Cross-Validated Functional Generalized Partially Linear Single-Functional Index Model

Abstract

In this paper, we have introduced a functional approach for approximating nonparametric functions and coefficients in the presence of multivariate and functional predictors. By utilizing the Fisher scoring algorithm and the cross-validation technique, we derived the necessary components that allow us to explain scalar responses, including the functional index, the nonlinear regression operator, the single-index component, and the systematic component. This approach effectively addresses the curse of dimensionality and can be applied to the analysis of multivariate and functional random variables in a separable Hilbert space. We employed an iterative Fisher scoring procedure with normalized B-splines to estimate the parameters, and both the theoretical and practical evaluations demonstrated its favorable performance. The results indicate that the nonparametric functions, the coefficients, and the regression operators can be estimated accurately, and our method exhibits strong predictive capabilities when applied to real or simulated data.

1. Introduction

Parametric regression models are a common tool in generalized linear models (GLMs), where the relationship between the average response and covariates is explored through a chosen link function, often with a canonical or predefined form, as noted by McCullagh [1] and Nelder [2]. However, in certain cases, this approach may not be suitable due to the lack of knowledge or the presence of a more intricate link function.

In order to address this challenge, various models have been developed, including nonparametric and semiparametric regression models. However, these models are often limited in their application due to the curse of dimensionality, which poses significant challenges when dealing with high-dimensional data.

Efforts have been made to overcome this limitation by employing two key approaches: (i) the approximation of the link functions and (ii) dimension reduction techniques. One approach that has been proposed is the generalized additive model (GAM), which was introduced by Hastie [3] and extensively discussed by Wood [4]. In the GAM, the nonparametric component is represented as a sum of univariate functions, allowing for the flexible modeling of the complex relationships. However, it is important to note that a potential limitation of the GAM is its inability to explicitly account for interactions between explanatory variables. This means that the model may not fully capture the effects of interactions, which could be important in certain contexts.

The single-index model (SIM) was introduced by Härdle et al. [5] and Hristache et al. [6] to address certain cases where dimensionality reduction and relaxation of restrictive parametric assumptions are needed. The SIM approach achieves this by transforming multiple covariates into a linear combination of the covariates. Building upon the SIM, Ait-Saïdi et al. [7] investigated the functional single-index model (FSIM), which extends the single-index model usefulness framework to incorporate functional predictors.

Longitudinal data scenarios involving discrete explanatory variables in the linear component were investigated by Liang et al. [8], and Chen et al. [9] extended the framework to develop the partially linear single-index models (PLSIMs). These models allow for the modeling of discrete variables within the single-index framework.

Partially generalized linear single-index models (PGLSIMs), introduced by Carroll et al. [10], employ kernel smoothing techniques to estimate the single-index link function. PGLSIMs offer increased flexibility and modeling capabilities. In a study by Wang et al. [11], a novel approach was proposed. This approach combines penalized spline smoothing of the quasi-likelihood with the Fisher scoring mechanism, resulting in improved theoretical robustness and suitability for PGLSIM modeling.

Overall, these models, including SIM, FSIM, PLSIM, and PGLSIM, provide valuable tools for addressing complex scenarios, accommodating functional predictors and discrete variables, and effectively handling high-dimensional data.

Several models have been developed to address the complexity of functional variables in the regression analysis. However, it is important to note that the models mentioned earlier may not fully capture the intricacies of the data when some covariates are of the functional kind. Researchers have dedicated their efforts to explore functional variables in the regression models, as evidenced by the works of Ramsay et al. [12] and Ferraty et al. [13].

Furthermore, the field has seen investigations into specific models such as semifunctional partial linear regression, as studied by Aneiros et al. [14], and partially linear modeling with multifunctional covariates, as explored by Aneiros et al. [15]. Various other works have contributed to our understanding of this topic, including studies on inference by Horváth et al. [16]; the introduction to this subject by Kokoszka et al. [17]; the spline approaches by Schumaker [18]; the functional principal component analysis (FPCA) by Cao et al. [19]; the lack-of-fit testing by Li et al. [20]; and the works by Ould-Saïd et al. [21], Laksaci et al. [22], and Ouassou et al. [23,24] on the regression analysis with functional covariates.

We can refer to specific studies concerned by different aspects of the regression models. For instance, Yu et al. [25] focus on the partially functional linear single-index regression model, while Yu et al. [26] provide a comprehensive review of the penalized spline-smoothing methodology for the partially linear single-index model (PLSIM), where the sub-regression function is assumed to be a spline function with a fixed number of knots. Regarding the FDA, Rachdi et al. [27] and Alahiane et al. [28] investigate partially linear generalized single-index models for functional data (PLGSIMF) using the B-spline expansion and quasi-likelihood function. In their studies, the functional model is assumed to be linear. On the other hand, Alahiane et al. [29] explore the high-dimensional case, where the functional model is nonlinear.

The projection pursuit regression, proposed by Friedman et al. [30], Hall [31], and Huber [32], introduces an additive model that operates on derived features rather than the original inputs. This approach aims to capture complex relationships between variables by projecting them onto a lower-dimensional space. In the context of FDA, Ferraty et al. [33] extended the projection pursuit regression to the functional framework. They achieved this by incorporating the cross-validation method in the Nadaraya–Watson estimation technique and the spline functions. This generalization allows for the extension of the FDA results to the case where both the covariates and the responses are curves.

In this paper, we present a novel model called cross-validated estimations for generalized partially linear single-functional index models (CVGPLSFIM). We therefore estimate: (i) the nonlinear regression operator comprising a functional index that is chosen by the cross-validation method; (ii) the nonlinear regression operator combining a spline approximation and the one-dimensional systematic component with unknown link; and (iii) the single-index function using an iterative algorithm based on smoothing by spline functions and the maximization of the quasi-likelihood function. We also provide the convergence rates of our different estimators of the different CVGPLSFIM parameters. In fact, our contributions include examining the performance of the single and functional index, the nonparametric regression operator, the systematic component, and the optimal direction. Through numerical calculations on simulated data and then on real data, we demonstrate that our model outperforms the models mentioned throughout this introduction.

The structure of this article is as follows. In Section 2.1, Section 2 and Section 3, we present our estimation methodology, discuss the asymptotic properties of the proposed estimators, and give an iterative algorithm that maximizes the quasi-likelihood function, allowing us to compute the estimators. Section 4 presents the results of a simulation study. Furthermore, in Section 5, we apply our methodology to a real dataset consisting of curves from the chemometrics field. The technical lemmas needed to prove Theorems 1–3 are provided in Appendix A. In order to save space, the detailed proofs of the various results have been compiled in an additional and available supplementary material.

2. Estimation Methodology

2.1. Preliminary Definitions

Let Y be a scalar response variable, Z be a functional random variable that is valued in $\mathcal{H}$ separable Hilbert space endowed with inner product ${\displaystyle \langle f,g\rangle ={\int }_{I}f\left(t\right)g\left(t\right)dt}$. Let $(X,Z)\in {\mathbb{R}}^d\times \mathcal{H}$ be the predictor vector where $X=(X_1,X_2,\dots ,X_d)$ and d are a fixed integer. For a fixed $(x,z)\in {\mathbb{R}}^d\times \mathcal{H}$, we assume that the conditional density function of the response Y given $(X,Z)=(x,z)$ belongs to the following canonical exponential family

$$f_{Y|X=x,Z=z}\left(y\right)=exp\left(y\xi (x,z)-B\left(\xi \right(x,z\left)\right)+C\left(y\right)\right),$$

(1)

where B and C are two known functions that are defined from $\mathbb{R}$ into $\mathbb{R}$, and $\xi :{\mathbb{R}}^d\times \mathcal{H}\to \mathbb{R}$ is the parameter in the generalized linear model that is linked to the dependent variable

$$\mu (x,z)=\mathrm{I}\mathrm{E}\left[Y|X=x,Z=z\right]=B^{\prime }\left(\xi (x,z)\right).$$

(2)

where $B^{\prime }$ denotes the first derivative of the function B.

In what follows, we modelize the scalar response Y as a cross-validated generalized partially linear functional single-index model (CVGPLFSIM) by

$$g\left(\mu (X,Z)\right)={\eta }_0\left({\alpha }^{\top }X\right)+R\left(\langle \beta ,Z\rangle \right)+\epsilon .$$

(3)

where $\alpha ={({\alpha }_1,\dots ,{\alpha }_d)}^{\top }\in {\mathbb{R}}^d$ is the d-dimensional single-index coefficient vector; ${\eta }_0$ is the unknown single-index link function (the systematic nonlinear component), which will be assumed to be sufficiently smooth; and R is the nonlinear regression operator to be estimated, where $x^{\top }$ denotes the transpose vector of x and $\beta \in \mathcal{H}$ is the so-called the functional index, which is such that ${\parallel \beta \parallel }^2=1$ and $\langle \beta ,Z\rangle$ is the functional single-index.

