scand. J. Stat. 30 241–255.
  • [6] Cardot, H., Ferraty, F. and Sarda, P. (2003). Spline estimators for the functional linear model. Statist. Sinica 13 571–591.
  • [7] Cardot, H., Goia, A. and Sarda, P. (2004). Testing for no effect in functional linear regression models, some computational approaches. Comm. Statist. Simulation Comput. 33 179–199.
  • [8] Cardot, H. and Johannes, J. (2010). Thresholding projection estimators in functional linear models. J. Multivariate Anal. 101 395–408.
  • [9] Cardot, H., Mas, A. and Sarda, P. (2007). CLT in functional linear regression models. Probab. Theory Related Fields 138 325–361.
  • [10] Cardot, H. and Sarda, P. (2011). Functional linear regression. In The Oxford Handbook of Functional Data Analysis 21–46. Oxford Univ. Press, Oxford.
  • [11] Comte, F. and Johannes, J. (2010). Adaptive estimation in circular functional linear models. Math. Methods Statist. 19 42–63.
  • [12] Comte, F. and Johannes, J. (2012). Adaptive functional linear regression. Ann. Statist. 40 2765–2797.
  • [13] Crambes, C., Kneip, A. and Sarda, P. (2009). Smoothing splines estimators for functional linear regression. Ann. Statist. 37 35–72.
  • [14] Cuevas, A. and Fraiman, R. (2004). On the bootstrap methodology for functional data. In COMPSTAT 2004—Proceedings in Computational Statistics 127–135. Physica, Heidelberg.
  • [15] Dauxois, J., Pousse, A. and Romain, Y. (1982). Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference. J. Multivariate Anal. 12 136–154.
  • [16] Donoho, D. and Jin, J. (2004). Higher criticism for detecting sparse heterogeneous mixtures. Ann. Statist. 32 962–994.
  • [17] Dunford, N. and Schwartz, J. T. (1988). Linear Operators, Part I: General Theory. Wiley, New York.
  • [18] Gohberg, I., Goldberg, S. and Kaashoek, M. A. (1990). Classes of Linear Operators. Vol. I. Operator Theory: Advances and Applications 49. Birkhäuser, Basel.
  • [19] Gohberg, I., Goldberg, S. and Kaashoek, M. A. (1993). Classes of Linear Operators. Vol. II. Operator Theory: Advances and Applications 63. Birkhäuser, Basel.
  • [20] González-Manteiga, W. and Martínez-Calvo, A. (2011). Bootstrap in functional linear regression. J. Statist. Plann. Inference 141 453–461.
  • [21] Hall, P. and Horowitz, J. L. (2007). Methodology and convergence rates for functional linear regression. Ann. Statist. 35 70–91.
  • [22] Hall, P. and Hosseini-Nasab, M. (2006). On properties of functional principal components analysis. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 109–126.
  • [23] Hall, P. and Hosseini-Nasab, M. (2009). Theory for high-order bounds in functional principal components analysis. Math. Proc. Cambridge Philos. Soc. 146 225–256.
  • [24] Hall, P. and Vial, C. (2006). Assessing extrema of empirical principal component functions. Ann. Statist. 34 1518–1544.
  • [25] Hilgert, N., Mas, A. and Verzelen, N. (2013). Supplement to “Minimax adaptive tests for the functional linear model.” DOI:10.1214/13-AOS1093SUPP.
  • [26] Ingster, Y. I. (1993). Asymptotically minimax hypothesis testing for nonparametric alternatives. I. Math. Methods Statist. 2 85–114.
  • [27] Ingster, Y. I. (1993). Asymptotically minimax hypothesis testing for nonparametric alternatives. II. Math. Methods Statist. 2 171–189.
  • [28] Ingster, Y. I. (1993). Asymptotically minimax hypothesis testing for nonparametric alternatives. III. Math. Methods Statist. 2 249–268.
  • [29] Mas, A. and Menneteau, L. (2003). Perturbation approach applied to the asymptotic study of random operators. In High Dimensional Probability, III (Sandjberg, 2002). Progress in Probability 55 127–134. Birkhäuser, Basel.
  • [30] Meister, A. (2011). Asymptotic equivalence of functional linear regression and a white noise inverse problem. Ann. Statist. 39 1471–1495.
  • [31] R Development Core Team (2009). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0. Available at http://www.R-project.org.
  • [32] Rudin, W. (1987). Real and Complex Analysis, 3rd ed. McGraw-Hill, New York.
  • [33] Spokoiny, V. G. (1996). Adaptive hypothesis testing using wavelets. Ann. Statist. 24 2477–2498.
  • [34] Verzelen, N. (2012). Minimax risks for sparse regressions: Ultra-high dimensional phenomenons. Electron. J. Stat. 6 38–90.
  • [35] Verzelen, N. and Villers, F. (2010). Goodness-of-fit tests for high-dimensional Gaussian linear models. Ann. Statist. 38 704–752.
  • [36] Yuan, M. and Cai, T. T. (2010). A reproducing kernel Hilbert space approach to functional linear regression. Ann. Statist. 38 3412–3444.

Supplemental materials