. @bullet-une-Étape, A) de détermination d'une fonction approchant la log-vraisemblance de la distribution cible f ? 0 correspondant à une distribution sans effet d'empilement à partir des mesures de temps de réponse de particules détectées avec effet d'empilement, c'est-à-dire des mesures de la distribution g ? 0

. @bullet-une-Étape, B) de maximisation de la log-vraisemblance approchée sur l'ensemble des paramètres ? afin de déterminer l'estimateur comme le vecteur ? qui maximise la log-vraisemblance approchée

C. and D. De-l-', INVENTION La présente invention concerne un procédé d'estimation de la distribution des temps de réponse de particules d'un système, dans le but notamment de le caractériser. Elle s'applique par exemple aux mesures de fluorescence résolues en temps

B. Balabdaoui, F. Wellner, and J. A. , Mixtures and monotonicity: a review of estimation under monotonicity constraints. IMS Lecture Notes -Monograph Series, 2004.

B. , M. N. Bodunov, E. N. Valeur, and B. , Mathematical functions for the analysis of luminescence decays with underlying distributions 1. Kohlrausch decay function (stretched exponential), Chemical Physics, vol.515, pp.171-182, 2005.

B. , M. N. Bodunov, E. N. Valeur, and B. , Mathematical functions for the analysis of luminescence decays with underlying distributions: 2. Becquerel (compressed hyperbola) and related decay functions, Chemical Physics, vol.317, pp.57-62, 2005.

B. , P. J. Klaassen, C. A. Ritov, Y. Wellner, and J. A. , Efficient and Adaptive Estimation for Semiparametric Models, 1998.

P. Billingsley, Convergence of Probability Measures, 1999.
DOI : 10.1002/9780470316962

T. Bruns, W. S. Strauss, and H. Schneckenburger, Cell Membrane Fluidity Determined by Fluorescence Lifetime and Polarization Screening, no. 114 in Springer Proceedings in Physics, 2007.
DOI : 10.1007/978-3-540-68764-1_61

C. , R. J. Ruppert, and D. , Transformation and weighting in regression, 1988.

C. , H. A. Steinkamp, and J. A. , Flow cytometric fluorescence lifetime measurements, 2000.

D. , A. P. Laird, N. M. Rubin, and D. B. , Maximum likelihood from incomplete data via the EM algorithm, Journal of the Royal Statistical Society. Series BMethodological), vol.39, issue.1, pp.1-38, 1977.

D. , R. A. Lorentz, and G. G. , Constructive Approximation, 1993.

G. , A. Modlin, R. L. Brand, and L. , Analysis of fluorescence decay curves by means of the Laplace transformation, Biophysical Journal, vol.15, issue.3, pp.263-280, 1975.

G. , A. Carlin, J. B. Stern, H. S. Rubin, and D. B. , Bayesian Data Analysis, 2003.

G. , R. Levit, and B. , Applications of the van Trees inequality: a Bayesian Cramér-Rao bound, Bernoulli, vol.1, issue.12, pp.59-79, 1995.

G. , E. Limkeman, M. Lakowicz, J. Maliwal, B. Cherek et al., Resolution of mixtures of fluorophores using variable-frequency phase and modulation data, Biophysical Journal, vol.46, issue.4, pp.479-486, 1984.

G. , A. Steinberg, and I. Z. , Fast relaxation processes in a protein revealed by the decay kinetics of tryptophan fluorescence, Biochemistry, vol.13, issue.25, pp.5170-5718, 1974.

G. , M. Philippe, A. Robert, and C. P. , MCMC control spreadsheets for exponential mixture estimation, Journal of Computational and Graphical Statistics, vol.8, issue.2, pp.298-317, 1999.

H. , P. Selinger, and B. , Better estimates of exponential decay parameters, Journal of Physical Chemistry, vol.85, issue.20, pp.2941-2946, 1981.

I. , I. Dyson, and R. , The analysis of fluorescence decay by a method of moments, Biophysical Journal, vol.9, pp.1337-1350, 1969.

J. , W. Visser, A. J. Herron, J. N. Sutter, and M. , Fluorescence spectroscopy, Methods for Structural Analysis of Protein Pharmaceuticals, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00137719

K. , J. Wolfowitz, and J. , Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. The Annals of, Mathematical Statistics, vol.27, issue.4, pp.887-906, 1956.

L. , J. Laczko, G. Cherek, H. Gratton, E. Limkeman et al., Analysis of fluorescence decay kinetics from variable-frequency phase shift and modulation data, Biophysical Journal, vol.46, issue.4, pp.463-477, 1984.