We suggest estimators for the unknown single-index vector $\alpha$, the unknown systematic component ${\eta }_0(·)$, the unknown functional index $\beta ,$ and the unknown nonlinear regression operator R. Then, we will derive their asymptotic distributions, and we also provide some illustrations of these models and their performances.

Remark 1.

• For identifiability purposes, we guess that ${\left|\right|\alpha \left|\right|}_d=1$ and the first component of α is non-negative, i.e., ${\alpha }_1>0$, where $\left|\right|·{\left|\right|}_d$ denotes the Euclidean norm on ${\mathbb{R}}^d$.
• In order to identify the function ${\eta }_0(·)$, we define its support as $[a,b]$, where $a=inf{\alpha }^{\top }X$ and $b=sup{\alpha }^{\top }X$.
• In the definition of the real canonical link function g, we will assume that the functional random variable $Z=\left\{Z\right(t),t\in [0,1\left]\right\}$ is valued in $\mathcal{H}$ such that:

  $$\mathbb{E}\left[Z\right]=0,\mathbb{E}\left(\epsilon \right|X,Z)=0\mathrm{and}var\left(\epsilon \right|X,Z)={\sigma }^2.$$
• If the conditional variance $var\left(Y\right|X=x,Z=z)={\sigma }^{2}V\left(\mu (x,z)\right)$ where $V(·)$ is an unknown positive function, then the estimation of the mean function $g\left(\mu \right)$ may be obtained by replacing the log-likelihood $f_{Y|X=x,Z=z}$ given by (1) by the quasi-likelihood $Q(u,v)$, which is given for any real numbers u and v by

  $${\displaystyle \frac{\partial Q(u,v)}{\partial u}}={\displaystyle \frac{v-u}{{\sigma }^{2}V\left(u\right)}}={\displaystyle \frac{v-u}{var\left(Y\right|X=x,Z=z)}}.$$

2.2. Methodology

Let $\left(X_i,Y_i,Z_i\right)$ for $i=1,\dots ,n$, be an independent and identically distributed (i.i.d.) n-sample of $(X,Y,Z)$ and, for each $i=1,\dots ,n$,

$$g\left(\mu \left(X_i,Z_i\right)\right)={\eta }_0\left({\alpha }^{\top }{X}_i\right)+R\left(\langle \beta ,Z_i\rangle \right)+{\epsilon }_i.$$

(4)

Let $\{B_j\left(u\right),j=1,\dots ,N_n\}$ the B-spline basis functions of order r; ${\displaystyle h=\frac{(b-a)}{J_n+1}}$ is the distance between the neighbors knots, where $J_n$ denote the number of interior knots in $[a,b]$, as in Wang et al. [11], Rachdi et al. [27], Alahiane et al. [28] and Alahiane et al. [29].

Let ${\mathcal{S}}_n$ be the space of polynomial splines on $[a,b]$ of order $r\ge 1$, $v\in {\mathbb{N}}^{*}$ and $e\in (0,1]$ such that $p=v+e>1.5$. We denote by $\mathcal{H}\left(p\right)$ the collection of functions g that are defined on $[a,b]$ whose v-th order derivative, $g^{\left(v\right)}$, exists and satisfies the following e-th order Lipschitz condition

$$\left|g^{\left(v\right)}\left(m_1\right)-g^{\left(v\right)}\left(m_2\right)\right|\le C{\left|m_1-m_2\right|}^e,\mathrm{for}\mathrm{all}a\le m_1,m_2\le b.$$

Using the method by De Boor [34], we can approximate $\eta ,$ which is assumed in $\mathcal{H}\left(p\right)$, by the function $\tilde{\eta }\in {\mathcal{S}}_n.$ So, we can write $\tilde{\eta }\left(u\right)={\tilde{\gamma }}^{\top }B\left(u\right)$, where $B\left(u\right)$ is the spline basis and $\tilde{\gamma }\in {\mathbb{R}}^{N_n}$ is the spline coefficient vector.

We introduce a new knot sequence $t_0<t_1<\dots <t_{k+1}$ in the range of R. Then, there exists $N^{\prime }=k+r+1$ functions in the B-splines basis, which are normalized and of order r, such that $g(·)\approx {\delta }^{\top }{B}_1(.)\mathrm{where}{B}_1(·)={\left(B_{11}(·),B_{12}(·),\dots ,B_{1N^{\prime }}(·)\right)}^{\top }$ and $\delta \in {\mathbb{R}}^{N^{\prime }}.$

Using the setting ${\displaystyle W_i=\langle \beta ,Z_i\rangle ,}$ the mean function estimator $\widehat{m}\left(x,z\right)$ is therefore obtained by assessing the parameter $\theta ={\left({\alpha }^{\top },{\gamma }^{\top },{\delta }^{\top }\right)}^{\top }$ and inverting the subsequent equation $g\left(\mu (X_i,Z_i)\right)={\widehat{\gamma }}^{\top }B\left({\widehat{\alpha }}^{\top }x\right)+{\widehat{\delta }}^{\top }{B}_1\left(W_i\right).$

We may notice that the parameter $\theta ={\left({\alpha }^{\top },{\gamma }^{\top },{\delta }^{\top }\right)}^{\top }$ is deduced by maximizing the following requirement of quasi-likelihood

$$\begin{array}{c}\hfill \widehat{\theta }={\left({\widehat{\alpha }}^{\top },{\widehat{\gamma }}^{\top },{\widehat{\delta }}^{\top }\right)}^{\top }=\underset{\theta =(\alpha ,\gamma ,\delta )\in {\mathbb{R}}^d\times {\mathbb{R}}^{N_n}\times {\mathbb{R}}^{N^{\prime }}}{argmax}L\left(\theta \right),\end{array}$$

(5)

where ${\displaystyle L\left(\theta \right):=L(\alpha ,\gamma ,\delta )={\displaystyle \frac{1}{n}}\sum _{i=1}^{n}Q\left(g^{-1}\left(m_i\right),Y_i\right)}$ and $m_i:={\gamma }^{\top }B\left({\alpha }^{\top }{X}_i\right)+{\delta }^{\top }{B}_1\left(W_i\right)$.

In order to overcome the constraints $\parallel \alpha \parallel =1$ and ${\alpha }_1>0$ that are imposed on the d-dimensional index $\alpha$, we adopt a reparametrization approach that is inspired by the methodology employed by Yu [26]

$$\alpha \left(\tau \right)={\left(\sqrt{1-{\parallel \tau \parallel }^2},{\tau }^{\top }\right)}^{\top }\mathrm{for}\tau \in {\mathbb{R}}^{d-1}.$$

The true value ${\tau }_0$ of $\tau$ must be $∥{\tau }_0∥\le 1$. Subsequently, we assume that $∥{\tau }_0∥<1$.

The Jacobian matrix of $\alpha \to \alpha \left(\tau \right)$ of dimension $d\times (d-1)$ is

$$J\left(\tau \right)=\left(\begin{array}{c}-{\displaystyle \frac{1}{\sqrt{1-{\parallel \tau \parallel }^2}}}{\tau }^{\top }\\ I_{(d-1)\times (d-1)}\end{array}\right)$$

Denote $l=1,2$, $q_l(m,y)={\displaystyle \frac{{\partial }^l}{\partial m^l}}Q\left(m,y\right)$ and ${\rho }_l={\displaystyle \frac{1}{{\sigma }^{2}V\left(g^{-1}\left(m\right)\right)}}{\left[{\displaystyle \frac{d}{dm}}\left(g^{-1}\left(m\right)\right)\right]}^l.$

So,

$$q_1(m,y)=\left(y-m\right){\rho }_1\left(m\right)\mathrm{and}{q}_2(m,y)=\left(y-m\right){\rho }_1^{\prime }\left(m\right)-{\rho }_2\left(m\right).$$

The score vector is

$${\displaystyle S\left({\theta }_{\tau }\right)={\displaystyle \frac{\partial l}{\partial {\theta }_{\tau }}}\left({\theta }_{\tau }\right)={\displaystyle \frac{1}{n}}\sum _{i=1}^n{q}_1\left(m_i,Y_i\right){\xi }_i(\tau ,\gamma ,\delta ),}$$

where

$${\xi }_i(\tau ,\gamma ,\delta )=\left(\begin{array}{c}{\gamma }^{\top }{B}^{\prime }\left({\alpha }^{\top }\left(\tau \right)X_i\right)J^{\top }\left(\tau \right)X_i\\ B\left({\alpha }^{\top }\left(\tau \right)X_i\right)\\ B_1\left(W_i\right)\end{array}\right).$$

