Semiparametric function-on-function quantile regression model with dynamic single-index interactions

Abstract

In this paper we propose a new semiparametric function-on-function quantile regression model with time-dynamic single-index interactions. Our model is very flexible in taking into account of the nonlinear time-dynamic interaction effects of the multivariate longitudinal/functional covariates on the longitudinal response, that most existing quantile regression models for longitudinal data are special cases of our proposed model. We propose to approximate the bivariate nonparametric coefficient functions by tensor product B-splines, and employ a check loss minimization approach to estimate the bivariate coefficient functions and the index parameter vector. Under some mild conditions, we establish the asymptotic normality of the estimated single-index coefficients using projection orthogonalization technique, and obtain the convergence rates of the estimated bivariate coefficient functions. Furthermore, we propose a score test to examine whether there exist interaction effects between the covariates. The finite sample performance of the proposed method is illustrated by Monte Carlo simulations and an empirical data analysis.

Introduction

The single-index varying coefficient model (SVCM) has attracted much attention since it was proposed by Xia and Li (1999). The model is given by

$$Y=\sum _{k=1}^q{g}_k\left({\mathbf{X}}^{\top }{\mathit{\beta }}_0\right)Z_k+\epsilon ,$$

where Y is the response, ${({\mathbf{X}}^{\top },{\mathbf{Z}}^{\top })}^{\top }$ is a vector of covariates consisting of a p-dimensional vector $\mathbf{X}={(X_1,\dots ,X_p)}^{\top }$ and a q-dimensional vector $\mathbf{Z}={(Z_1,\dots ,Z_q)}^{\top }$ with $Z_1=1$. As a semiparametric regression model, SVCM is a popular way to accommodate multivariate covariates while retaining model flexibility. There has been extensive literature on statistical inference of SVCM. For example, Fan et al. (2003) proposed an efficient backfitting algorithm to estimate SVCM. Xue and Wang (2012) developed statistical inference for SVCM using empirical likelihood method. Ma and Song (2015) studied a more general single-index varying coefficient model in which ${\mathit{\beta }}_0$ can be different for different function $g_k$. To assess how multiple environmental factors act jointly to modify individual genetic risk on complex disease, Liu et al. (2016) considered a partial linear varying multi-index coefficient model which includes SVCM as a special case. Zhao et al. (2017b) and Lv and Li (2022) studied quantile regression for SVCM.

The aforementioned works studied SVCM in the context of independent observations. As far as we know, there is limited literature on SVCM for longitudinal/functional data. However, with modern technology related to data collection and storage, functional data have become increasingly available in many scientific fields, such as meteorology, chemistry, economics and epidemiology, and thus have gained considerable attention in the literature in recent years; see Wang et al. (2016), Zhu et al. (2019) and Li et al. (2021). Recently, to capture the dynamic interaction effects between population aging and socio-economic variables on COVID-19 mortality rate, Liu et al. (2021) extended SVCM to the longitudinal data framework, and proposed a novel semiparametric single-index varying coefficient mean regression model with longitudinal response data:

$$Y(t)=\sum _{k=1}^q{g}_k({\mathbf{X}}^{\top }{\mathit{\beta }}_0,t)Z_k+\epsilon (t),$$

where $g_k(\cdot ,\cdot )$ are unknown bivariate coefficient functions and the random error $\epsilon (\cdot )$ is assumed to be a stochastic process with mean zero. Model (1) is a natural extension of SVCM by allowing SVCM to hold for any given time t through time-dynamic bivariate coefficient functions, and covers many existing semiparametric models; see Jiang and Wang (2011), Zhu et al. (2012), and Li et al. (2017) for details on the relationship between model (1) and existing models.

Mean regression only provides limited information about the conditional distribution of the response given covariates, and is well known to be very sensitive to heavy-tailed error distributions or outliers in response measurements. On the contrary, quantile regression is more robust to outliers and heavy-tailed distributions. More importantly, quantile regression can produce a more complete description of the conditional response distribution, and uncover different structural relationships between the response and covariates at the upper or lower tails, which is often of significant interest in a variety of fields (Chen and Müller, 2012; Schaumburg, 2012; Zhang, 2018; Zhu et al., 2022). Hence, in this paper, we focus on the following quantile regression model:

$$Y(t)=\sum _{k=1}^q{g}_{\tau k}({\mathbf{X}}^{\top }(t){\mathit{\beta }}_{\tau 0},t)Z_k(t)+e_{\tau }(t),$$

where $\mathbf{X}(t)={(X_1(t),\dots ,X_p(t))}^{\top }$, $\mathbf{Z}(t)={(Z_1(t),\dots ,Z_q(t))}^{\top }$ with $Z_1(t)\equiv 1$, and $\mathbb{P}(e_{\tau }(t)\le 0|\mathbf{X}(t),\mathbf{Z}(t),t)=\tau$, for $\tau \in (0,1)$. We call (2) semiparametric function-on-function quantile regression model with dynamic single-index interactions. Model (2) includes many existing quantile regression models for longitudinal data as special cases; see Wang et al. (2009), Leng and Zhang (2014), Zhao et al. (2017a), and Lin et al. (2020).

To estimate the index parameter vector ${\mathit{\beta }}_{\tau 0}$ as well as the bivariate coefficient functions $g_{\tau k}$ in model (2), we first use tensor product B-splines to approximate the bivariate coefficient functions, and then employ a check loss minimization approach to estimate them. Theoretically, our study is complicated due to the challenge of dealing with the time-dynamic single-index semiparametric structure and the nonsmooth loss function. Under some mild conditions, we establish the asymptotic normality of the estimated single-index coefficients, and obtain the convergence rates of the estimated bivariate coefficient functions. In addition, we propose a score test to examine whether there exist interaction effects between the covariates.

The main contributions of our work or the key differences between our paper and Liu et al. (2021) are three-fold. First, we extend model (1) to quantile regression framework. Quantile regression is a valuable alternative to mean regression for analyzing longitudinal data. Compared to mean regression, quantile regression for model (1) is more technically challenging and has not yet been considered in the literature. Second, we consider the sparse and unbalanced longitudinal data case. Liu et al. (2021) limited their discussion to the dense and balanced longitudinal response data case. However, longitudinal data are often highly unbalanced because data were collected at irregular and possibly subject-specific time points. Third, we study a function-on-function model in which the response and covariates are all longitudinal/functional data. Liu et al. (2021) focused on a function-on-scalar model in which the covariates are scalar variables.

The rest of this paper is organized as follows. In Section 2, we introduce the estimation procedure, and establish the asymptotic properties of the resulting estimators. We propose a score test for testing the bivariate coefficient functions in Section 3. Section 4 gives the estimation algorithm and the tuning parameter selection method. Monte Carlo studies and analysis of a real dataset are presented in Sections 5 and 6, respectively. Some concluding remarks are given in Section 7. All proofs are deferred to Appendix A.