Asymptotic bias reduction of maximum likelihood estimates via penalized likelihoods with differential geometry

A procedure for asymptotic bias reduction of maximum likelihood estimates of generic estimands was developed. The estimator is realized as a plug-in estimator, where the parameter maximizes the penalized likelihood with a penalty function that satisfies a quasi-linear partial differential equation of the first order. The integration of the partial differential equation with the aid of differential geometry is discussed. Applications to generalized linear models, linear mixed-effects models, and a location-scale family are presented.

[1]  Partha Lahiri,et al.  Multi-Goal Prior Selection: A Way to Reconcile Bayesian and Classical Approaches for Random Effects Models , 2019, Journal of the American Statistical Association.

[2]  P. Lahiri,et al.  Estimating variance of random effects to solve multiple problems simultaneously , 2018, The Annals of Statistics.

[3]  Ioannis Kosmidis,et al.  Bias in parametric estimation: reduction and useful side‐effects , 2013, 1311.6311.

[4]  S. Eguchi,et al.  Asymptotical improvement of maximum likelihood estimators on Kullback–Leibler loss , 2008 .

[5]  A. Dawid.  Invariant Prior Distributions , 2006 .

[6]  Jiming Jiang,et al.  Mean squared error of empirical predictor , 2004, math/0406455.

[7]  John M. Lee.  Introduction to Smooth Manifolds , 2002 .

[8]  Eric R. Ziegel,et al.  Generalized Linear Models , 2002, Technometrics.

[9]  森田 茂之,et al.  Geometry of differential forms , 2001 .

[10]  M. C. Chaki.  ON STATISTICAL MANIFOLDS , 2000 .

[11]  Shun-ichi Amari,et al.  Methods of information geometry , 2000 .

[12]  J. Hartigan.  The maximum likelihood prior , 1998 .

[13]  F. Komaki.  On asymptotic properties of predictive distributions , 1996 .

[14]  D. Firth.  Bias reduction of maximum likelihood estimates , 1993 .

[15]  J. Copas.  Binary Regression Models for Contaminated Data , 1988 .

[16]  P. McCullagh.  Tensor Methods in Statistics , 1987 .

[17]  Mogens Flensted‐Jensen,et al.  GROUPS AND GEOMETRIC ANALYSIS Integral Geometry, Invariant Differential Operators, and Spherical Functions (Pure and Applied Mathematics: A Series of Monographs and Textbooks) , 1985 .

[18]  Su Buqing,et al.  Affine differential geometry , 1983 .

[19]  B. Efron,et al.  Empirical Bayes on vector observations: An extension of Stein's method , 1972 .

[20]  D. Cox,et al.  A General Definition of Residuals , 1968 .

[21]  K. Nomizu,et al.  Foundations of Differential Geometry, Volume I. , 1965 .

[22]  M. Riesz.  L'intégrale de Riemann-Liouville et le problème de Cauchy , 1949 .