The conditional expectation of the Hessian matrix given X and Z is ${\displaystyle H\left({\theta }_{\tau }\right)=-{\displaystyle \frac{1}{n}}\sum _{i=1}^n{\rho }_2\left(m_i\right){\xi }_i(\tau ,\gamma ,\delta ){\xi }_i^{\top }(\tau ,\gamma ,\delta ).}$

The Fisher scoring update equations ${\displaystyle {\theta }_{\tau }^{(k+1)}={\theta }_{\tau }^{\left(k\right)}-{\left[H\left({\theta }_{\tau }^{\left(k\right)}\right)\right]}^{-1}S\left({\theta }_{\tau }^{\left(k\right)}\right)}$ becomes

$$\begin{array}{ccc}\hfill {\theta }_{\tau }^{(k+1)}& =& {\theta }_{\tau }^{\left(k\right)}+{\left[\sum _{i=1}^n{\rho }_2\left(m_i^{\left(k\right)}\right){\xi }_i\left({\tau }^{\left(k\right)},{\gamma }^{\left(k\right)},{\delta }^{\left(k\right)}\right){\xi }_i^{\top }\left({\tau }^{\left(k\right)},{\gamma }^{\left(k\right)},{\delta }^{\left(k\right)}\right)\right]}^{-1}\hfill \\ & & \times \left[\sum _{i=1}^n\left(Y_i-{\mu }_i^{\left(k\right)}\right){\rho }_1\left(m_i^{\left(k\right)}\right){\xi }_i\left({\tau }^{\left(k\right)},{\gamma }^{\left(k\right)},{\delta }^{\left(k\right)}\right)\right],\hfill \end{array}$$

where $m_i^{\left(k\right)}={\gamma }^{\left(k\right)\top }B\left({\alpha }^{\left(k\right)\top }\left({\tau }^{\left(k\right)}\right)X_i\right)+{\delta }^{\left(k\right)\top }{B}_1\left(W_i\right)$ and ${\mu }_i^{\left(k\right)}=g^{-1}\left(m_i^{\left(k\right)}\right)$, for $1\le i\le n$.

So, we obtain

$$\begin{array}{ccc}\hfill \widehat{\eta }\left(t\right)& =& {\widehat{\gamma }}^{\top }B\left(t\right)={\gamma }^{\left(k\right)\top }B\left(t\right),\hfill \\ \hfill {\widehat{m}}_i& =& {\widehat{\gamma }}^{\top }B\left({\alpha }^{\top }\left(\widehat{\tau }\right)X_i\right)+{\widehat{\delta }}^{\top }{B}_1\left(W_i\right)={\gamma }^{\left(k\right)\top }B\left({\alpha }^{\top }\left({\tau }^k\right)\right)X_i+{\delta }^{\left(k\right)\top }{B}_1\left(W_i\right),\hfill \\ \hfill \widehat{R}\left(Z_i\right)& =& {\widehat{\delta }}^{\top }{B}_1\left(W_i\right)={\delta }^{\left(k\right)\top }{B}_1\left(W_i\right),\hfill \end{array}$$

where $\widehat{\alpha }=\alpha \left({\tau }^{\left(k\right)}\right)$ is the estimator of the single-index coefficient vector $\alpha$ and ${\widehat{\mu }}_i=g^{-1}\left({\widehat{m}}_i\right)$.

Through the utilization of this procedure, we will employ the cross-validation method (see the Section 2.3) to determine the optimal direction $\beta (·)$. Additionally, we will estimate in Section 2.4 the various components of our CVGPLSFIM model using a plug-in approach.

2.3. Cross-Validated Estimation of the Functional Index

We use the cross-validation principle based on the leave-one-out statistical sample $\{(X_j,Z_j),j=1,\dots ,n,j\ne i\}$ to estimate the functional index $\beta$ by:

$$\widehat{\beta }=\underset{\beta \in \Xi }{argmin}CV\left(\beta \right),$$

where

$${\displaystyle CV\left(\beta \right)={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(Y_i-g^{-1}\left({\widehat{m}}_i^{(-i)}\right)\right)}^2{I}_{Z_i\in \mathcal{G}},}$$

(6)

with ${\widehat{m}}_i^{(-i)}={\widehat{\gamma }}^{\top }B\left({\alpha }^{\top }\left(\widehat{\tau }\right)X_i^{(-i)}\right)+{\widehat{\delta }}^{\top }{B}_1\left(W_i^{(-i)}\right)$; $\mathcal{G}$ is a subset of $\mathcal{H}$ introduced for usual technical boundedness reasons, and $\Xi ={\Xi }_n$ with $\Xi \subset \mathcal{H}\bigcap \left\{{\beta :\parallel \beta \parallel }^2=1\right\}$ is constructed in a similar way as in Ait-Saïdi et al. [7]. Specifically:

• Each direction $\beta \in {\Xi }_n$ is obtained from an $l_n$-dimensional space generated by the B-spline basis functions $\left\{e_1,\dots ,e_{l_n}\right\}$. Therefore, we focus on directions

  $${\displaystyle \beta (.)=\sum _{k=1}^{l_n}{\lambda }_k{e}_k,\mathrm{where}\left({\lambda }_1,\dots ,{\lambda }_{l_n}\right)\in \mathcal{V}.}$$

  (7)
• The set of coefficients vectors $\mathcal{V}$ in (7) is obtained by the following procedure:

  –

  Step 1: For each $\left(b_1,\dots ,b_{l_n}\right)\in {\mathcal{C}}^{l_n}$, where $\mathcal{C}=\left\{c_1,\dots ,c_K\right\}\subset {\mathbb{R}}^K$ denotes a set of K “seed-coefficients”, we construct the initial functional direction ${\displaystyle {\beta }_{\mathtt{init}}(·)=\sum _{k=1}^{l_n}{b}_k{e}_k(·).}$

  –

  Step 2: For each ${\beta }_{\mathtt{init}}$ selected in Step 1 that verifies the condition ${\beta }_{\mathtt{init}}\left(t_0\right)>0$, we construct $\left({\lambda }_1,\dots ,{\lambda }_{l_n}\right)=\left(b_1,\dots ,b_{l_n}\right)/{\langle {\beta }_{\mathtt{init}},{\beta }_{\mathtt{init}}\rangle }^{1/2}.$

  –

  Step 3: Construct $\mathcal{V}$ as the set of vectors $(b_1,\dots ,b_{l_n})$ obtained in Step 2. Therefore, the final set of eligible functional direction is

  $${\Xi }_n=\left\{{\displaystyle \beta (·)=\sum _{k=1}^{l_n}{\lambda }_k{e}_k(·)\mathrm{such}\mathrm{that}\left({\lambda }_1,\dots ,{\lambda }_{l_n}\right)\in \mathcal{V}}\right\}.$$

Using the method by De Boor [34], if ${\beta }_0$ is sufficiently smooth, then it is well approximated by some function in the $l_n$-dimensional space, which is generated by the B-spline basis. From the construction (see Step 2), each $\beta \in {\Xi }_n$ satisfies $\langle \beta ,\beta \rangle =1$ and $\beta \left(t_0\right)>0$. So, the identifiability of the CVGPLFSIM model is guaranteed. Like in Ait-Saïdi et al. [7], we consider the cubic B-spline functions and $\mathcal{V}=\left\{-1,0,1\right\}$.

