Christophe Crambes

Chayma Daayeb

Ali Gannoun

Yousri Henchiri
email: yousri.henchiri@umontpellier.fr.

Functional linear model with partially observed covariate and missing values in the response

Keywords: Functional linear model, Functional Principal Components, Missing data, Missing At Random, Missing Completely At Random, Regression imputation

Dealing with missing values is an important issue in data observation or data record-Christophe Crambes et al.

ing process. In this paper, we consider a functional linear regression model with partially observed covariate and missing values in the response. We use a reconstruction operator that aims at recovering the missing parts of the explanatory curves, then we are interested in regression imputation method of missing data on the response variable, using functional principal component regression to estimate the functional coefficient of the model. We study the asymptotic behavior of the prediction error when missing data are replaced by the imputed values in the original dataset. The practical behavior of the method is also studied on simulated data and a real dataset.

Introduction

The analysis of functional data has grown very significantly in recent years, as evidenced by the numerous literatures on the subject: [START_REF] Ramsay | Statistical analysis with missing data[END_REF], [START_REF] Ferraty | Nonparametric functional data analysis: Theory and practice[END_REF], [START_REF] Hsing | Theoretical foundations of functional data analysis, with an introduction to linear operators[END_REF], [START_REF] Horváth | Inference for functional data with applications[END_REF] provide a non-exhaustive list of monographs giving an overview of this topic. One of the most popular model in functional data analysis is the functional linear model, when one is interested in considering a relationship between a real-valued variable Y and a covariate X = (X(t), t ∈ [a, b]) valued in a real separable Hilbert space H of functions defined on a compact interval [a, b] of R. We assume that X is centered, that is E(X(t)) = 0 for all t ∈ [a, b]. In the following, we consider the space H = L 2 ([a, b]) of square integrable functions defined on [a, b], endowed with its usual inner product defined by u, v = b a u(t)v(t)dt for all functions u, v ∈ H, and its associated norm . . This model, studied by many authors as for instance [START_REF] Cardot | Functional linear model[END_REF], [START_REF] Cai | Prediction in functional linear regression[END_REF], [START_REF] Hall | Methodology and convergence rates for functional linear regression[END_REF], [START_REF] Crambes | Smoothing splines estimators for functional linear regression[END_REF], is defined by

Y = θ 0 + b a θ(t)X(t)dt + ε, (1.1)
where θ 0 ∈ R and θ is a square integrable function defined on [a, b] modeling the relationship between the real random variable Y and the square integrable random function X. The error of the model ε is a centered real random variable independent of X with finite variance E(ε 2 ) = σ 2 ε . We can also write the functional linear regression model (1.1) as

Y = θ 0 + ΘX + ε,
where Θ : H → R is a linear continuous operator defined by Θu = θ, u for any function u ∈ H. The existence and unicity of this regression function θ is discussed in [START_REF] Cardot | Spline estimators for the functional linear model[END_REF]. A smooth version of the functional principal components regression (SPCR) is introduced. It consists in considering the empirical covariance operator of the predictor X and diagonalizing it to select the eigenfunctions associated to the highest eigenvalues. Then, a least squares regression is performed with the response Y and the coordinates of the functional covariate X projected on the space spanned by the selected eigenfunctions.

Considering a sample (X i , Y i ) i=1,...,n of independent and identically distributed couples with the same distribution as (X, Y ), we define the empirical cross covari-ance operator ∆ n given by ∆ n u = 1 n n i=1 X i , u Y i for all u ∈ H, the empirical covariance operator Γ n given by Γ n u = 1 n n i=1 X i , u X i for all u ∈ H. Denoting ( φ j ) j=1,...,kn the eigenfunctions associated to Γ n corresponding to the k n highest eigenvalues λ 1 > . . . > λ kn > 0 (where k n is an integer depending on n), we define the orthogonal projection operator Π kn onto the subspace Span( φ 1 , . . . , φ kn ) by Π kn u = kn j=1 φ j , u φ j for all u ∈ H. Then, the functional principal component regression estimator Θ of Θ is defined by

Θ = θ, . = Π kn ∆ n ( Π kn Γ n Π kn ) -1 .
The corresponding estimator of θ is given by

θ = 1 n n i=1 kn j=1 X i , φ j Y i λ j φ j = kn j=1 s j φ j , (1.2)
with s j = 1 n λ j n i=1 X i , φ j Y i . In addition, the estimator of θ 0 is θ

0 = Y = 1 n n i=1 Y i .
Now, given θ 0 and θ, it is easy to obtain the residuals of the fit, given by ε i,kn = Y i -θ 0 -X i , θ , for i = 1, . . . , n, that can be used to estimate the error variance, σ 2 ε , through

σ 2 ε,kn = 1 n -k n -1 n i=1 ε 2 i,kn .
In the previously cited works on the functional linear model, data is fully observed.

