Efficient estimation for a subclass of shape invariant models. (English) Zbl 1189.62057

Summary: We observe a fixed number of unknown \(2\pi \)-periodic functions differing from each other by both phase and amplitude. This semiparametric model appears in the literature under the name “shape invariant model.” While the common shape is unknown, we introduce an asymptotically efficient estimator of the finite-dimensional parameters (phase and amplitude) using profile likelihood and Fourier basis. Moreover, this estimation method leads to a consistent and asymptotically linear estimator for the common shape.


62G05 Nonparametric estimation
62F12 Asymptotic properties of parametric estimators
62G20 Asymptotic properties of nonparametric inference
62J02 General nonlinear regression
Full Text: DOI arXiv


[1] Dalalyan, A. S., Golubev, G. K. and Tsybakov, A. B. (2006). Penalized maximum likelihood and semiparametric second order efficiency. Ann. Statist. 34 169-201. · Zbl 1091.62020
[2] Gamboa, F., Loubes, J. and Maza, E. (2007). Semi-parametric estimation of shifts. Electron. J. Stat. 1 616-640. · Zbl 1141.62313
[3] Gassiat, E. and Lévy-Leduc, C. (2006). Efficient semiparametric estimation of the periods in a superposition of periodic functions with unknown shape. J. Time Ser. Anal. 27 877-910. · Zbl 1150.62047
[4] Guardabasso, V., Rodbard, D. and Munson, P. J. (1988). A versatile method for simultaneous analysis of families of curves. FASEB J. 2 209-215.
[5] Härdle, W. and Marron, J. S. (1990). Semiparametric comparison of regression curves. Ann. Statist. 18 63-59. · Zbl 0703.62053
[6] Kneip, A. and Gasser, T. (1988). Convergence and consistency results for self-modeling nonlinear regression. Ann. Statist. 16 82-112. · Zbl 0725.62060
[7] Lawton, W., Sylvestre, E. and Maggio, M. (1972). Self modeling nonlinear regression. Technometrics 14 513-532. · Zbl 0239.62045
[8] Loubes, J. M., Maza, E., Lavielle, M. and Rodriguez, L. (2006). Road trafficking description and short term travel time forecasting with a classification method. Canad. J. Statist. 34 475-491. · Zbl 1104.62127
[9] Luan, Y. and Li, H. (2004). Model-based methods for identifying periodically expressed genes based on time course microarray gene expression data. Bioinformatics 20 332-339.
[10] McNeney, B. and Wellner, J. A. (2000). Application of convolution theorems in semiparametric models with non-i.i.d. data. J. Statist. Plann. Inference 91 441-480. · Zbl 0970.62031
[11] Murphy, S. A. and Van der Vaart, A. W. (2000). On profile likelihood. J. Amer. Statist. Assoc. 95 449-485. JSTOR: · Zbl 0995.62033
[12] Van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3 . Cambridge Univ. Press, Cambridge. · Zbl 0910.62001
[13] Van der Vaart, A. W. (2002). Semiparametric statistics. In Lectures on Probability Theory and Statistics (Saint-Flour, 1999). Lecture Notes in Math. 1781 331-457. Springer, Berlin. · Zbl 1013.62031
[14] Vimond, M. (2007). Inférence statistique par des transformées de Fourier pour des modèles de régression semi-paramétriques. Ph.D. thesis, Institut de Mathématiques de Toulouse, Univ. Paul Sabatier.
[15] Wang, Y. and Brown, M. M. (1996). A flexible model for human circadian rhythms. Biometrics 52 588-596. · Zbl 0875.62525
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.