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Poisson inverse problems. (English) Zbl 1106.62035
Summary: We focus on nonparametric estimators in inverse problems for Poisson processes involving the use of wavelet decompositions. Adopting an adaptive wavelet Galerkin discretization, we find that our method combines the well-known theoretical advantages of wavelet-vaguelette decompositions for inverse problems in terms of optimally adapting to the unknown smoothness of the solution, together with the remarkably simple closed-form expressions of Galerkin inversion methods. Adapting the results of A. R. Barron and C.-H. Sheu [ibid. 19, No. 3, 1347–1369 (1991; Zbl 0739.62027)] to the context of log-intensity functions approximated by wavelet series with the use of the Kullback-Leibler distance between two point processes, we also present an asymptotic analysis of convergence rates that justifies our approach. In order to shed some light on the theoretical results obtained and to examine the accuracy of our estimates in finite samples, we illustrate our method by the analysis of some simulated examples.

62G07 Density estimation
62M05 Markov processes: estimation; hidden Markov models
65T60 Numerical methods for wavelets
65J10 Numerical solutions to equations with linear operators
62M09 Non-Markovian processes: estimation
65C60 Computational problems in statistics (MSC2010)
46N30 Applications of functional analysis in probability theory and statistics
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[1] Abramovitch, F. and Silverman, B. W. (1998). Wavelet decomposition approaches to statistical inverse problems. Biometrika 85 115–129. JSTOR: · Zbl 0908.62095
[2] Barron, A. R. and Sheu, C.-H. (1991). Approximation of density functions by sequences of exponential families. Ann. Statist. 19 1347–1369. · Zbl 0739.62027
[3] Cavalier, L. and Koo, J.-Y. (2002). Poisson intensity estimation for tomographic data using a wavelet shrinkage approach. IEEE Trans. Inform. Theory 48 2794–2802. · Zbl 1062.92042
[4] Cohen, A., DeVore, R., Kerkyacharian, G. and Picard, D. (2001). Maximal spaces with given rate of convergence for thresholding algorithms. Appl. Comput. Harmon. Anal. 11 167–191. · Zbl 0997.62025
[5] Cohen, A., Hoffmann, M. and Reiss, M. (2004). Adaptive wavelet-Galerkin methods for inverse problems. SIAM J. Numer. Anal. 42 1479–1501. · Zbl 1077.65054
[6] Csiszár, I. (1975). I-divergence geometry of probability distributions and minimization problems. Ann. Probab. 3 146–158. JSTOR: · Zbl 0318.60013
[7] Donoho, D. L. (1995). Nonlinear solution of linear inverse problems by wavelet–vaguelette decomposition. Appl. Comput. Harmon. Anal. 2 101–126. · Zbl 0826.65117
[8] Donoho, D. L., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1996). Density estimation by wavelet thresholding. Ann. Statist. 24 508–539. · Zbl 0860.62032
[9] Fan, J. and Koo, J.-Y. (2002). Wavelet deconvolution. IEEE Trans. Inform. Theory 48 734–747. · Zbl 1071.94511
[10] Figueiredo, M. and Nowak, R. (2003). An EM algorithm for wavelet-based image restoration. IEEE Trans. Image Process. 12 906–916. · Zbl 1279.94015
[11] Johnstone, I. M. and Silverman, B. W. (1990). Speed of estimation in positron emission tomography and related inverse problems. Ann. Statist. 18 251–280. · Zbl 0699.62043
[12] Hall, P. and Smith, R. L. (1988). The kernel method for unfolding sphere size distributions. J. Comput. Phys. 74 409–421. · Zbl 0637.65148
[13] Kim, W.-C. and Koo, J.-Y. (2002). Inhomogeneous Poisson intensity estimation via information projections onto wavelet subspaces. J. Korean Statist. Soc. 31 343–357.
[14] Kolaczyk, E. D. (1999). Bayesian multiscale models for Poisson processes. J. Amer. Statist. Assoc. 94 920–933. JSTOR: · Zbl 1072.62630
[15] Kolaczyk, E. D. (1999). Wavelet shrinkage estimation of certain Poisson intensity signals using corrected thresholds. Statist. Sinica 9 119–135. · Zbl 0927.62081
[16] Kolaczyk, E. D. and Nowak, R. D. (2004). Multiscale likelihood analysis and complexity penalized estimation. Ann. Statist. 32 500–527. · Zbl 1048.62036
[17] Koo, J.-Y. and Kim, W.-C. (1996). Wavelet density estimation by approximation of log-densities. Statist. Probab. Lett. 26 271–278. · Zbl 0843.62040
[18] Kovac, A. and Silverman, B. W. (2000). Extending the scope of wavelet regression methods by coefficient-dependent thresholding. J. Amer. Statist. Assoc. 95 172–183.
[19] Mase, S. (1995). Stereological estimation of particle size distributions. Adv. in Appl. Probab. 27 350–366 JSTOR: · Zbl 0822.62028
[20] Nowak, R. D. and Kolaczyk, E. D. (2000). A statistical multiscale framework for Poisson inverse problems. Information-theoretic imaging. IEEE Trans. Inform. Theory 46 1811–1825. · Zbl 0999.94004
[21] Nychka, D. and Cox, D. D. (1989). Convergence rates for regularized solutions of integral equations from discrete noisy data. Ann. Statist. 17 556–572. · Zbl 0672.62054
[22] Nychka, D., Wahba, G., Goldfarb, S. and Pugh, T. (1984). Cross-validated spline methods for the estimation of three-dimensional tumor size distributions from observations on two-dimensional cross sections. J. Amer. Statist. Assoc. 79 832–846. JSTOR: · Zbl 0572.62085
[23] O’Sullivan, F. (1986). A statistical perspective on ill-posed problems (with discussion). Statist. Sci. 1 502–527. · Zbl 0625.62110
[24] Reynaud-Bouret, P. (2003). Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities. Probab. Theory Related Fields 126 103–153. · Zbl 1019.62079
[25] Silverman, B. W., Jones, M. C., Nychka, D. and Wilson, J. D. (1990). A smoothed EM approach to indirect estimation problems, with particular reference to stereology and emission tomography (with discussion). J. Roy. Statist. Soc. Ser. B 52 271–324. JSTOR: · Zbl 0703.62105
[26] Szkutnik, Z. (2000). Unfolding intensity function of a Poisson process in models with approximately specified folding operator. Metrika 52 1–26. · Zbl 1093.62570
[27] Vannucci, M. and Corradi, F. (1999). Covariance structure of wavelet coefficients: Theory and models in a Bayesian perspective. J. R. Stat. Soc. Ser. B Stat. Methodol. 61 971–986. JSTOR: · Zbl 0940.62023
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