ESTIMATION IN FUNCTIONAL REGRESSION FOR GENERAL EXPONENTIAL FAMILIES

This paper studies a class of exponential family models whose canonical parameters are specified as linear functionals of an unknown infinite-dimensional slope function. The optimal minimax rates of convergence for slope function estimation are established. The estimators that achieve the optimal ra...

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Bibliographic Details
Published in:The Annals of statistics 2012-10, Vol.40 (5), p.2421-2451
Main Authors: Dou, Winston Wei, Pollard, David, Zhou, Harrison H.
Format: Article
Language:English
Subjects:
60K35
62G20
62J05
Approximation
Approximation of compact operators
Assouad’s lemma
Covariance
Eigenvalues
Estimators
exponential families
functional estimation
Generalized linear model
Linear regression
Mathematical functions
Maximum likelihood estimation
Minimax
minimax rates of convergence
Parameter estimation
Parametric models
Regression analysis
Studies
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Summary:This paper studies a class of exponential family models whose canonical parameters are specified as linear functionals of an unknown infinite-dimensional slope function. The optimal minimax rates of convergence for slope function estimation are established. The estimators that achieve the optimal rates are constructed by constrained maximum likelihood estimation with parameters whose dimension grows with sample size. A change-of-measure argument, inspired by Le Cam's theory of asymptotic equivalence, is used to eliminate the bias caused by the nonlinearity of exponential family models.
ISSN:0090-5364
2168-8966
DOI:10.1214/12-AOS1027