This may not always be the case, and missing data appear in many situations, for example when the measuring device breaks down or when an observation interval is not available. This topic has to be studied a lot in the multivariate framework, for example we refer the reader to [START_REF] Little | Statistical analysis with missing data[END_REF] and [START_REF] Graham | Missing data analysis and design[END_REF]. For functional data, the literature only starts developing. In functional linear regression, the work of [START_REF] Crambes | Regression imputation in the functional linear model with missing values in the response[END_REF] considers a missing data mechanism on the response Y while the functional covariate is completely observed. A regression imputation methodology for the missing data is proposed and the authors propose an estimation of the functional parameter θ with the reconstructed dataset, as well as the prediction of new values. The method consistency is studied both from a theoretical and a practical point of view. The same problematic is studied in another paper from [START_REF] Febrero-Bande | Estimation, imputation and prediction for the functional linear model with scalar response with responses missing at random[END_REF], although not exploring theoretical results. Other works explore the context of missing data in the response while the response is missing at random in a nonparametric setting (see [START_REF] Ferraty | Mean estimation with data missing at random for functional covariables[END_REF], [START_REF] Ling | Nonparametric regression estimation for functional stationary ergodic data with missing at random[END_REF]) or in a functional partial linear regression setting (see [START_REF] Ling | Semi-functional partially linear regression model with responses missing at random[END_REF], [START_REF] Zhou | Estimation for functional partial linear models with missing responses[END_REF]) or while the response is not missing at random (see [START_REF] Li | Functional linear regression models for nonignorable missing scalar responses[END_REF]). In our work, we want to consider the functional linear model where some observations of the real response are affected with missing data and the covariate is partially observed, which is an unexplored topic as far as we know.

For the missing data mechanism in the response, we consider a dichotomous random variable δ [Y ] leading to the sample (δ

[Y ] i ) i=1,...,n such that δ [Y ] i = 1 if the value Y i is available and δ [Y ] i = 0 if the value Y i is missing, for all i = 1, . . . , n.
Here, we consider that the data in the response is missing at random (MAR): the fact that the value Y is missing does not depend on the response of the model, but can possibly depend on the covariate, that is,

P(δ [Y ] = 1 | X, Y ) = P(δ [Y ] = 1 | X).
As a consequence of this MAR assumption, the variable δ [Y ] (the fact that an observation is missing) is independent of the error of the model ε. In the following, the number of missing values among Y 1 , . . . , Y n is denoted

m [Y ] n = n i=1 1 {δ [Y ] i =0} .
For the missing data mechanism of the functional covariate, we adopt the paradigm of partially observed functions as in [START_REF] Kneip | On the optimal reconstruction of partially observed functional data[END_REF] or [START_REF] Kraus | Components and completion of partially observed functional data[END_REF]. We also refer the reader to [START_REF] Delaigle | Estimating the covariance of fragmented and other related types of functional data[END_REF] or [START_REF] Kraus | Ridge reconstruction of partially observed functional data is asymptotically optimal[END_REF] for recent contributions on this topic. More precisely, for each curve X i , i = 1, . . . , n, we

consider the observed part O i ⊆ [a, b] of X i and the missing part M i = [a, b] O i .
The observed part O i refers to an interval (or several intervals) where the curve X i is observed at some measure points of O i . Based on the punctual observations, the whole curve can be reconstructed on O i with usual methods (e.g. smoothing splines, regression splines, local polynomial smoothing, . . . ). On the contrary, no information is available on the missing part M i . An example of such partially observed functions is given in section 5 of the paper.

The objective of this paper is to predict a new value of the response Y given a new test observation on the explanatory variable X once the partially observed curves X have been reconstructed and the missing data Y have been imputed. More precisely, we want to obtain convergence rates for this prediction error, and we want to analyse how these convergence rates depend on the convergence rates of the reconstruction of the missing parts of the covariate and the convergence rates of the imputation error.

Moreover, we want to explore the interest of the imputation methodology compared to other methods, for example the naive method which would consist in simply ignoring the missing data and only using the observations when both X and Y are observed, or other imputation methods.

In the following, we give in section 2 theoretical results when the covariate is partially observed. Then, in section 3, we extend these results when the covariate is partially observed and some observations of the real response are affected with missing data.

In section 4, we present some simulation results to show the behaviour of the method in practice. Section 5 is devoted to a real dataset application. Finally, all the proofs are postponed to section 6.

