Estimation in functional regression for general exponential families

TL;DR

This paper develops optimal estimation methods for infinite-dimensional slope functions in exponential family models, establishing minimax convergence rates and constructing estimators that achieve these rates.

Contribution

It introduces a constrained maximum likelihood approach with increasing parameter dimension and uses a change-of-measure technique to handle nonlinearity bias.

Findings

Established minimax convergence rates for slope function estimation.

Constructed estimators that attain optimal rates.

Applied change-of-measure to address nonlinearity bias.

Abstract

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.