Nonlinearly Smoothed EM Density Estimation with Automated Smoothing Parameter Selection for Nonparametric Deconvolution Problems

References

• Adams , R. A. 1975 . Sobolev Spaces , New York : Academic Press .
   Google Scholar
• Anselone , P. M. and Sloan , I. H. 1985 . “Integral Equations on the Half Line,” . Journal of Integral Equations , 9 : 3 – 23 .
   Web of Science ®Google Scholar
• Carroll , R. J. and Hall , P. 1988 . “Optimal Rates of Convergence for Deconvolving a Density,” . Journal of the American Statistical Association , 83 : 1184 – 1186 .
   Web of Science ®Google Scholar
• Cox , D. D. 1988 . “Approximation of the Method of Regularization Estimators,” . Annals of Statistics , 16 : 694 – 712 .
   Web of Science ®Google Scholar
• Cox , D. D. and O'Sullivan , F. 1990 . “Asymptotic Analysis of Penalized Likelihood and Related Estimators,” . Annals of Statistics , 18 : 1676 – 1695 .
   Web of Science ®Google Scholar
• Devroye , L. and Györfi , L. 1985 . Nonparametric Density Estimation , New York : Wiley and Sons .
   Google Scholar
• Diggle , P. J. and Hall , P. 1993 . “A Fourier Approach to Nonparametric Deconvolution of a Density Estimate,” . Journal of the Royal Statistical Society , 55 : 1184 – 1186 . Ser. B
   Google Scholar
• de Montricher , G. M. , Tapia , R. A. and Thompson , J. R. 1975 . “Nonparametric Maximum Likelihood Estimation of Probability Densities by Penalty Function Methods,” . Annals of Statistics , 3 : 1329 – 1348 .
   Web of Science ®Google Scholar
• Eggermont , P. P. B. 1992 . “Nonlinear Smoothing and the EM Algorithm for Positive Integral Equations of the First Kind,” . Applied Mathematics and Optimization , (to appear)
   Google Scholar
• Eggermont , P. P. B. and LaRiccia , V. N. 1995 . “Maximum Smoothed Likelihood Density Estimation for Inverse Problems,” . Annals of Statistics , 23 : 199 – 220 .
   Web of Science ®Google Scholar
• Eggermont , P. P. B. and LaRiccia , V. N. 1996a . Almost Sure Convergence Rates for Entropy Estimation , University of Delaware . Manuscript
   Google Scholar
• Eggermont , P. P. B. and LaRiccia , V. N. 1996b . “A Simple and Effective Bandwidth Selector for Kernel Density Estimation,” . Scandinavian Journal of Statistics , 23 : 285 – 301 .
   Web of Science ®Google Scholar
• Fan , J. 1991a . “Global Behavior of Deconvolution Kernel Estimates,” . Statistica Sinica , 1 : 541 – 555 .
   Web of Science ®Google Scholar
• Fan , J. 1991b . “On the Optimal Rates of Convergence for Nonparametric Deconvolution Problems,” . Annals of Statistics , 19 : 1257 – 1272 .
   Web of Science ®Google Scholar
• Good , I. J. and Gaskins , R. A. 1971 . “Nonparametric Roughness Penalties for Probability Densities,” . Biometrika , 58 : 255 – 257 .
   Web of Science ®Google Scholar
• Groeneboom , P. and Wellner , J. A. 1992 . Information Bounds and Nonparametric Maximum Likelihood Estimation , Boston : Birkhäuser Verlag .
   Google Scholar
• Hardy , G. H. , Littlewood , J. E. and Polya , G. 1952 . Inequalities , Cambridge : Cambridge University Press .
   Google Scholar
• Kemperman , J. H. B. 1967 . “On the Optimum Rate of Transmitting Information,” . Springer Lecture Notes in Mathematics , 23 : 126 – 169 .
   Google Scholar
• Mendelsohn , J. and Rice , J. 1982 . “Deconvolution of Microfluorometric Histograms with B-Splines,” . Journal of the American Statistical Association , 77 : 748 – 753 .
   Web of Science ®Google Scholar
• Morozov , A. 1966 . “On the Solution of Functional Equations by the Method of Regularization,” . Soviet Mathematics Doklady , 7 : 414 – 417 .
   Google Scholar
• Parzen , E. 1979 . “Nonparametric Statistical Data Modeling” (with discussion) . Journal of the American Statistical Association , 74 : 105 – 131 .
   Web of Science ®Google Scholar
• Shepp , L. A. and Vardi , Y. 1982 . “Maximum Likelihood Reconstruction in Emission Tomography,” . IEEE Transactions on Medical Imaging , MI-1 : 113 – 122 .
   Google Scholar
• Shorack , G. R. and Wellner , J. A. 1986 . Empirical Processes With Applications to Statistics , New York : Wiley and Sons .
   Google Scholar
• Silverman , B. W. 1982 . “On the Estimation of a Probability Density Function by the Maximum Penalized Likelihood Method,” . Annals of Statistics , 10 : 795 – 810 .
   Web of Science ®Google Scholar
• Silverman , B. W. , Jones , M. C. , Wilson , J. D. and Nychka , D. W. 1990 . “A Smoothed EM Approach to Indirect Estimation Problems, with Particular Reference to Stereology and Emission Tomography,” . Journal of the Royal Statistical Society , 52 : 271 – 324 . Ser. B
   Google Scholar
• Stefanski , L. A. 1990 . “Rates of Convergence of Some Estimators in a Class of Deconvolution Problems,” . Statistics and Probability Letters , 9 : 229 – 235 .
   Web of Science ®Google Scholar
• Stefanski , L. A. and Carroll , R. J. 1990 . “Deconvoluting Kernel Density Estimators,” . Statistica , 21 : 169 – 184 .
   Google Scholar
• Taylor , R. L. and Zhang , H. M. 1990 . “On a Strongly Consistent Non-parametric Density Estimator for the Deconvolution Problem,” . Communications in Statistics, A. Theory and Methods , 19 : 3325 – 3342 .
   Web of Science ®Google Scholar
• Thompson , J. R. and Tapia , R. A. 1990 . Nonparametric Function Estimation, Modelling, and Simulation , Philadelphia , PA : SIAM .
   Google Scholar
• van Es , A. J. 1991 . Uniform Deconvolution: Nonparametric Maximum Likelihood and Inverse Problems, Nonparametric functional estimation and related topics , Edited by: Roussas , G. 191 – 198 . Dordrecht : Kluwer Academic Publishing .
   Google Scholar
• Vardi , Y. and Lee , D. 1993 . “From Image Deblurring to Optimal Investments: Maximum Likelihood Solutions for Positive Linear Inverse Problems,” . Journal of the Royal Statistical Society , 55 : 569 – 612 . Ser. B
   Google Scholar
• Vardi , Y. , Shepp , L. A. and Kaufman , L. 1985 . “A Statistical Model for Positron Emission Tomography,” . Journal of the American Statistical Association , 80 : 8 – 38 .
   Web of Science ®Google Scholar
• Wilson , J. D. 1989 . “A Smoothed EM Algorithm for the Solution of Wicksell's Corpuscle Problem,” . Journal of Statistical and Computational Simulation , 31 : 195 – 221 .
   Google Scholar