2 Partially observed covariate

Curve reconstruction

We write "O" and "M " to denote a given production of O i and M i . In addition, we denote the observed and missing parts of X i by X O i and X M i . As noticed in Kneip and Liebl (2020, p. 7) all the following remains valid if we consider the more general case of several observed subintervals, that is

O i = ∪ J j=1 O J i where O 1 i , . . . , O J i are J
disjoint intervals where the curve X i is observed. For the sake of simplicity, we will take J = 1 and O i = O 1 i . We write the Karhunen-Loève (KL) decomposition of

X O i in L 2 (O) X O i (t) = +∞ k=1 ξ O ik φ O k (t),
where t ∈ O. In this decomposition, the principal component scores are defined for all i = 1, . . . , n and k ≥ 1 by

ξ O ik = φ O k , X O i , where E(ξ O ik ) = 0 and E(ξ O ik ξ O i ) = λ O k for
all k = and zero for all k = . Moreover, the eigenfunctions satisfy

φ O k (t) = φ O k , γ O t λ O k , (2.1) for all t ∈ O and k ≥ 1, where γ O t (s) = γ O (t, s) = E X O i (t)X O i (s)
, and the decreas-

ing eigenvalues λ O 1 > λ O 2 > . . . > 0 are tending to zero.
We consider a reconstruction problem relating the missing part of the curves to the observed part, writing

X M i (s) = L(X O i (t)) + Z i (s),
for all t ∈ O and s ∈ M , where L : L 2 (O) → L 2 (M ) is a linear reconstruction operator and Z i ∈ L 2 (M ) is the reconstruction error. This reconstruction estimator is estimated in [START_REF] Kneip | On the optimal reconstruction of partially observed functional data[END_REF] by

L(X O i )(s) = +∞ k=1 ξ O ik φ O k (s) = +∞ k=1 ξ O ik φ O k , γ s λ O k , ( 2

Estimation of the reconstruction in practice

We consider a discretization without measurement errors, that is ((W i1 , t i1 ), . . . , (W ip , t ip ))

denote the observable data pairs of the function

X O i , namely W ij = X O i (t ij ), for i = 1, . . .
γ(t, u; h γ ) φ O k (u)du = λ O k φ O k (t),
for all t ∈ O. For the functional principal component scores ξ

O ik = O X O i (t)φ k (t)dt,
the estimator is defined by

ξ O ik = p j=1 φ O k (t ij )W ij (t ij -t i,j-1 ), with t i0 = a.
Finally, to estimate L(X O i ) in (2.2), considering a positive integer k n , we define

L kn (X O i )(s) = kn k=1 ξ O ik φ O k , γ s λ O k ,
where γ s = γ(., s; h γ ). At this step we are able to find the estimator of the missing parts of

X O i X M i (s) = L kn (X O i )(t),
for all t ∈ O and s ∈ M . In the following, we denote

X i (t) =      X O i (t) if t ∈ O, L kn (X O i )(t) if t ∈ M .

Estimation of θ and prediction

For estimating θ, we set

θ = 1 n n i=1 kn j=1 X i , φ j,rec Y i λ j,rec φ j,rec = kn j=1 s j φ j,rec , with s j = 1 n λ j,rec n i=1 X i , φ j,rec Y i .
The estimation of the operator Θ is given by Θ = θ, . = Π kn,rec ∆ n,rec ( Π kn,rec Γ n,rec Π kn,rec ) -1 , where ∆ n,rec is the reconstructed cross covariance operator given by ∆ n,rec = 1

n n i=1 X i , . Y i ,
Γ n,rec is the reconstructed covariance operator given by Γ n,rec = 1 n n i=1 X i , . X i , and Π kn,rec is the projection operator onto the subspace Span( φ 1,rec , . . . , φ kn,rec ), that is the subspace spanned by the k n first eigenfunctions of the covariance operator Γ n,rec .

The eigenvalues of the covariance operator Γ n,rec are denoted λ 1,rec , . . . , λ kn,rec . Moreover, the estimator of θ 0 is defined by θ 0 = Y . Given θ 0 and θ, the residuals of the fit,

ε i,kn = Y i -θ 0 -X i , θ , for i = 1, .
. . , n, can be used to estimate the error variance as follows

σ 2 ε,kn = 1 n -k n -1 n i=1 ε 2 i,kn .
Finally, given a new observation of the covariate X, denoted X new , possibly partially observed, we predict the corresponding value of the response Y by

Y new = θ 0 + θ, X new .

Assumptions

We present in this part the assumptions needed for our results. These assumptions are used in [START_REF] Kneip | On the optimal reconstruction of partially observed functional data[END_REF] in order to control the curve reconstruction for the covariate.

(A.1) The variable X has a finite four moment order, that is E X 4 < ∞.

