Statistics and Its Interface

Volume 15 (2022)

Number 2

Estimation of Hilbertian varying coefficient models

Pages: 129 – 149



Hyerim Hong (Seoul National University)

Dongwoo Kim (Seoul National University)

Young Kyung Lee (Kangwon National University)

Byeong U. Park (Seoul National University)


In this paper we discuss the estimation of a fairly general type of varying coefficient model. The model is for a response variable that takes values in a general Hilbert space and allows for various types of additive interaction terms in representing the effects of predictors. It also accommodates both continuous and discrete predictors. We develop a powerful technique of estimating the very general model. Our approach may be used in a variety of situations where one needs to analyze the relation between a set of predictors and a Hilbertian response. We prove the existence of the estimators of the model itself and of its components, and also the convergence of a backfitting algorithm that realizes the estimators. We derive the rates of convergence of the estimators and their asymptotic distributions. We also demonstrate via simulation study that our approach works efficiently, and illustrate its usefulness through a real data application.


Hilbertian response, varying coefficient model, additive regression, smooth backfitting, compact operator

Research of Young Kyung Lee was supported by a research grant of Kangwon National University in 2020. Research of Byeong U. Park was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2019R1A2C3007355).

Received 4 March 2020

Accepted 18 October 2020

Published 11 January 2022