# Scalar-on-Function Relative Error Regression for Weak Dependent Case
## 2 The Re-Regression Model and Its Estimation  
As discussed in the introduction, we aim to evaluate the relationship between an exogenous Hilbertian variable \(X\) and a real endogenous variable \(Y\). Specifically, the variables \((X,Y)\) belong in \(\mathcal{H}\times\mathbb{R}\). The set \(\mathcal{H}\) constitutes a separable Hilbert space. We assume that the norm \(\|\cdot\|\) in \(\mathcal{H}\) is associated with the inner product \(\langle\cdot,\cdot\rangle\). Furthermore, we define on \(\mathcal{H}\) a complete orthonormal basis \((e_{k})_{k\geq 1}\). In addition, we suppose that \(Y\) is strictly positive, and we suppose that the Hilbertian operators \(\mathbf{E}[Y^{-1}|X]\) and \(\mathbf{E}[Y^{-2}|X]\) exist and are, almost surely, finite. The RE-regression is defined by  
\[R(x)=\arg\min_{\theta}\mathbf{E}\Bigg{(}\bigg{(}\frac{Y-\theta}{Y}\bigg{)}^{2 }\big{|}X=x\Bigg{)}. \tag{1}\]  
By differentiating with respect to \(\theta\), we prove that  
\[R(x)=\frac{\mathbf{E}\big{[}Y^{-1}\big{|}X=x\big{]}}{\mathbf{E}\big{[}Y^{-2} \big{|}X=x\big{]}}. \tag{2}\]  
Clearly, the RE-regression \(R(\cdot)\) is a good alternative to the traditional regression, in the sense that, the traditional regression, based on the least square error, treats all variables with equal weight. This is inadequate when the observations contain some outliers. Thus,the traditional regression can lead to irrelevant results in the presence of outliers. Thus, the main advantage of the RE-regression \(R(\cdot)\) compared to the traditional regression is the possibility to reduce the effect of the outliers (see Equation (1)). So, we can say that the robustness feature is one of the main advantages of the RE-regression. Additionally, unlike the classical robust regression (the M-egression), the RE-regression is very easy to implement in practice. It has an explicit definition based on the ratio of the first and the second inverted conditional moments (see Equation (2)).  
Now, consider \((X_{i},Y_{i})_{i-1,...,n}\) strictly stationary observations, as copies of a couple \((X,Y)\). The Hilbertian time series framework of the present contribution is carried out using the quasi-association setting (see [PERSON] [16] for the definition of the Hilbert space). We use the kernel estimators of the inverse moments \(\mathbf{E}[Y^{-1}|X]\) and \(\mathbf{E}[Y^{-2}|X]\) as conditional expectations of \(Y^{-\gamma}\) (\(\gamma=1,2\)), given \(X=x\), to estimate \(R(x)\) by  
\[\widetilde{R}(x)=\frac{\sum_{i=1}^{n}Y_{i}^{-1}K\Big{(}\frac{\|X-X_{i}\|}{h_{n} }\Big{)}}{\sum_{i=1}^{n}Y_{i}^{-2}K\Big{(}\frac{\|X-X_{i}\|}{h_{n}}\Big{)}}, \tag{3}\]  
where \(h_{n}\) is a positive sequence of real numbers, and \(K\) is a real-function so-called kernel. The choice of \(h_{n}\) is the determining issue of the applicability of the estimator \(\widetilde{R}\). A common solution is to utilize kernel smoothing with the kNN estimation, for which  
\[\widehat{R}(x)=\frac{\sum_{i=1}^{n}Y_{i}^{-1}K\Big{(}\frac{\|X-X_{i}\|}{A_{n}^ {k}}\Big{)}}{\sum_{i=1}^{n}Y_{i}^{-2}K\Big{(}\frac{\|X-X_{i}\|}{A_{n}^{k}} \Big{)}}, \tag{4}\]  
where  
\[A_{n}^{k}(x)=\min\Bigg{\{}a_{n}>0;\quad\sum_{i=1}^{n}\mathbf{I}_{B(x,A_{n})}(X _{i})=k\Bigg{\}},\]  
where \(B(x,a_{n})\) is an open ball of radius \(a_{n}>0\) centered \(x\). In \(\widetilde{R}\), the smoothing parameter is the number \(k\). Once again, the selection of \(k\) is crucial.