2.4. The CVGPLFSIM Model

By plug-in, first, the functional index $\widehat{\beta }$ in the model is

$$g\left(\mu \left(X_i,Z_i\right)\right)=\eta \left({\alpha }^{\top }{X}_i\right)+R\left(W_i\right)\mathrm{for}i=1,\dots ,n,$$

(8)

where $W_i=\langle \widehat{\beta },Z_i\rangle$ denotes the functional index component. We seek a function $\eta \in {\mathcal{S}}_n$ along with a value of $\alpha$ that minimizes the following quasi-likelihood function

$$L(\eta ,\alpha )={\displaystyle \frac{1}{n}}\sum _{i=1}^{n}Q\left(g^{-1}\left\{\eta \left({\alpha }^{\top }{X}_i\right)+{\delta }^{\top }{B}_1\left(W_i\right)\right\},Y_i\right).$$

(9)

By denoting $\theta =({\alpha }^{\top },{\gamma }^{\top },{\delta }^{\top })$, the maximization problem (9) is equivalent to find a value $\theta$ maximizing

$$l\left(\theta \right)={\displaystyle \frac{1}{n}}\sum _{i=1}^{n}Q\left(g^{-1}\left\{{\gamma }^{\top }B\left({\alpha }^{\top }{X}_i\right)+{\delta }^{\top }{B}_1\left(W_i\right)\right\},Y_i\right),$$

(10)

where

$$g\left(\mu \left(X_i,Z_i\right)\right)={\gamma }^{\top }B\left({\alpha }^{\top }{X}_i\right)+{\delta }^{\top }{B}_1\left(W_i\right),\mathrm{for}i=1,\dots ,n.$$

(11)

The mean function estimator $\widehat{\mu }$ is given by the evaluation of the parameters $\widehat{\theta }={\left({\widehat{\alpha }}^{\top },{\widehat{\gamma }}^{\top },{\widehat{\delta }}^{\top }\right)}^{\top }$ and inverting Equation (11). In fact, $\widehat{\theta }={\left({\widehat{\alpha }}^{\top },{\widehat{\gamma }}^{\top },{\widehat{\delta }}^{\top }\right)}^{\top }$ is determined by maximizing the following quasi-likelihood

$$\widehat{\theta }={\left({\widehat{\alpha }}^{\top },{\widehat{\gamma }}^{\top },{\widehat{\delta }}^{\top }\right)}^{\top }=\underset{\theta ={({\alpha }^{\top },{\gamma }^{\top },{\delta }^{\top })}^{\top }\in {\mathbb{R}}^d\times {\mathbb{R}}^{N_n}\times {\mathbb{R}}^{N^{\prime }}}{argmax}l\left(\theta \right),$$

(12)

where

$$l\left(\theta \right):=l(\alpha ,\gamma ,\delta )={\displaystyle \frac{1}{n}}\sum _{i=1}^{n}Q\left(g^{-1}\left\{{\gamma }^{\top }B\left({\alpha }^{\top }{X}_i\right)+{\delta }^{\top }{B}_1\left(W_i\right)\right\},Y_i\right)={\displaystyle \frac{1}{n}}\sum _{i=1}^{n}Q\left(g^{-1}\left\{m_i\right\},Y_i\right),$$

with $m_i={\gamma }^{\top }B\left(U_i\right)+{\delta }^{\top }{B}_1\left(W_i\right),\mathrm{where}{U}_i={\alpha }^{\top }{X}_i$, ${\alpha }_0,{\gamma }_0,{\delta }_0$, and ${\eta }_0(·)$ denote the true values, respectively, of $\alpha ,\gamma ,\delta$, and $\eta (·)$. So, the spline estimator of ${\eta }_0(·)$ is $\widehat{\eta }(·)={\widehat{\gamma }}^{\top }B(·)$.

Let $R\left(\tau \right)=\left(\begin{array}{cc}J\left(\tau \right)& 0\\ 0& I_{N^{\prime }\times N^{\prime }}\end{array}\right)$ be the Jacobian matrix of ${({\alpha }^{\top }\left(\tau \right),{\delta }^{\top })}^{\top }$ and

$$(\tilde{\alpha },\tilde{\delta })=\underset{{\parallel \alpha \parallel }_d=1,\delta \in {\mathbb{R}}^{N^{\prime }}}{argmax}{\displaystyle \frac{1}{n}}\sum _{i=1}^{n}Q\left(g^{-1}\left\{\tilde{\eta }\left({\alpha }^{\top }{X}_i\right)+{\delta }^{\top }{B}_1\left(W_i\right)\right\},Y_i\right).$$

(13)

Then,

$$(\tilde{\tau },\tilde{\delta })=\underset{\tau \in {\mathbb{R}}^{d-1},\delta \in {\mathbb{R}}^{N^{\prime }}}{argmax}\tilde{l}(\tau ,\delta ),$$

(14)

where ${\displaystyle \tilde{l}(\tau ,\delta )={\displaystyle \frac{1}{n}}\sum _{i=1}^{n}Q\left(g^{-1}\left\{\tilde{\eta }\left({\alpha }^{\top }\left(\tau \right)X_i\right)+{\delta }^{\top }{B}_1\left(W_i\right)\right\},Y_i\right)}$ and $\eta (.)$ was replaced by $\tilde{\eta }(.).$

We define ${\tilde{\theta }}_{\tau }={\left({\tilde{\tau }}^{\top },{\tilde{\gamma }}^{\top },{\tilde{\delta }}^{\top }\right)}^{\top }$ such that

$${\left({\tilde{\tau }}^{\top },{\tilde{\gamma }}^{\top },{\tilde{\delta }}^{\top }\right)}^{\top }=\underset{\tau \in {\mathbb{R}}^{d-1},\gamma \in {\mathbb{R}}^{N_n},\delta \in {\mathbb{R}}^{N^{\prime }}}{argmax}{\displaystyle \frac{1}{n}}\sum _{i=1}^{n}Q\left(g^{-1}\left\{{\gamma }^{\top }B\left({\alpha }^{\top }\left(\tau \right)X_i\right)+{\delta }^{\top }{B}_1\left(W_i\right)\right\},Y_i\right).$$

(15)

Then, $l\left({\theta }_{\tau }\right)$ becomes

$${\displaystyle l\left({\theta }_{\tau }\right)={\displaystyle \frac{1}{n}}\sum _{i=1}^{n}Q\left(g^{-1}\left\{{\gamma }^{\top }B\left({\alpha }^{\top }\left(\tau \right)X_i\right)+{\delta }^{\top }{B}_1\left(W_i\right)\right\},Y_i\right)={\displaystyle \frac{1}{n}}\sum _{i=1}^{n}Q\left(g^{-1}\left\{m_i\right\},Y_i\right).}$$

The score vector is given by

$$\begin{array}{c}\hfill S\left({\theta }_{\tau }\right)={\displaystyle \frac{\partial l}{\partial {\theta }_{\tau }}}\left(\theta \right){|}_{\theta ={\theta }_{\tau }}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{q}_1\left(m_i,Y_i\right){\xi }_i(\tau ,\gamma ,\delta ),\end{array}$$

(16)

where ${\xi }_i(\tau ,\gamma ,\delta )=\left(\begin{array}{c}{\gamma }^{\top }{B}^{\prime }\left({\alpha }^{\top }\left(\tau \right)X_i\right)J^{\top }\left(\tau \right)X_i\\ B\left({\alpha }^{\top }\left(\tau \right)X_i\right)\\ B_1\left(W_i\right)\end{array}\right).$

Then, the Hessian matrix of the quasi-likelihood function is

$${\displaystyle H\left({\theta }_{\tau }\right)={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\rho }_2\left(m_i\right){\xi }_i(\tau ,\gamma ,\delta ){\xi }_i^{\top }(\tau ,\gamma ,\delta ).}$$

We have ${\tilde{\theta }}_{\tau }={\left({\tilde{\tau }}^{\top },{\tilde{\gamma }}^{\top },{\tilde{\delta }}^{\top }\right)}^{\top }=\underset{{\theta }_{\tau }=(\tau ,\gamma ,\delta )\in {\mathbb{R}}^{d-1}\times {\mathbb{R}}^N\times {\mathbb{R}}^{N^{\prime }}}{argmax}l\left({\theta }_{\tau }\right).$ Then, the Fisher scoring update equations become

$$\begin{array}{ccc}\hfill {\theta }_{\tau }^{(k+1)}& =& {\theta }_{\tau }^{\left(k\right)}+{\left[H\left({\theta }_{\tau }^{\left(k\right)}\right)\right]}^{-1}S\left({\theta }_{\tau }^{\left(k\right)}\right)\hfill \\ & =& {\theta }_{\tau }^{\left(k\right)}+{\left[\sum _{i=1}^n{\rho }_2\left(m_i^{\left(k\right)}\right){\xi }_i\left({\tau }^{\left(k\right)},{\gamma }^{\left(k\right)},{\delta }^{\left(k\right)}\right){\xi }_i^{\top }\left({\tau }^{\left(k\right)},{\gamma }^{\left(k\right)},{\delta }^{\left(k\right)}\right)\right]}^{-1}\hfill \\ & & \times \left[\sum _{i=1}^n\left(Y_i-{\mu }_i^{\left(k\right)}\right){\rho }_1\left(m_i^{\left(k\right)}\right){\xi }_i\left({\tau }^{\left(k\right)},{\gamma }^{\left(k\right)},{\delta }^{\left(k\right)}\right)\right],\hfill \end{array}$$