(A.2) Let np → ∞ when n → ∞ and p = p(n). We assume p = n η 1 with 0 < η 1 < ∞ in the following.

(A.3) The bandwidth h X satisfies h X → 0 and (ph X ) → ∞ as p → ∞. For instance, we assume that h X = 1 n η 2 with 0 < η 2 < η 1 . The bandwidth h γ satisfies h γ → 0 and (n(p 2 -p)h γ ) → ∞ as n(p 2 -p) → ∞. For example, we can take h γ = 1 n η 3 with 0 < η 3 < 2η 1 + 1.
(A.4) Let κ 1 and κ 2 be nonnegative, second order univariate and bivariate kernel functions with support [-1, 1]. For example, we can use univariate and bivariate

Epanechnikov kernel functions with compact support

[-1, 1], namely κ 1 (x) = 3 4 (1 -x 2 )1 [-1,1] (x) and κ 2 (x, y) = 9 16 (1 -x 2 )(1 -y 2 )1 [-1,1] (x)1 [-1,1] (y). (A.5) For any subinterval O ⊆ [a, b],
we assume that the eigenvalues λ 1 > λ 2 > . . . > 0 have multiplicity one. Moreover, we assume that there exist a O > 1 and

0 < c O < ∞ such that (i) λ O k -λ O k+1 ≥ c O k -a O -1 , (ii) λ O k = O(k -a O ), (iii) 1/λ O k = O(k a O ) as k → ∞. (A.6) For any subinterval O ⊆ [a, b], we assume that there exists 0 < D O < ∞ such that the eigenfunctions satisfy sup t∈[a,b] sup k≥1 φ O k (t) ≤ D O .
Assumption (A.1) holds for many processes X (Gaussian processes, bounded processes). Assumption (A.2) is mild and can be satisfied even if the number of observation points p does not go fast to infinity. As in [START_REF] Kneip | On the optimal reconstruction of partially observed functional data[END_REF], we assume that p = n η 1 with 0 < η 1 < ∞. Assumptions (A.3) and (A.4) are classic in the context of local polynomials smoothers. Assumptions (A.5) and (A.6), related to eigenvalues and eigenfunctions of the covariance operator of X, are given in [START_REF] Kneip | On the optimal reconstruction of partially observed functional data[END_REF]. In particular, a polynomial decrease of the eigenvalues is required, allowing a large class of eigenvalues for the covariance operator of X.

Asymptotic results

Under assumptions (A.1)-(A.6), it is proved in [START_REF] Kneip | On the optimal reconstruction of partially observed functional data[END_REF] that, in the case where

p ∼ n η 1 with η 1 ≤ 1/2, we have for any t ∈ [a, b] |X i (t) -X i (t)| = O p p -(a O -1)/(2(a O +2)) .
(2.

3)

The previous result allows to obtain some bounds between quantities related to functional principal components analysis with the constructed curves and with the original curves. These bounds are given in the following proposition. For any linear continuous operator T : H → H or any linear continuous operator S : H → R, we define the operator norm of T as T ∞ = sup x =1 T x , and the operator norm of S as

S ∞ = sup x =1 |Sx|.
Proposition 2.1 Under assumptions (A.1)-(A.6), we have

(i) Γ n,rec -Γ n ∞ = O p p -(a O -1)/(2(a O +2)) , (ii) ∆ n,rec -∆ n ∞ = O p p -(a O -1)/(2(a O +2)) , (iii) ∀k ≥ 1, φ k,rec -φ k = O p k a O +1 p -(a O -1)/(2(a O +2)) , (iv) ∀k ≥ 1, λ k,rec -λ k = O p p -(a O -1)/(2(a O +2)) .
We finish this section with the main result giving a bound for the prediction error of Y new with a new value of the covariate X new .

Theorem 2.2 Under assumptions (A.1)-(A.6), if we take k n ∼ p 1/(a O +2) and p ∼ n η 1 with η 1 ≤ 1/2, the prediction error is

E θ 0 + θ, X new -θ 0 -θ, X new 2 = O p n -η 1 (a O -1)/(2(a O +2)) .
This prediction error rate

O p n -η 1 (a O -1)/(2(a O +2))
is related to the rate given in Corollary 4.1 in [START_REF] Kneip | On the optimal reconstruction of partially observed functional data[END_REF] (in the particular case where η 1 = 1/2). This means that, provided with some conditions on the number of observation points p and the number of principal components k n are fulfilled, the prediction error rate has the same order as the curve reconstruction error rate. In other words, this means that, when reconstructing missing parts of the explanatory curves in a functional linear model and then predicting a new value of the response, the most important step is the curve reconstruction. This step is going to fix the convergence rate of the prediction.

