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Risk bounds for new M-estimation problems∗

Published online by Cambridge University Press:  04 November 2013

Nabil Rachdi ,
Jean-Claude Fort  and
Thierry Klein
Nabil Rachdi
Affiliation:
EADS Innovation Works, 12 rue Pasteur, 92152 Suresnes, France. nabil.rachdi@eads.net Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse, France
Jean-Claude Fort
Affiliation:
Université Paris Descartes, SPC, MAP5, 45 rue des Saints-Pères, 75006 Paris, France; jean-claude.fort@parisdescartes.fr
Thierry Klein
Affiliation:
Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse, France; thierry.klein@math.univ-toulouse.fr
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Abstract

In this paper, we consider a new framework where two types of data are available: experimental data Y1,...,Yn supposed to be i.i.d from Y and outputs from a simulated reduced model. We develop a procedure for parameter estimation to characterize a feature of the phenomenon Y. We prove a risk bound qualifying the proposed procedure in terms of the number of experimental data n, reduced model complexity and computing budget m. The method we present is general enough to cover a wide range of applications. To illustrate our procedure we provide a numerical example.

Keywords

M-estimationinverse problemsempirical processesoracle inequalitiesmodel selection
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 740 - 766
Copyright
© EDP Sciences, SMAI, 2013

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References

Barbillon, P., Celeux, G., Grimaud, A., Lefebvre, Y., and De Rocquigny, E., Nonlinear methods for inverse statistical problems. Comput. Stat. Data Anal. 55 (2011) 132–142. Google Scholar
P. Billingsley, Convergence of probability measures. Wiley New York (1968).
E.de Rocquigny, N. Devictor and S. Tarantola, editors. Uncertainty in industrial practice. John Wiley.
M.D. Donsker, Justification and extension of Doob’s heuristic approach to the Kolmogorov-Smirnov theorems. Annal. Math. Stat. (1952) 277–281.
Dudley, R.M., Weak convergence of measures on nonseparable metric spaces and empirical measures on euclidian spaces. Illinois J. Math. 11 (1966) 109–126. Google Scholar
P. Gaenssler, Empirical Processes. Instit. Math. Stat., Hayward, CA (1983).
A. Goldenshluger and O. Lepski, Uniform bounds for norms of sums of independent random functions (2009) Preprint: arXiv:0904.1950.
P.J. Huber, Robust estimation of a location parameter. Annal. Math. Stat. (1964) 73–101.
P.J. Huber, Robust statistics. Wiley-Interscience (1981).
J.P.C. Kleijnen, Design and analysis of simulation experiments. Springer Verlag (2007).
Klein, T. and Rio, E., Concentration around the mean for maxima of empirical processes. Ann. Prob. 33 (2005) 1060–1077. Google Scholar
M.R. Kosorok, Introduction to empirical processes and semiparametric inference. Springer Series in Statistics (2008).
M. Ledoux, The concentration of measure phenomenon. AMS (2001).
P. Massart, Concentration inequalities and model selection: Ecole d’Eté de Probabilités de Saint-Flour XXXIII-2003. Springer Verlag (2007).
Massart, P. and Nédélec, É., Risk bounds for statistical learning. Annal. Stat. 34 (2006) 2326–2366. Google Scholar
D. Pollard, Empirical processes: theory and applications. Regional Conference Series in Probability and Statistics Hayward (1990).
Rachdi, N., Fort, J.C. and Klein, T., Stochastic inverse problem with noisy simulator- an application to aeronautic model. Annal. Facult. Sci. Toulouse 21.
T.J. Santner, B.J. Williams and W. Notz, The design and analysis of computer experiments. Springer Verlag (2003).
G.R Shorack and J.A Wellner. Empirical processes with applications to statistics. Wiley Series in Probability and Statistics (1986).
Soize, C. and Ghanem, R., Physical systems with random uncertainties: chaos representations with arbitrary probability measure. SIAM J. Sci. Comput. 26 (2004) 395–410. Google Scholar
Talagrand, M., Sharper bounds for Gaussian and empirical processes. Annal. Prob. 22 (1994) 28–76. Google Scholar
S. van de Geer, Empirical processes in M-estimation. Cambridge University Press (2000).
A.W. van der Vaart, Asymptotic statistics. Cambridge University Press (2000).
A.W. van der Vaart and J.A. Wellner, Weak Convergence and Empirical Processes. Springer Series in Statistics (1996).
E. Vazquez. Modélisation comportementale de systèmes non-linéaires multivariables par méthodes à noyaux et applications. Ph.D. thesis (2005).