Estimating linear functionals in Poisson mixture models

Journal of Nonparametric Statistics

Volumn 21, Issue 6, 2009, Pages 713-728

  Cavalier, Laurent ^a   Hengartner, Nicolas W ^b  

^a AIX MARSEILLE UNIVERSITY   (France)

^b LOS ALAMOS NATIONAL LABORATORY   (United States)

Author keywords

Inverse problems; Linear functionals; Poisson mixtures

Indexed keywords

EID: 70449413668     PISSN: 10485252     EISSN: 10290311     Source Type: Journal    
DOI: 10.1080/10485250902971716     Document Type: Article

References (35)

1. N.W. Hengartner, L. Talbot, I. Shepard, and P.J. Bickel, Estimating the probability density of the scattering cross section from Rayleigh scattering experiments, J. Opt. Soc. Am. A 12 (1994), pp. 1316-1323.
2. J.-P. Wang and B.G. Lindsay, A penalized nonparametric maximum likelihood approach to species richness estimation, J. Am. Statist. Assoc. 100 (2005), pp. 942-959.
3. J.L. Norris and K.H. Pollack, Nonparametric MLE incorporation of heterogeneity and model testing into premarked cohort studies, Environ. Ecol. Stat. 27 (2001), pp. 21-32.
4. H.G. Tucker, An estimate of the compounding distribution of a compound Poisson distribution, Theory Probab. Appl. 8(1963), pp. 195-200.
5. G. Walter, Orthogonal polynomial estimators of the prior distribution of a compound Poisson distribution, Sankhya Ser. A 47 (1985), pp. 222-230.
6. R.S. Pilla and B.G. Lindsay, Alternative EM methods for nonparametric finite mixture models, Biometrika 88 (2001), pp. 535-550.
7. H. Holzmann, A. Munk, and B. Stratmann, Identifiability of finite mixtures-with applications to circular distributions, Sankhya 66 (2004), pp. 440-449.
8. J.-P. Wang, A linearization procedure and a VDM/ECM algorithm for penalized and constrained nonparametric maximum likelihood estimation for mixture models, Comput. Stat. Data Anal. 51 (2007), pp. 2946-2957.
9. C.-H. Zhang, On estimating mixing densities in exponential family models for discrete variables, Ann. Stat. 23 (1995), pp. 929-945.
10. W.-L. Loh and C.-H. Zhang, Global properties of kernel estimators for mixing densities in discrete exponential family models, Statist. Sinica 6 (1996), pp. 561-578.
11. N.W. Hengartner, Adaptive demixing in Poisson mixture models, Ann. Stat. 25 (1997), pp. 917-928.
12. A.C.M. van Rooij and F.H. Ruymgaart, Asymptotic minimax rates for abstract linear estimators, J. Statist. Plann. Inference 53 (1996), pp. 389-402.
13. L. Cavalier, G.K. Golubev, D. Picard, and A.B. Tsybakov, Oracle inequalities in inverse problems, Ann. Stat. 30 (2002), pp. 843-874.
14. L. Cavalier and A.B. Tsybakov, Sharp adaptation for inverse problems with random noise, Probab. Theory Related Fields 123 (2002), pp. 323-354.
15. L. Cavalier, Nonparametric estimation in inverse problems, Inverse Probl. 24 (2008), pp. 1-19.
16. J. Fan, On the optimal rates of convergence for nonparametric deconvolution problems, Ann. Stat. 19 (1991), pp. 1257-1272.
17. [17] M.H. Neumann, On the effect of estimating the error density in nonparametric deconvolution, J. Nonparametr. Stat. 7(1997), pp. 307-330.
18. M. Pensky and B. Vidakovic, Adaptive wavelet estimator for nonparametric density deconvolution, Ann. Stat. 27 (1999), pp. 2033-2053.
19. J. Fan, Deconvolution with supersmooth distributions, Can. J. Stat. 20 (1992), pp. 155-169.
20. S. Efromovich, Density estimation for the case of supersmooth measurement error, J. Am. Statist. Assoc. 92 (1997), pp. 526-535.
21. P.J. Bickel and Y. Ritov, Nonparametric estimators which can be 'plugged-in', Ann. Stat. 31 (2003), pp. 1033-1053.
22. E. Giné and R. Nickl, Uniform central limit theorems for kernel density estimators, Probab. Theory Related Fields 141 (2008), pp. 333-387.
23. A. Van der Vaart, On differential functionals, Ann. Stat. 19 (1991), pp. 178-204.
24. K.L. Chung, A Course in Probability Theory, 2nd ed., Academic Press, NewYork, 1974.
25. Y.S. Chow and H. Teicher, Probability Theory: Independence, Interchangeability, Martingales, 2nd ed., Springer Verlag, NewYork, 1988.
26. D. Rainville, Special Functions, 4th ed., Macmillan Co., NewYork, 1967.
27. W.H. Beyer, CRC Standard Mathematical Tables, 27th ed., CRC Press, Boca Raton, FL, 1985.
28. L. Cavalier, G.K. Golubev, O. Lepski, and A.B. Tsybakov, Block thresholding and sharp adaptive estimation in severely ill-posed inverse problems, Theory Probab. Appl. 48 (2003), pp. 534-556
29. SIAM version 48 (2004), pp. 426-446.
30. I. Steinwart, D. Hush, and C. Scovel, An explicit description of the reproducing kernel Hilbert spaces of Gaussian RBF kernels, IEEE Trans. Inform. Theory 52 (2006), pp. 4635-4643.
31. I.A. Ibragimov and R.Z. Khas'minskii, Estimation of distribution density belonging to a class of entire functions, Theory Probab. Appl. 27 (1982), pp. 551-562.
32. S. Loustau, Performances statistiques de méthodes à noyaux, PhD thesis, Université Aix-Marseille 1, 2008.
33. P.J. Bickel and Y. Ritov, Estimating linear functionals of a PET image, IEEE J. Med. Imaging 14 (1995), pp. 81-87.
34. B. Scholkopf and A.J. Smola, Learning with kernels: Support vector machines, regularization, optimization, and beyond, MIT Press, Cambridge, USA, 2001.
35. R.C. Williamson, A.J. Smola, and B. Scholkopf, Generalization performance of regularization networks and support vector machines via entropy numbers of compact operators, IEEE Trans. Inform. Theory 47 (2001), pp. 2516-2532.