Remark 1 Due to the bound (2.3), the result of Theorem 2.2 remains valid if we replace X new with X new .

Corollary 2.3 Under the hypotheses of Theorem 2.2, in the favorable situation where

η 1 = 1/2, the prediction error is E θ 0 + θ, X new -θ 0 -θ, X new 2 = O p n -(a O -1)/(4(a O +2)) .
3 Partially observed covariate and missing data on the response

In this section, we are interested in the most general case of missing data in functional linear regression: when the covariate is partially observed and when the response is affected by missing data. We have seen in the previous section the methodology for reconstructing the missing parts of the explanatory curves. Concerning missing data on the response, we are going to apply the methodology presented in [START_REF] Crambes | Regression imputation in the functional linear model with missing values in the response[END_REF], imputing missing values on the response using a regression imputation. Next, once the initial sample is completed, we will present the estimation of the functional parameter θ and predict new values for the response.

Regression imputation on the response

In this subsection, we use the methodology to impute a missing value of Y as in [START_REF] Crambes | Regression imputation in the functional linear model with missing values in the response[END_REF]. We consider the whole data, possibly with reconstructed explanatory curves, except the ones for which the value of Y is not available.

We define the covariance operator with the reconstructed curves

Γ obs n,rec = 1 n -m [Y ] n n i=1 X i , . δ [Y ] i X i .
Let Π obs kn,rec be the projection operator onto the subspace Span( φ obs 1,rec , . . . , φ obs kn,rec ) where

φ
variates θ = 1 n -m [Y ] n n-m [Y ] n i=1 kn j=1 X i , φ obs j,rec δ [Y ] i Y i λ obs j,rec φ obs j,rec = kn j=1 s j φ obs j,rec , with s j = 1 (n-m [Y ] n ) λ obs j,rec n-m [Y ] n i=1 X i , φ obs j,rec δ [Y ]
i Y i . We also estimate the intercept θ 0

with θ 0 = Y obs = 1 n-m [Y ] n n i=1 δ [Y ] i Y i . Now, the residuals of the fit, ε i,kn = Y i -θ 0 - X i , θ for i = 1, .
. . , n, can be used to estimate the error variance as follows

σ 2 ε,kn = 1 n -m [Y ] n -k n -1 n i=1 δ [Y ] i ε 2 i,kn .
Then, considering a missing value on the response, say Y such that δ

[Y ] = 0, we define the imputed value Y ,imp by

Y ,imp = θ 0 + θ, X = θ 0 + kn j=1 s j X , φ obs j,rec , with s j = 1 (n-m [Y ] n ) λ obs j,rec n i=1 i = X i , φ obs j,rec δ [Y ] i Y i .
Let us remark that the imputation Y ,imp can also be written

Y ,imp = Π obs kn,rec ∆ obs n,rec Π obs kn,rec Γ obs kn,rec Π obs kn,rec -1 X , where ∆ obs n,rec = 1 n-m [Y ] n n i=1 X i , . δ [Y ] i Y i .

Estimation of θ and prediction

Once the whole database has been reconstructed, we estimate the functional coeffi-

cient θ with θ = 1 n n i=1 kn j=1 X i , φ j,rec Y i λ j,rec φ j,rec = kn j=1 s j φ j,rec ,
where

s j = 1 n λ j,rec n i=1 X i , φ j,rec Y i and Y i = Y i δ [Y ] i + Y i,imp (1 -δ [Y ]
i ) for all i = 1, . . . , n. The estimation of the operator Θ is similarly given by

Θ = θ , . = Π kn,rec ∆ n,rec Π kn,rec Γ n,rec Π kn,rec -1
, where the cross covariance operator is ∆

n,rec = 1 n n i=1 X i , . Y i , the covariance oper- ator is Γ n,rec = 1 n n i=1 X i , . X i ,
Y new = θ 0 + θ , X new = θ 0 + 1 n n i=1 kn j=1 X i , φ j,rec X new , φ j,rec Y i λ j,rec = θ 0 + kn j=1 s j X new , φ j,rec ,
where

θ 0 = Y = 1 n n i=1 Y i .
Then, the residuals of the fit,

ε i,kn = Y i -θ 0 -X i , θ
for i = 1, . . . , n, allow to estimate the error variance writing

( σ ε,kn ) 2 = 1 n -k n -1 n i=1 ( ε i,kn ) 2 .

Asymptotic results

The first result gives an error rate of the imputed values.

