Estimation in Functional Regression for General Exponential Families

TL;DR

This paper investigates the estimation of infinite-dimensional slope functions in exponential family models, establishing optimal convergence rates and constructing estimators that achieve these rates using advanced statistical techniques.

Contribution

It introduces a new approach to estimate slope functions in exponential family models with optimal convergence rates, employing a change-of-measure technique to handle non-linearity bias.

Findings

Established minimax convergence rates for slope estimation.

Constructed estimators that attain these optimal rates.

Applied Le Cam's theory to address non-linearity 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 non-linearity of exponential family models.