(17)

where for $1\le i\le n,$

$$\begin{array}{ccc}\hfill m_i^{\left(k\right)}& =& {\gamma }^{\left(k\right)\top }B\left({\alpha }^{\left(k\right)\top }\left({\tau }^{\left(k\right)}\right)X_i\right)+{\delta }^{\left(k\right)\top }{B}_1\left(W_i\right),\hfill \\ \hfill {\mu }_i^{\left(k\right)}& =& g^{-1}\left(m_i^{\left(k\right)}\right),\hfill \\ \hfill \widehat{\eta }\left(t\right)& =& {\displaystyle {\widehat{\gamma }}^{\top }B\left(t\right)\approx {\gamma }^{k)\top }B\left(t\right)=\sum _{j=1}^{N_n}{\gamma }_j^{\left(k\right)}{B}_j\left(t\right),\widehat{R}(·)=\sum _{j=1}^{N^{\prime }}{\delta }_j^{\left(k\right)}{B}_{1,j}\left(.\right),}\hfill \\ \hfill {\widehat{m}}_i& =& {\displaystyle {\widehat{\gamma }}^{\top }B\left({\alpha }^{\top }\left(\widehat{\tau }\right)X_i\right)+{\delta }^{\left(k\right)\top }{B}_1\left(W_i\right)\approx \sum _{j=1}^{N_n}{\gamma }_j^{\left(k\right)}{B}_j\left({\alpha }^{\top }{\left({\tau }^k\right)}^{ }{X}_i\right)+\sum _{j=1}^{N^{\prime }}{\delta }_j^{\left(k\right)}{B}_{1,j}\left(W_i\right).}\hfill \end{array}$$

Then, ${\displaystyle {\widehat{\mu }}_i=g^{-1}\left({\widehat{m}}_i\right)}$ and ${\displaystyle \widehat{\alpha }=\alpha \left({\tau }^{\left(k\right)}\right)}$ is the estimator of the single-index coefficient vector of the CVGPLSFIM model. The statistic $\widehat{\beta }$ is the estimated functional single-index, and $\widehat{R}$ is the estimated nonlinear regression operator R obtained by the CVGPLSFIM model.

3. Main Asymptotic Properties

In this section, we establish the asymptotic properties of the estimators of (i) the nonparametric components, (ii) the parametric components, (iii) the unique index, (iv) the nonlinear regression operator, and (v) the convergence of the estimators of the univariate components. These properties are established under a set of specific assumptions.

3.1. Assumptions

Let $\phi$, ${\phi }_1$, and ${\phi }_2$ be measurable functions on $[a,b]$. We define the empirical inner product and its corresponding norm as follows:

$${\displaystyle {〈{\phi }_1,{\phi }_2〉}_n={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\phi }_1\left(U_i\right){\phi }_2\left(U_i\right)\mathrm{and}{\parallel \phi \parallel }_n^2={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\phi }^2\left(U_i\right),\mathrm{where}{U}_i={\alpha }^{\top }{X}_i.}$$

If $\phi$, ${\phi }_1$, and ${\phi }_2$ are $L^2$-integrable, we define the theoretical inner product and its corresponding norm as follows:

$$\left({\phi }_1,{\phi }_2\right.〉=\mathbb{E}\left[{\phi }_1\left(U\right){\phi }_2\left(U\right)\right]\mathrm{and}{\parallel \phi \parallel }_2^2=\mathbb{E}\left[{\phi }^2\left(U\right)\right]={\int }_a^b{\phi }^2\left(u\right)f\left(u\right)du.$$

Let $\epsilon =Y-g^{-1}\left(m_0\left(T\right)\right)$, where $T={\left(X^{\top },{\overline{W}}^{\top }\right)}^{\top }\mathrm{and}\overline{W}=B_1\left(W\right)$. We assume that

(C1)

The single-index link function ${\eta }_0(·)\in \mathcal{H}\left(p\right),$ where $\mathcal{H}\left(p\right)$ is defined as above.

(C2)

For all $m\in \mathbb{R}$ and for all y in the range of the response variable Y, we have for $k=1,2$ that

$$q_2(m,y)<0\mathrm{and}{c}_q<\left|q_2^k(m,y)\right|<C_q,$$

for some positive constants $c_q$ and $C_q.$

(C3)

The $\nu$-th order partial derivative of the joint density function of X satisfies the Lipschitz condition of order $\kappa (\kappa \in (0,1\left]\right).$

The marginal density function of ${\alpha }^{\top }X$ is continuous and bounded away from zero and supported within $[a,b]$.

(C4)

For any vector $\tau$, there exists positive constants $c_{\tau }$ and $C_{\tau }$ such that

$$c_{\tau }{I}_{t\times t}\le \mathbb{E}\left[\left(\begin{array}{c}1\\ T\end{array}\right){\left(\begin{array}{c}1\\ T\end{array}\right)}^{\top }|{\alpha }^{\top }\left(\tau \right)X={\alpha }^{\top }\left(\tau \right)x\right]\le C_{\tau }{I}_{t\times t},$$

where $t=1+d+N_n+N^{\prime }$ and $T={(X^{\top },{\overline{W}}^{\top })}^{\top }.$

(C5)

The number $N_n$ of knots satisfies $n^{{\displaystyle \frac{1}{2(p+1)}}}\ll N_n\ll n^{{\displaystyle \frac{1}{8}}}(p>3).$

(C6)

The fourth-order moment of the random variable Z is finite, i.e., ${\mathbb{E}\parallel Z(.)\parallel }^4\le C$, where C denotes a generic positive constant.

(C7)

The covariance function $K(t,s)=Cov\left(Z\right(t),Z(s\left)\right)$ is positive definite.

(C8)

For some finite positive constants $C_{\rho }$, $C_{\rho }^{*}$, and $M_0$,

$$\left|{\rho }_1\left(m_0\right)\right|\le C_{\rho }\mathrm{and}\left|{\rho }_1\left(m\right)-{\rho }_1\left(m_0\right)\right|\le C_{\rho }^{*}\left|m-m_0\right|\mathrm{for}\mathrm{all}\left|m-m_0\right|\le M_0.$$

(C9)

For some finite positive constants $C_g$, $C_g^{*}$, and $M_1$, the link function g in the model (3) satisfies:

${\displaystyle \left|\frac{d}{dm}g\left(m\right){|}_{m=m_0}\right|\le C_g}$ and, for all $\left|m-m_0\right|\le M_1$,

$$\left|{\displaystyle \frac{d}{dm}}{g}^{-1}\left(m\right)-{\displaystyle \frac{d}{dm}}{g}^{-1}\left(m\right){|}_{m=m_0}\right|\le C_g^{*}\left|m-m_0\right|.$$

(C10)

There exists a positive constant $C_0$ such that $\mathbb{E}\left({\epsilon }^2\right|U_{\tau ,0})\le C_0.$

(C11)

We assume that all the random variables $\langle \beta ,Z\rangle$ for all $\beta \in \mathcal{H}$ have values on a set $\mathcal{C}$, where $\mathcal{C}$ is a compact subset of $\mathbb{R}.$

(C12)

The nonlinear regression operator $R(·)\in \mathcal{H}\left(p\right).$

Comments on the Assumptions: The smoothness condition in (C1) describes that the single-index function ${\eta }_0(·)$ can be approximated by functions in the B-spline space with a normalized basis. On the other hand, condition (C2) ensures the uniqueness of the solution, where condition (C3) is a smoothness condition on the joint and marginal density functions of ${\alpha }^{\top }X$ and X. Condition (C5) allows us to obtain the rate of growth of the dimension of the spline spaces relative to the sample size. Conditions (C6) and (C7) are required for the covariate function Z, whereas conditions (C4) and (C8)–(C10) are technical hypotheses that will be needed for the results’s proofs. Conditions (C11) and (C12) are smoothness conditions of the nonlinear operator regression R.

3.2. The Consistency Study

3.2.1. Technical Lemmas

In this subsection, we present the needed lemmas to prove Theorems 2 and 3, for which the proofs take into account the behaviors of all the components involved in the model (3).

Lemma 1.

Under assumptions (C1)–(C4) and (C6)–(C8), we have

$$\sqrt{n}\left(\begin{array}{c}\tilde{\tau }-{\tau }_0\\ \tilde{\delta }-{\delta }_0\end{array}\right)\stackrel{D}{⟶}\mathcal{N}\left(0,A^{-1}{\Sigma }_1{A}^{-1}\right),$$

where ${\Sigma }_1$ and A will be defined below in the Appendix for more details; the symbol $\stackrel{D}{⟶}$ denotes the convergence in distribution and $\overline{W}=B_1\left(W\right)$, $A=\left(\begin{array}{cc}{A}_{11}& A_{12}\\ A_{12}^{\top }& A_{22}\end{array}\right)$ with

$$\begin{array}{ccc}\hfill A_{11}& =& \mathrm{I}\mathrm{E}\left[{\rho }_2\left(m_0\left(T\right)\right){\left\{{\eta }_0^{\prime }\left(U_{\tau ,0}\right)\right\}}^2{J}^{\top }\left({\tau }_0\right)X{\overline{W}}^{\top }J\left({\tau }_0\right)\right],\hfill \\ \hfill A_{22}& =& \mathrm{I}\mathrm{E}\left[{\rho }_2\left(m_0\left(T\right)\right)\overline{W}{\overline{W}}^{\top }\right],\hfill \\ \hfill A_{12}& =& \mathrm{I}\mathrm{E}\left[q_1^2(m_0\left(T\right),Y).\left(\begin{array}{c}{\eta }_0^{\prime }\left(U_{\tau ,0}\right)J^{\top }\left({\tau }_0\right)X\\ \overline{W}\end{array}\right){\left(\begin{array}{c}{\eta }_0^{\prime }\left(U_{\tau ,0}\right)J^{\top }\left({\tau }_0\right)X\\ \overline{W}\end{array}\right)}^{\top }\right]\hfill \end{array}$$

By applying the $\delta$-method, we obtain the following lemma.