Theorem 3.1 Under assumptions (A.1)-(A.6), if we take k n ∼ p 1/(a O +2) and p ∼ n η 1 with η 1 ≤ 1/2, we have

E (Y ,imp -θ 0 -θ, X ) 2 = O p n -η 1 (a O -1)/(2(a O +2)) + n η 1 /(a O +2) n -m [Y ] n .
Moreover, the aggregate error for all the imputed values is given by

n =1 (1-δ [Y ] )E (Y ,imp -θ 0 -θ, X ) 2 = O p m [Y ] n n -η 1 (a O -1)/(2(a O +2)) + m [Y ] n n η 1 /(a O +2) n -m [Y ] n
Table 1: Single and aggregate imputation mean square error convergence rates.

single error aggregate error

(i) m [Y ] n = a n n O p n -η 1 (a O -1)/(2(a O +2)) O p a n n 1-η 1 (a O -1)/(2(a O +2)) (ii) m [Y ] n ∼ ρn O p n -η 1 (a O -1)/(2(a O +2)) O p n 1-η 1 (a O -1)/(2(a O +2)) (iii) n -m [Y ] n = n γ γ ≥ η 1 (a O +1) 2(a O +2) O p n -η 1 (a O -1)/(2(a O +2)) O p n 1-η 1 (a O -1)/(2(a O +2)) γ < η 1 (a O +1) 2(a O +2) O p n η 1 /(a O +2)-γ O p n 1+η 1 /(a O +2)-γ
The following corollary explores some specific cases of the above error rates. The given results simply come from a comparison between the convergence rates of the above result, hence the proof is ommited.

Corollary 3.2 We consider cases where the number of missing values on the response are (i) negligeable with respect to the sample size, (ii) proportional to the sample size, (iii) of the same order than the sample size. More precisely

(i) m [Y ]
n = a n n where a n goes to zero when n goes to infinity,

(ii) m [Y ] n ∼ ρn with 0 < ρ < 1, (iii) n -m [Y ] n = n γ with 0 < γ < 1.
We summarize the error rate for a single imputed value and the aggregate error in Table 1.

We finish the theoretical results with the prediction error of Y new with a new value of the covariate X new . The proof of this result is ommited as it uses previous results of Theorems 2.2 and 3.1 and follows exactly the same lines as the proof of Theorem 2.2.

Theorem 3.3 Under assumptions (A.1)-(A.6), and k n ∼ p 1/(a O +2) and p ∼ n η 1 with η 1 ≤ 1/2, the prediction error is

E θ 0 + θ , X new -θ 0 -θ, X new 2 = O p n -η 1 (a O -1)/(2(a O +2)) + n η 1 /(a O +2) n -m [Y ] n .
In the particular case where η 1 = 1/2, the first term in the convergence rate is

O p n -(a O -1)/(4(a O +2)) .
All our convergence rates depend in particular on the parameter a O > 1, which is directly linked to the smoothness of the stochastic process X. The larger a O is, the smoother X is. When a O tends to 1 (non-smooth processes, for example a standard

Brownian motion corresponds to a O = 2), the convergence rate deteriorates. When a O tends to infinity (very smooth processes), the convergence rate n -η 1 (a O -1)/(2(a O +2))

is equivalent to n -η 1 /2 .

As before, we consider cases in the corollary below where the number of missing values on the response are (i) negligeable with respect to the sample size, (ii) proportional to the sample size, (iii) of the same order than the sample size.

Corollary 3.4 In the cases (i), (ii) and (iii) ) , the prediction error of a new value of the response is

with γ ≥ η 1 (a O +1) 2(a O +2
E θ 0 + θ , X new -θ 0 -θ, X new 2 = O p n -η 1 (a O -1)/(2(a O +2)) .
In the case (iii) with γ < η 1 (a O +1) 2(a O +2) , the prediction error of a new value of the response is

E θ 0 + θ , X new -θ 0 -θ, X new 2 = O p n η 1 /(a O +2)-γ .
In other words, in situations where the number of missing values on the response is negligeable or moderate with respect to the sample size, the convergence rate of the prediction error is given by the convergence rate obtained in [START_REF] Kneip | On the optimal reconstruction of partially observed functional data[END_REF] for the curve reconstruction. As a conclusion, when dealing with a functional linear model with a partially observed covariate and missing values in the response, the convergence rate of the prediction error strongly depends on the curve reconstruction error, with respect to the response imputation error.

Remark 2 As noticed at the end of the previous section, all the results obtained in this section remain valid if we replace X with X.

Simulations

Model and samples

All the procedures described below were implemented with the R software. In the simulations, we deal with functions defined on the interval [0, 1]. We consider the model

Y = θ 0 + θ, X + ε, (4.1)
where the error ε is either a Gaussian noise ε ∼ N (0, σ 2 ε ) with σ ε = 0.2 and σ ε = 1.5, or drawn from a centered Beta(2,2) law. We derived different models from (4.1), simulating more or less smooth processes X. For the sake of concision, we only give the results for the model presented below. Results for other models are available on demand to the authors.

