Risk bounds for new M-estimation problems
ESAIM: Probability and Statistics, Tome 17 (2013), pp. 740-766.

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.

DOI : 10.1051/ps/2012025
Classification : 65C60, 60F05, 62F12, 60G20, 65J22
Mots clés : M-estimation, inverse problems, empirical processes, oracle inequalities, model selection
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Rachdi, Nabil; Fort, Jean-Claude; Klein, Thierry. Risk bounds for new M-estimation problems. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 740-766. doi : 10.1051/ps/2012025. http://archive.numdam.org/articles/10.1051/ps/2012025/

[1] P. Barbillon, G. Celeux, A. Grimaud, Y. Lefebvre, and E. De Rocquigny, Nonlinear methods for inverse statistical problems. Comput. Stat. Data Anal. 55 (2011) 132-142. | MR | Zbl

[2] P. Billingsley, Convergence of probability measures. Wiley New York (1968). | MR | Zbl

[3] E. De Rocquigny, N. Devictor and S. Tarantola, editors. Uncertainty in industrial practice. John Wiley.

[4] M.D. Donsker, Justification and extension of Doob's heuristic approach to the Kolmogorov-Smirnov theorems. Annal. Math. Stat. (1952) 277-281. | MR | Zbl

[5] R.M. Dudley, Weak convergence of measures on nonseparable metric spaces and empirical measures on euclidian spaces. Illinois J. Math. 11 (1966) 109-126. | MR | Zbl

[6] P. Gaenssler, Empirical Processes. Instit. Math. Stat., Hayward, CA (1983). | MR

[7] A. Goldenshluger and O. Lepski, Uniform bounds for norms of sums of independent random functions (2009) Preprint: arXiv:0904.1950. | MR | Zbl

[8] P.J. Huber, Robust estimation of a location parameter. Annal. Math. Stat. (1964) 73-101. | MR | Zbl

[9] P.J. Huber, Robust statistics. Wiley-Interscience (1981). | MR | Zbl

[10] J.P.C. Kleijnen, Design and analysis of simulation experiments. Springer Verlag (2007). | MR | Zbl

[11] T. Klein and E. Rio, Concentration around the mean for maxima of empirical processes. Ann. Prob. 33 (2005) 1060-1077. | MR | Zbl

[12] M.R. Kosorok, Introduction to empirical processes and semiparametric inference. Springer Series in Statistics (2008). | MR | Zbl

[13] M. Ledoux, The concentration of measure phenomenon. AMS (2001). | MR | Zbl

[14] P. Massart, Concentration inequalities and model selection: Ecole d'Eté de Probabilités de Saint-Flour XXXIII-2003. Springer Verlag (2007). | MR | Zbl

[15] P. Massart and É. Nédélec, Risk bounds for statistical learning. Annal. Stat. 34 (2006) 2326-2366. | MR | Zbl

[16] D. Pollard, Empirical processes: theory and applications. Regional Conference Series in Probability and Statistics Hayward (1990). | MR | Zbl

[17] N. Rachdi, J.C. Fort and T. Klein, Stochastic inverse problem with noisy simulator- an application to aeronautic model. Annal. Facult. Sci. Toulouse 21. | Numdam | MR | Zbl

[18] T.J. Santner, B.J. Williams and W. Notz, The design and analysis of computer experiments. Springer Verlag (2003). | MR | Zbl

[19] G.R Shorack and J.A Wellner. Empirical processes with applications to statistics. Wiley Series in Probability and Statistics (1986). | MR | Zbl

[20] C. Soize and R. Ghanem, Physical systems with random uncertainties: chaos representations with arbitrary probability measure. SIAM J. Sci. Comput. 26 (2004) 395-410. | MR | Zbl

[21] M. Talagrand, Sharper bounds for Gaussian and empirical processes. Annal. Prob. 22 (1994) 28-76. | MR | Zbl

[22] S. Van De Geer, Empirical processes in M-estimation. Cambridge University Press (2000). | MR | Zbl

[23] A.W. Van Der Vaart, Asymptotic statistics. Cambridge University Press (2000). | MR | Zbl

[24] A.W. Van Der Vaart and J.A. Wellner, Weak Convergence and Empirical Processes. Springer Series in Statistics (1996). | MR | Zbl

[25] E. Vazquez. Modélisation comportementale de systèmes non-linéaires multivariables par méthodes à noyaux et applications. Ph.D. thesis (2005).

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