Lemma 2.

Under assumptions (C1)–(C4) and (C6)–(C8), we have

$$\sqrt{n}\left(\begin{array}{c}\alpha \left(\tilde{\tau }\right)-\alpha \left({\tau }_0\right)\\ \tilde{\delta }-{\delta }_0\end{array}\right)\stackrel{D}{⟶}\mathcal{N}\left(0,R\left({\tau }_0\right)A^{-1}{\Sigma }_1{A}^{-1}{R}^{\top }\left({\tau }_0\right)\right),$$

where

$$R\left(\tau \right)=\left(\begin{array}{cc}J\left(\tau \right)& 0\\ 0& I_{N^{\prime }\times N^{\prime }}\end{array}\right),$$

and

$${\Sigma }_1=\mathrm{I}\mathrm{E}\left[q_1^2\left(m_0\left(T\right)\right)\left(\begin{array}{c}{\eta }_0^{\prime }\left(U_{\tau ,0}\right)J^{\top }\left({\tau }_0\right)X\\ \overline{W}\end{array}\right){\left(\begin{array}{c}{\eta }_0^{\prime }\left(U_{\tau ,0}\right)J^{\top }\left({\tau }_0\right)X\\ \overline{W}\end{array}\right)}^{\top }\right].$$

Furthermore, $\alpha \left(\tilde{\tau }\right)-\alpha \left({\tau }_0\right)=o_{\mathbb{P}}\left({\displaystyle \frac{1}{\sqrt{n}}}\right)$ and $\tilde{\delta }-{\delta }_0=o_{\mathbb{P}}\left({\displaystyle \frac{1}{\sqrt{n}}}\right)$.

Lemma 3.

Under assumptions (C1)–(C5), we have

$$\parallel \widehat{\theta }-\tilde{\theta }\parallel =O_{\mathbb{P}}\left\{\sqrt{N_n}\left(h^p+{\displaystyle \frac{1}{\sqrt{nh}}}\right)\right\}.$$

(18)

where $N_n$ is the number of B-splines basis functions of order r.

The proofs of the previous results are supported by the following lemmas.

3.2.2. Convergence of the Estimated Univariate Components

So, for the nonlinear regression operator R behavior, we have the following theorem:

Theorem 1.

Under assumptions (C1)–(C8) and (C11)–(C12), we have   

$$\parallel \widehat{R}{-R\parallel }_2=O_{\mathbb{P}}\left\{\sqrt{N_n}\left({\displaystyle \frac{1}{\sqrt{n}h}}+h^p\right)\right\}.$$

The proof of Theorem 1 will be given in the Supplementary Materials.

3.2.3. Estimation of the Systematic Component Function

Theorem 2.

Under assumptions (C1)–(C7), we have

$${∥\widehat{\eta }-{\eta }_0∥}_2=O_{\mathbb{P}}\left\{\sqrt{N_n}\left({\displaystyle \frac{1}{\sqrt{n}h}}+h^p\right)\right\}.$$

and

$${∥\widehat{\eta }-{\eta }_0∥}_n=O_{\mathbb{P}}\left\{\sqrt{N_n}\left({\displaystyle \frac{1}{\sqrt{n}h}}+h^p\right)\right\}.$$

The proof of Theorem 2 will be given in the Supplementary Materials. This proof takes into account the components of our model, which makes it different from the results obtained by Alahiane et al. [28,29].

3.2.4. Estimation of the Parametric Components

The next theorem shows that the maximum quasi-likelihood estimator is root-n-consistent and is asymptotically normal, although the convergence rate of the nonparametric component $\widehat{\eta }$ is slower than root-n. Before enouncing the theorem, let us denote

$$\begin{array}{ccc}\hfill {\rm Y}\left(u_{\tau ,0}\right)={\displaystyle \frac{\mathbb{E}\left[X{\rho }_2\left(m_0\left(T\right)\right)|U_{\tau ,0}=u_{\tau ,0}\right]}{\mathbb{E}\left[{\rho }_2\left(m_0\left(T\right)\right)|U_{\tau ,0}=u_{\tau ,0}\right]}}& ,& \Gamma \left(u_{\tau ,0}\right)={\displaystyle \frac{\mathbb{E}\left[W{\rho }_2\left(m_0\left(T\right)\right)|U_{\tau ,0}=u_{\tau ,0}\right]}{\mathbb{E}\left[{\rho }_2\left(m_0\left(T\right)\right)|U_{\tau ,0}=u_{\tau ,0}\right]}},\hfill \\ \hfill \Phi \left(x\right)=\Phi \left(U_{\tau ,0},x\right)=x-{\rm Y}\left(u_{\tau ,0}\right)& \mathrm{and}& \Psi \left(w\right)=\Psi \left(U_{\tau ,0},w\right)=w-\Gamma \left(u_{\tau ,0}\right).\hfill \end{array}$$

Theorem 3.

Under assumptions (C1)–(C10), the constrained quasi-likelihood estimators $\widehat{\alpha }$ and $\widehat{\delta }$ with $\parallel \widehat{\alpha }{\parallel }_d=1$ are jointly asymptotically normally distributed, i.e.,

$$\sqrt{n}\left(\begin{array}{c}\widehat{\alpha }-{\alpha }_0\\ \widehat{\delta }-{\delta }_0\end{array}\right)\stackrel{\mathcal{D}}{⟶}\mathcal{N}\left(0,R\left({\tau }_0\right)D^{-1}{R}^{\top }\left({\tau }_0\right)\right),$$

where $\stackrel{\mathcal{D}}{\to }$ denotes the convergence in distribution,

$$D=\mathbb{E}\left[{\rho }_2\left(m_0\left(T\right)\right)\left(\begin{array}{c}{\eta }_0^{\prime }\left(U_{\tau ,0}\right)J^{\top }\left({\tau }_0\right)\Phi \left(X\right)\\ \Psi \left(\overline{W}\right)\end{array}\right){\left(\begin{array}{c}{\eta }_0^{\prime }\left(U_{\tau ,0}\right)J^{\top }\left({\tau }_0\right)\Phi \left(X\right)\\ \Psi \left(\overline{W}\right)\end{array}\right)}^{\top }\right],$$

and

$$\alpha \left(\widehat{\tau }\right)-\alpha \left({\tau }_0\right)=O_{\mathbb{P}}\left({\displaystyle \frac{1}{\sqrt{n}}}\right),\widehat{\delta }-{\delta }_0=O_{\mathbb{P}}\left({\displaystyle \frac{1}{\sqrt{n}}}\right).$$

where $O_{\mathbb{P}}$ denotes the Bachmann–Landau notation “in probability”.

Proof of Theorem 3.

The proof of Theorem 3 is given in the Supplementary Materials. This proof takes into account all the components involved in the model (3), which makes it different from the results obtained by Alahiane et al. [28,29].    □

4. A Simulation Study

We aim to show the performance of various estimators of the parameters $\tau$, $\gamma$, $\delta$, the nonparametric function $\eta$, the functional index $\beta$, and the nonlinear regression operator R of the model (3) through numerical simulations under both the Gaussian and the logistic cases. The conditional density of Y given $X=x,Z=z$ is described by Equation (1).