In this model, as in [START_REF] Hall | Methodology and convergence rates for functional linear regression[END_REF], the functional covariate X is generated by a set of cosine basis functions φ 1 ≡ 1 and φ j+1 = √ 2 cos(jπt) for j > 1, such that

X(t) = 150 j=1 j ζ j φ j (t) for all t ∈ [0, 1],
where the ζ j 's are independently sampled from the uniform distribution on [-√ 3, √ 3] and the j 's are defined by j = (-1) j+1 (j) -β/2

with β = 4. The covariance function writes cov(X(t), X(s)) = 150 j=1 2 j β cos(jπt) cos(jπs).

The true parameters of the model are θ 0 = 3 and θ defined for all t ∈ [0, 1] by

θ(t) = 50 j=1 b j φ j (t),
with b 1 = 0.3 and b j = 4(-1) j+1 j -2 for all j > 1.

The trajectories of X i for i = 1, . . . , N are discretized in p = 100 equidistant points.

We consider n = 4 5 N for the training sets (X 1 , Y 1 ), . . . , (X n , Y n ) and n 1 = 1 5 N for the Christophe Crambes et al.

test sets (X n+1 , Y n+1 ), . . . , (X n+n 1 , Y n+n 1 ), where N = 1400. In each simulation, we replicated S = 400 samples.

Criteria

We used the following criteria, related to the prediction step with the test samples.

• Criterion 1: the mean square errors (M SE) averaged over S samples

M SE = 1 S S j=1 M SE(j), where M SE(j) = 1 n 1 n+n 1 =n+1 Y j -θ 0 -θ, X j 2
is the mean square error computed on the j th simulated sample, j ∈ {1, . . . , S}.

• Criterion 2: the ratio respect to truth between the mean square prediction error and the mean square prediction error when the true mean is known averaged over S samples

RT = 1 S S j=1 RT (j),
where

RT (j) = n+n 1 =n+1 (Y j -θ 0 -θ,X j ) 2 n+n 1 =n+1 ( j ) 2
is the ratio between the mean square prediction error and the mean square prediction error when the true mean is known, computed on the j th simulated sample.

Notice that all the criteria tend to zero when the sample size tends to infinity. Criterion RT is a rescaled version of M SE if we substitute the denominator by its limit (specifically, M SE(j) = RT (j)σ 2 ).

Methodology

As in [START_REF] Crambes | Regression imputation in the functional linear model with missing values in the response[END_REF], we use a smoothed version of the estimator (1.2) based on the SPCR. We use a regression spline basis with parameters: the number κ of knots of the spline functions, the degree q of spline functions and the order m of derivative. Let us remark that, with appropriate conditions, all the theoretical results obtained in our work will also apply when using the SPCR estimation. We take κ = 20, q = 3 and m = 2. The choice of these parameters is not crucial in our study, especially in comparison with the choice of the number of principal components (see [START_REF] Crambes | Regression imputation in the functional linear model with missing values in the response[END_REF] for more details). In this subsection, we firstly present the missing data simulation scenarios for the response and functional covariate. Secondly, we give a procedure to choose the optimal tuning parameter on a growing sequence of dimension k n = 2, . . . , 22.

Missing data simulation scenario

In our simulations, we have adopted the following scenario to determine the number of missing data on the response Y as in [START_REF] Crambes | Regression imputation in the functional linear model with missing values in the response[END_REF]: we simulate δ [Y ] according to the logistic functional regression. The variable δ follows the Bernoulli law with parameter p(X) such that log p(X)

1 -p(X) = α 0 , X + c,
where α 0 = sin(2πt) for all t ∈ [0, 1] and c is a constant allowing to take different levels of missing data. For exemple c = 1 for around 26.97% of missing data, c = 0.2 for around 44.99% of missing data and c = -0.2 for around 45.087% of missing data.

To deal with partially observed curves for the covariate, we adopted the missing data simulation scenario from [START_REF] Kneip | On the optimal reconstruction of partially observed functional data[END_REF] such that

• 70% (respectively 55%) of the curves are fully observed on [0, 1],

• for the 30% (respectively 45%) of partially observed curves, the curve X i is fully observed on [A i , B i ] ⊂ [0, 1] with A i drawn with uniform law on the interval [0, A] and B i = A i + B, with A = 3/8 and B = 6/8.

