Wavelet penalized likelihood estimation in generalized functional models

TL;DR

This paper introduces a wavelet penalized likelihood approach for generalized functional regression, enabling flexible estimation of covariate effects with proven asymptotic optimality and adaptive properties.

Contribution

It develops a novel penalized likelihood method using wavelet estimators for semiparametric generalized functional models, with theoretical guarantees and an efficient algorithm.

Findings

Achieves quasi-minimax optimal estimation rates.

Demonstrates adaptive estimation with LASSO penalty.

Shows competitive finite sample performance in simulations.

Abstract

The paper deals with generalized functional regression. The aim is to estimate the influence of covariates on observations, drawn from an exponential distribution. The link considered has a semiparametric expression: if we are interested in a functional influence of some covariates, we authorize others to be modeled linearly. We thus consider a generalized partially linear regression model with unknown regression coefficients and an unknown nonparametric function. We present a maximum penalized likelihood procedure to estimate the components of the model introducing penalty based wavelet estimators. Asymptotic rates of the estimates of both the parametric and the nonparametric part of the model are given and quasi-minimax optimality is obtained under usual conditions in literature. We establish in particular that the LASSO penalty leads to an adaptive estimation with respect to the regularity of the estimated function. An algorithm based on backfitting and Fisher-scoring is also proposed for implementation. Simulations are used to illustrate the finite sample behaviour, including a comparison with kernel and splines based methods.