L. , R. J. Rubin, and D. B. , Statistical Analysis with Missing Data, Population (French Edition), vol.43, issue.6, 2002.
DOI : 10.2307/1533221

L. , A. K. Brochon, and J. C. , Analyzing the distribution of decay constants in pulse-fluorimetry using the maximum entropy method, Biophysical Journal, vol.52, issue.5, pp.693-706, 1987.

M. , J. Mengersen, K. Robert, and C. P. , Bayesian modelling and inference on mixtures of distributions, Handbook of Statistics, 2005.

M. , J. &. Robert, and C. P. , Bayesian core: a practical approach to computational Bayesian statistics, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00450489

M. , J. S. Wand, and M. P. , Exact mean integrated squared error, The Annals of Statistics, vol.20, issue.2, pp.712-736, 1992.

A. E. Mckinnon, A. G. Szabo, and D. R. Miller, The deconvolution of photoluminescence data, The Journal of Physical Chemistry, vol.81, issue.16, pp.1564-1570, 1977.
DOI : 10.1021/j100531a009

M. , G. Peel, and D. , Finite Mixture Models, 2000.

O. , D. V. Phillips, and D. , Time-correlated single photon counting, 1984.

O. , D. V. Ware, W. R. Andre, and J. C. , Deconvolution of fluorescence decay curves. A critical comparison of techniques, Journal of Physical Chemistry, vol.83, issue.10, pp.1333-1343, 1979.

O. , L. Tedesco, A. Touso, A. Oliveira, D. Maeder et al., Abstracting reliable parameters from time-correlated single photon counting experiments, Journal of Photochemistry and Photobiology A: Chemistry, vol.175, pp.221-225, 2005.

P. , M. Wahl, M. Kapusta, P. Erdmann, and R. , Dead-time effects in TCSPC data analysis, Proceedings of SPIE, 2007.

P. , R. S. Bartolucci, F. Lindsay, and B. G. , Model building for semiparametric mixtures, 2006.

P. , R. S. Lindsay, and B. G. , Alternative EM methods for nonparametric finite mixture models, Biometrika, vol.8, issue.2, pp.535-550, 2001.

P. , J. Ayoub, M. Maurel, D. Perroy, J. Trinquet et al., Energy transfer technologies to monitor the dynamics and signaling properties of G-protein-coupled receptors in living cells. In: Biophysical analysis of membrane proteins: investigating structure and function, 2007.

R. , V. Popescu, and A. , Integrated Molecular and Cellular Biophysics, 2008.

R. , T. Roueff, and F. , Minimax estimation of the mixing density of a mixture of exponential distributions, 2009.

R. , T. Roueff, F. Souloumiac, and A. , Information bounds and MCMC parameter estimation for the pile-up model, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00448726

R. , T. Roueff, F. Souloumiac, and A. , A corrected likelihood approach for the pile-up model with application to fluorescence lifetime measurements using exponential mixtures, 2009.

R. , T. Roueff, F. Souloumiac, and A. , Désempilement de mesures de temps de réponse par un algorithme E.M. modifié, 2009.

R. , T. Roueff, F. Souloumiac, and A. , Procédé d'estimation des paramètres de la distribution des temps de réponse de particules d'un système, appliqué notamment aux mesures de fluorescence, 2009.

R. , R. Walker, and H. , Mixture densities, maximum likelihood and the em algorithm, SIAM Review, vol.26, issue.2, pp.195-239, 1984.

R. , C. P. Casella, and G. , Monte Carlo Statistical Methods, 2004.

R. , F. Ryden, and T. , Nonparametric estimation of mixing densities for discrete distributions, The Annals of Statistics, vol.33, pp.2066-2108, 2005.

S. , I. N. Zaccai, N. R. Zaccai, and J. , Methods in molecular biophysics: structure, dynamics, function, 2007.

J. Shao, Mathematical Statistics, 2003.
DOI : 10.1007/b97553

T. , M. A. Wong, and W. H. , The calculation of posterior distributions by data augmentation, Journal of the American Statistical Association, vol.82, issue.398, pp.528-540, 1987.

T. , D. Smith, A. Makov, and U. , Statistical Analysis of Finite Mixture Distributions, 1985.

V. De-ven, M. Ameloot, M. Valeur, B. Boens, and N. , Pitfalls and their remedies in time-resolved fluorescence spectroscopy and microscopy, Journal of Fluorescence, vol.15, issue.3, pp.377-412, 2005.

. Van-der and A. W. Vaart, Asymptotic Statistics, 1998.
DOI : 10.1017/CBO9780511802256

W. , W. R. Doemeny, L. J. Nemzek, and T. L. , Deconvolution of fluorescence and phosphorescence decay curves. Least-squares method, The Journal of Physical Chemistry, vol.77, issue.17, pp.2038-2048, 1973.