On nonparametric maximum likelihood for a class of stochastic inverse problems

We establish the consistency of a nonparametric maximum likelihood estimator for a class of stochastic inverse problems. We proceed by embedding the framework into the general settings of early results of Pfanzagl related to mixtures [Pfanzagl, J., 1998a. Consistency of maximum likelihood estimators for certain nonparametric families, in particular: mixtures. J. Statist. Plann. Inference 19(2), 137-158, MR 89g:62063; Pfanzagl, J., 1998b. Large deviation inequality for maximum likelihood estimators for certain nonparametric families, in particular: mixtures. Ann. Stats. 19(2), 137-158, MR 89g:62063].

[1]  Marie Davidian,et al.  Nonlinear models for repeated measurement data: An overview and update , 2003 .

[2]  C. J. Stone,et al.  Large-Sample Inference for Log-Spline Models , 1990 .

[3]  Lancelot F. James,et al.  Consistent estimation of mixture complexity , 2001 .

[4]  B. Lindsay.  Mixture models : theory, geometry, and applications , 1995 .

[5]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[6]  B. Lindsay.  The Geometry of Mixture Likelihoods: A General Theory , 1983 .

[7]  A. Schumitzky.  Nonparametric EM Algorithms for estimating prior distributions , 1991 .

[8]  A. Mallet,et al.  Modeling INR data to predict maintenance fluindione dosage. , 1998, Therapeutic drug monitoring.

[9]  D. Böhning.  A review of reliable maximum likelihood algorithms for semiparametric mixture models , 1995 .

[10]  É. Moulines,et al.  Convergence of a stochastic approximation version of the EM algorithm , 1999 .

[11]  F. O’Sullivan.  A Statistical Perspective on Ill-posed Inverse Problems , 1986 .

[12]  Tze Leung Lai,et al.  Nonparametric estimation in nonlinear mixed effects models , 2003 .

[13]  荒井保明.  Pharmacokinetics , 1993 .

[14]  A. V. D. Vaart,et al.  Posterior convergence rates of Dirichlet mixtures at smooth densities , 2007, 0708.1885.

[15]  France Mentré,et al.  Nonparametric maximum likelihood estimation for population pharmacokinetics, with application to cyclosporine , 1988, Journal of Pharmacokinetics and Biopharmaceutics.

[16]  June.  The support reduction algorithm for computing nonparametric function estimates in mixture models June 25 , 2003 , 2022 .

[17]  Dankmar Böhning,et al.  Computer-Assisted Analysis of Mixtures and Applications: Meta-Analysis, Disease Mapping, and Others , 1999 .

[18]  Jean-Michel Loubes,et al.  Estimation of the distribution of random shifts deformation , 2008, 0812.3253.

[19]  Bruce G. Lindsay,et al.  A review of semiparametric mixture models , 1995 .

[20]  W. W. Daniel.  Applied Nonparametric Statistics , 1979 .

[21]  Dankmar Böhning,et al.  Computer-Assisted Analysis of Mixtures and Applications , 2000, Technometrics.

[22]  P. Eggermont,et al.  Maximum penalized likelihood estimation , 2001 .

[23]  J. Kiefer,et al.  CONSISTENCY OF THE MAXIMUM LIKELIHOOD ESTIMATOR IN THE PRESENCE OF INFINITELY MANY INCIDENTAL PARAMETERS , 1956 .

[24]  J. Pfanzagl,et al.  Consistency of maximum likelihood estimators for certain nonparametric families, in particular: mixtures , 1988 .

[25]  C. Priebe,et al.  Alternating kernel and mixture density estimates , 2000 .

[26]  Sara van de Geer,et al.  Rates of convergence for the maximum likelihood estimator in mixture models , 1996 .

[27]  Alain Mallet,et al.  Optimal design in random-effects regression models , 1997 .

[28]  B. Lindsay.  The Geometry of Mixture Likelihoods, Part II: The Exponential Family , 1983 .

[29]  A. Tsybakov,et al.  Oracle inequalities for inverse problems , 2002 .

[30]  E. Kuhn,et al.  Coupling a stochastic approximation version of EM with an MCMC procedure , 2004 .

[31]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[32]  F Mentré,et al.  Handling covariates in population pharmacokinetics. , 1994, International journal of bio-medical computing.

[33]  Djalil CHAFAÏ,et al.  On the strong consistency of approximated M-estimators , 2005 .