We believe that the model is given by the following equation:

$$g\left(\mu (X_i,Z_i)\right)=sin\left\{{\displaystyle \frac{\pi \left({\alpha }^{\top }{X}_i-A\right)}{B-A}}\right\}+R\left(\langle \beta ,Z_i\rangle \right)+{\epsilon }_i,\mathrm{for}i=1,\dots ,n.$$

(19)

The responses $Y_i$ are simulated according to the Equation (19); $X_i$ are taken uniformly over the interval $[-0.5,0.5]$, whereas the errors ${\epsilon }_i\sim \mathcal{N}(0,0.025)$. Moreover, we take the following coefficients:

$$\alpha ={\displaystyle \frac{1}{\sqrt{3}}}{(1,1,1)}^{\top },A={\displaystyle \frac{\sqrt{3}}{2}}-{\displaystyle \frac{1.645}{\sqrt{12}}}\mathrm{and}B={\displaystyle \frac{\sqrt{3}}{2}}+{\displaystyle \frac{1.645}{\sqrt{12}}}.$$

The functional real variable $Z_i(·)$ is taken as $Z\left(t\right)=acos\left(2\pi t\right)+bsin\left(4\pi t\right)+2c(t-0.25)(t-0.5),t\in [-1,1]$, where $a\sim \mathcal{U}(0,1)$, $b\sim \mathcal{U}(0,1)$ and $c\sim \mathcal{U}(0,1)$. A set of 1000 independent curves is generated according to the following model: $Z_i\left(t\right)=a_{i}cos\left(2\pi t\right)+b_{i}sin\left(4\pi t\right)+2c_i(t-0.25)(t-0.5),t\in [-1,1]$, where $a_i$, $b_i$, and $c_i$ are uniformly distributed on $[0,1]$, respectively. Curves are discretized over a very fine mesh of one thousand equispaced points; the set of curves is stored in the matrix $\mathcal{Z}=\left[Z_i\left(t_j\right)\right],$$i=1,\dots ,1000,j=1,\dots ,1000$. A random selection of 30 of these functional data is plotted in Figure 1.

We consider the functional index $\beta \left(t\right)={\displaystyle \frac{1}{\sqrt{2}}}\left[sin\left({\displaystyle \frac{3}{2}}\pi t\right)+sin\left({\displaystyle \frac{1}{2}}\pi t\right)\right]$ and the regression operator $R\left(u\right)={\left\{{\displaystyle \frac{1}{\sqrt{2}}{\int }_{-1}^1\left[sin\left(\frac{3}{2}\pi t\right)+sin\left(\frac{1}{2}\pi t\right)\right]Z\left(t\right)\mathrm{d}t}\right\}}^3$.

We set directions, and we use the first to the fourth eigenfunctions of the variance operator of Z; note that the cumulative explained variances of the corresponding linear principal components are $71.3\%$, $87.5\%$, $98.8\%$, and $99.6\%$.

In the first step of our algorithm, we base our simulation experiments on samples of 300 couples $(Z_i,Y_i)$ that we randomly extracted; we use the first 200 couples as the training set and the remaining 100 couples as the test set. We do not use all the columns of $\mathcal{Z}$, only a selection. At each step, we use cubic splines, and the number of knots is set to six.

Concerning the nonparametric functional estimator, the semimetric used is the standard distance $L_2$ between the curves.

In the second step, the nodes are selected according to the formula

$$C{n}^{{\displaystyle \frac{1}{2r}}}log\left(n\right),$$

where $C\in [0.3,1]$ (see [11]).

We choose $C=0.6$, and we perform replications of 3000 samples of sizes $n=500$.

Through the plug-in process (the second step), we estimate the model parameters (8) using the GPLFSIM algorithm as described previously.

$$g\left(\mu \left(X_i,Z_i\right)\right)=\eta \left({\alpha }^{\top }{X}_i\right)+R\left(\langle \widehat{\beta },Z_i\rangle \right)\mathrm{for}i=1,\dots ,n.$$

(20)

Then, the computed bias, the standard deviation (SD), and the mean squared error (MSE) with respect to the parameter $\tau$, the parameter $\gamma$, and the parameter $\delta$ are summarized in

Table 1. Bias, SD, and MSE according to the parameter $\tau$ for CVGPLFSIM with the identity link function and $n=500$ in both the Gaussian and the logistic cases.

	Gaussian Case 1		Logistic Case 2
Sample Size	${\mathbf{\tau }}_{\mathbf{1}}$	${\mathbf{\tau }}_{\mathbf{2}}$	${\mathbf{\tau }}_{\mathbf{1}}$	${\mathbf{\tau }}_{\mathbf{2}}$
Bias	0.0013	−0.0012	0.0027	−0.0061
SD	0.0012	0.0034	0.0031	0.0201
MSE	3.13 $\times {10}^{-6}$	1.3 $\times {10}^{-5}$	4.45 $\times {10}^{-5}$	4.14 $\times {10}^{-4}$

Note: This table summarizes the bias, SD, and MSE of $\tau$ with sample size $n=500$. ¹ $\tau$ for the Gaussian case. ² $\tau$ for the logistic case.

,

Table 2. Bias, SD, and MSE evolutions with respect to the parameter $\gamma$ variation for CVPGPLSFIM with the identity link function and $n=500$.

	${\mathit{\gamma }}_1$	${\mathit{\gamma }}_2$	${\mathit{\gamma }}_3$	${\mathit{\gamma }}_4$	${\mathit{\gamma }}_5$
Bias	−0.0026	0.0142	−0.0231	0.0242	−0.0037
SD	0.0102	0.0165	0.0141	0.0140	0.0071
MSE	1.1080 $\times {10}^{-4}$	4.7389 $\times {10}^{-4}$	7.3242 $\times {10}^{-4}$	7.8164 $\times {10}^{-4}$	6.4100 $\times {10}^{-5}$

,

Table 3. Bias, SD, and MSE evolutions with respect to the parameter $\gamma$ variation for CVGPLFSIM with the identity link function and $n=500$.

	${\mathit{\gamma }}_6$	${\mathit{\gamma }}_7$	${\mathit{\gamma }}_8$	${\mathit{\gamma }}_9$	${\mathit{\gamma }}_{10}$
Bias	0.0032	0.0023	−0.0021	0.0012	−0.0045
SD	0.0033	0.0041	0.0014	0.0027	0.0042
MSE	2.113 $\times {10}^{-5}$	2.21 $\times {10}^{-5}$	6.37 $\times {10}^{-6}$	8.73 $\times {10}^{-6}$	3.789 $\times {10}^{-5}$

,

Table 4. Bias, SD, and MSE evolutions with respect to the parameter $\delta$ variation for CVGPLFSIM with the identity link function and $n=500$.

	${\mathit{\delta }}_1$	${\mathit{\delta }}_2$	${\mathit{\delta }}_3$	${\mathit{\delta }}_4$	${\mathit{\delta }}_5$
Bias	0.0009	0.0037	−0.0045	0.0082	−0.0035
SD	0.0036	0.0012	0.0071	0.0038	0.0091
MSE	1.377 $\times {10}^{-5}$	1.513 $\times {10}^{-5}$	7.066 $\times {10}^{-5}$	8.168 $\times {10}^{-5}$	2.450 $\times {10}^{-5}$

,

Table 5. Bias, SD, and MSE evolutions with respect to the parameter $\gamma$ variation for CVGPLFSIM with the logistic link function and $n=500$.

	${\mathit{\gamma }}_1$	${\mathit{\gamma }}_2$	${\mathit{\gamma }}_3$	${\mathit{\gamma }}_4$	${\mathit{\gamma }}_5$
Bias	−0.0032	0.0156	−0.0233	0.0341	−0.0079
SD	0.0107	0.0126	0.0306	0.0321	0.0042
MSE	1.2473 $\times {10}^{-4}$	4.0212 $\times {10}^{-4}$	1.4792 $\times {10}^{-3}$	2.1932 $\times {10}^{-3}$	8.005 $\times {10}^{-5}$

,

Table 6. Bias, SD, and MSE evolutions with respect to the parameter $\gamma$ variation for CVGPLFSIM with the logistic link function and $n=500$.

	${\mathit{\gamma }}_6$	${\mathit{\gamma }}_7$	${\mathit{\gamma }}_8$	${\mathit{\gamma }}_9$	${\mathit{\gamma }}_{10}$
Bias	0.0063	0.0022	0.0327	0.0005	−0.0045
SD	0.0051	0.0062	0.0027	0.0091	0.0021
MSE	6.57 $\times {10}^{-5}$	4.32 $\times {10}^{-5}$	1.076 $\times {10}^{-3}$	8.306 $\times {10}^{-5}$	2.466 $\times {10}^{-3}$

and

Table 7. Bias, SD, and MSE evolutions with respect to the parameter $\delta$ variation for CVGPLFSIM with the logistic link function and $n=500$.

	${\mathit{\delta }}_1$	${\mathit{\delta }}_2$	${\mathit{\delta }}_3$	${\mathit{\delta }}_4$	${\mathit{\delta }}_5$
Bias	0.0036	0.0038	−0.0029	0.0145	−0.0169
SD	0.0054	0.0037	0.0122	0.0231	0.0132
MSE $\times {10}^{-4}$	4.212 $\times {10}^{-1}$	2.813 $\times {10}^{-1}$	1.5725	7.4386	4.5985

.

We present below in Figure 2 and Figure 3 the functional index and nonlinear regression operator obtained by the GPLFSIM algorithm in both cases: the Gaussian and the logistic cases. We chose the two distributions (Gaussian and logistic) because of their popularity and because of the use of activation functions (sigmoid functions) in neural networks, especially when the input data are high-dimensional.

In order to compute the bias, SD, and MSE, we recorded 3000 replications of the CVGPLFSIM algorithm in the Gaussian case and in the logistic case with $n=500$ as follows (see Table 1, Table 2, Table 3, Table 4, Table 5, Table 6 and Table 7).

It is obvious to remark that the quality of the estimators is illustrated through these simulations, as the method works quite well. The bias, SD, and MSE are generally reasonably low. The parametric and the nonparametric components, the single-index $\alpha$, the functional index $\beta (·)$, and the nonlinear regression operator R of Y over $X,Z$ are calculated by the procedure described above. The two tables therefore indicate the consistency of $\widehat{\alpha }$ and $\widehat{\delta }$ as the bias, SD, and MSE decrease as the sample size increases.

We have developed our algorithm in both cases, the identity link function and the logistic link function. The simulations show that the CVPGPLSFIM algorithm works well in both cases. We present below in Figure 4 the single index estimated by the model in both cases, the Gaussian and logistic cases.

We observe that the single-index estimated by our model fits well with the single-index.

We present below in Figure 5 the systematic component $\eta$ estimated by the model in both cases: the Gaussian and the logistic cases.

We consider the root of the averaged squared error criterion (see [35]) in both the Gaussian and the logistic cases:

$${\displaystyle {\mathtt{RASE}}_1={\left(\frac{1}{n}\sum _{i=1}^n{\left(\widehat{\eta }\left(u_i\right)-\eta \left(u_i\right)\right)}^2\right)}^{\frac{1}{2}}\mathrm{and}{\mathtt{RASE}}_2={\left(\frac{1}{n}\sum _{i=1}^n{\left(\widehat{R}\left(u_i\right)-R\left(u_i\right)\right)}^2\right)}^{\frac{1}{2}}}$$

Table 8. The RASE criterion for the function $\eta$ for both cases and for $n=500$ and $n=1000$.

	Gaussian Case			Logistic Case
Sample Size	Mean	Median	Variance	Mean	Median	Variance
$n=500$	$0.033$	$0.035$	$0.012$	$0.033$	$0.035$	$0.012$
$n=1000$	$0.031$	$0.032$	$0.011$	$0.031$	$0.032$	$0.011$

and

Table 9. The ${\mathtt{RASE}}_2$ criterion with the nonlinear regression operator R in the both cases and for $n=500$ and $n=1000$.

	Gaussian Case			Logistic Case
Sample Size	Mean	Median	Variance	Mean	Median	Variance
$n=500$	$0.035$	$0.037$	$0.006$	$0.046$	$0.039$	$0.031$
$n=1000$	$0.031$	$0.032$	$0.004$	$0.042$	$0.033$	$0.025$

summarize the samples means, medians, and variances of the ${\mathtt{RASE}}_1$ and ${\mathtt{RASE}}_2$ with different sample sizes in both the Gaussian and logistic cases.

We conclude that as the sample size n increases from 500 to $1000,$ the sample mean, median and variance of ${\mathtt{RASE}}_1$ decreases.

We conclude that when the sample size n increases from 500 to $1000,$ the sample mean, the median, and the variance of ${\mathtt{RASE}}_2$ decreases.

5. Application for Tecator Data

In this section, we employ the CVGPLFSIM model to analyze the Tecator data, which is a well-known dataset in the field of FDA. The dataset can be obtained from the following link: http://lib.stat.cmu.edu/datasets/tecator (accessed on 1 March 2024). It consists of 215 finely chopped meat samples, each associated with its corresponding fat content ($Y_i$ for $i=1,\dots ,215$), near-infrared absorbance spectra ($Z_i$ for $i=1,\dots ,215$) measured at 100 wavelengths ranging from 850 to 1050 nm, as well as the protein content $X_{1,i}$ and moisture content $X_{2,i}$ of the meat samples. For more comprehensive information and insights, we refer readers to Ferraty et al.  [13]. We aim to forecast the fat content of the finely chopped meat samples. Figure 6 shows a sample of the absorbance curves.

In order to evaluate the effectiveness of the model (3), we employ a random splitting of the sample into two subsets: a training subset, denoted as $I_1$, consisting of 160 observations, and a test subset, denoted as $I_2$, consisting of 55 observations. The purpose of the training subset is to estimate the model parameters, while the test subset is used to assess the accuracy of the predictors. We utilize the mean square error of prediction (MSEP), as defined in Aneiros et al. [14] and given by

$$\mathtt{MSEP}={\displaystyle \frac{1}{55}}\sum _{i\in I_2}{(Y_i-{\widehat{Y}}_i)}^2/{\mathrm{var}}_{I_2}\left(Y_i\right).$$

where ${\widehat{Y}}_i$ represents the predicted value based on the training subset and ${\mathrm{var}}_{I_2}\left(Y_i\right)$ denotes the variance of the response variables in the test subset. This indicator allows us to assess the accuracy of our predictions with respect to the variability in the test dataset.

The performance comparison of the CVGPLFSIM model with other models is presented in

Table 10. MSEP for different models in the Gaussian case.

Functional Models		MSEP
M-1 (CVGPLFSIM)	$g\left({\mu }_i(X_i,Z_i)\right)=\eta ({\alpha }_1{X}_{1,i}+{\alpha }_2{X}_{2,i})+R\left(\langle \beta ,Z_i\rangle \right)+{\epsilon }_i$	0.056
M-2 (GNP-FPLSIM)	$g\left({\mu }_i(X_i,Z_i)\right)=\eta ({\alpha }_1{X}_{1,i}+{\alpha }_2{X}_{2,i})+R\left(Z_i\right)+{\epsilon }_i$	0.082
M-3 (FPLSIM)	$Y_i=\eta ({\alpha }_1{X}_{1,i}+{\alpha }_2{X}_{2,i})+R\left(Z_i\right)+{\epsilon }_i$	0.102
M-4 (SIM)	$Y_i=\eta ({\alpha }_1{X}_{1,i}+{\alpha }_2{X}_{2,i})+{\epsilon }_i$	1.102

and

Table 11. MSEP for different models in the logistic case.

Functional Models		MSEP
M-1 (CVGPLFSIM)	$g\left({\mu }_i(X_i,Z_i)\right)=\eta ({\alpha }_1{X}_{1,i}+{\alpha }_2{X}_{2,i})+R\left(\langle \beta ,Z_i\rangle \right)+{\epsilon }_i$	0.045
M-2 (GNP-FPLSIM)	$g\left({\mu }_i(X_i,Z_i)\right)=\eta ({\alpha }_1{X}_{1,i}+{\alpha }_2{X}_{2,i})+R\left(Z_i\right)+{\epsilon }_i$	0.067
M-3 (FPLSIM)	$Y_i=\eta ({\alpha }_1{X}_{1,i}+{\alpha }_2{X}_{2,i})+R\left(Z_i\right)+{\epsilon }_i$	0.102
M-4 (SIM)	$Y_i=\eta ({\alpha }_1{X}_{1,i}+{\alpha }_2{X}_{2,i})+{\epsilon }_i$	1.102

. Based on the obtained results, we can infer that the CVGPLFSIM model demonstrates competitiveness and effectiveness in analyzing the given dataset.

Table 10 and Table 11 show the performance of the CVGPLFSIM model by comparing it with other models. The CVGPLFSIM model is a competitive one for such data.

Moreover, Figure 7 shows the nonparametric estimator of the function $\eta$, in both the Gaussian and the logistic cases.

Figure 8 and Figure 9 show the estimator functional index $\widehat{\beta }(·)$ and the estimator of the nonlinear regression operator $\widehat{R}$.

Figure 10 illustrates the difference between the fat content and its estimation from the model for both the Gaussian and the logistic cases.

We can see that our model fits well the content of fatness “215 pieces of meat”.

Possible extensions to our manuscript include hypothesis testing and confidence bands using the bootstrap method, the extension of the activation function (sigmoid functions) in the context of neural networks (deep learning) when the input data are high-dimensional, and functional projection pursuit regression for GFPLSIM models $g\left(\mu (X,Z)\right)=\eta \left({\alpha }^{\top }X\right)+r\left(Z\right)$, where ${\displaystyle r\left(Z\right)=\sum _{i=1}^m\langle {\beta }_i,Z\rangle }$-type models take advantage of the information required from the various revealing directions.