Choice of the optimal parameter

Theoretical results are generally obtained under assumptions concerning the rate of convergence of the integer k n . In practice, this integer is selected by minimizing a certain empirical criterion, for example the Generalized Cross Validation (GCV) criterion, the Cross Validation (CV) criterion and the K-fold Cross Validation (Kfold CV) criterion (see [START_REF] Crambes | Regression imputation in the functional linear model with missing values in the response[END_REF]). In our simulations, we chose the GCV procedure, known to be computationally fast. The GCV criterion is given below for imputation

GCV(k n ) = (n -m [Y ] n ) n i=1 ( Y i -θ 0 -θ, X i ) 2 δ i ((n -m [Y ] n ) -k n ) 2
, and the analogous criterion for prediction

GCV(k n ) = n n i=1 ( Y i -θ 0 -θ, X i ) 2 (n -k n ) 2 .

Analysis of results

The criteria were computed according to the different cases listed below.

• Case 1: FULL: X and Y are fully observed, this corresponds to the complete reference dataset,

•

and (DEL X DEL Y).

There is a more clear-cut difference between our method and the other ones when the percentage of missing data increases. We can empirically see the advantage of reconstructing the missing parts of the covariate. This echoes to our theoretical results where we remark that the prediction error rate is subordinate to the reconstruction error of the covariate.

Real dataset study

In this section, we are interested in a model involving electricity production, demand and prices of the German power market. [START_REF] Kneip | On the optimal reconstruction of partially observed functional data[END_REF] were already interested in the curve reconstruction problem of electricity prices curves (function be explained by the model and were set to the value 120. Negative prices are not impossible in this situation: electricity producers prefer to sell electricity at negative prices (meaning that they are paying for delivering electricity), it is sometimes more profitable than shutting off and restarting a central plant. Figure 1 shows the prices curves (in EUR/MWh) in function of the residual demand (in MWh), and Figure 2 shows the reconstructed curves with the method from [START_REF] Kneip | On the optimal reconstruction of partially observed functional data[END_REF]. Price curves can be seen as partially observed curves, as some prices cannot be observed with respect to some residual demand values.

Here, the price-demand functions are observed on different domains. This distinguishes our functional data set from classical functional data sets, where all functions are observed on a common domain. We consider a standardized domain where the standardization can be achieved as follows: for i = 1, . . . , n, we consider a se- In the second step, we reconstructed the missing parts of the differents curves.

We introduce now the model and

Y i = θ 0 + θ, X i + ε i , for i = 1, . . . ,
X i -X i , x (X i -X i ) = O p p -(a O -1)/(a O +2) ,
which gives the first result (i). The result (ii) can be shown exactly the same way.

Finally, we notice that α k = O p (k -a O -1 ) where we set α 1 = λ 1 -λ 2 and α k = min λ k-1 -λ k ; λ k -λ k+1 for all k ≥ 2. This allows to show results (iii) and (iv) from (i) and respectively Lemma 2.3 and Lemma 2.2 in [START_REF] Horváth | Inference for functional data with applications[END_REF].

Proof of Theorem 2.2

We start with the decomposition

E θ, X new -θ, X new 2 = E Π kn,rec ∆ n,rec Π kn,rec Γ n,rec Π kn,rec -1 X new -ΘX new 2 ≤ 2E Π kn,rec Θ Γ n,rec Π kn,rec Γ n,rec Π kn,rec -1 X new 2 + 2E Π kn,rec 1 n n i=1 X i , . ε i Π kn,rec Γ n,rec Π kn,rec -1 X new -ΘX new 2 .
Applying several times the identity (a + b) 2 ≤ 2a 2 + 2b 2 for any a, b ∈ R, we get

A 3 := 16E Θ Π kn X new -ΘΠ kn X new 2 = O λ kn k 2 n n + k n n .
We go on with A 4 := 8E (ΘΠ kn X new -ΘX new ) 2 . With Lemma 5.3 in Crambes and Henchiri (2019), we get

A 4 = 8 +∞ j=kn+1 ΘΓ 1/2 φ j 2 .
Next, using again (2.3), we can write

A 5 := 4E (ΘX new -ΘX new ) 2 = O p p -(a O -1)/(2(a O +2)) .
Finally, the last term of the decomposition comes from Lemma 5.2 in Crambes and Henchiri ( 2019) and gives

A 6 := 2E 1 n n i=1 X i , Π kn,rec Γ n,rec Π kn,rec -1 X new ε i 2 = 2σ 2 ε k n n + O k n n .
We can now conclude the proof of Theorem 2.2. The decomposition from the beginning of the proof gives

E θ, X new -θ, X new 2 = O p +∞ j=kn+1 ΘΓ 1/2 φ j 2 + p -(a O -1)/(2(a O +2)) + σ 2 ε k n n + O λ kn k 2 n n + k n n .
The first term in the convergence rate is

+∞ j=kn+1 ΘΓ 1/2 φ j 2 = +∞ j=kn+1 λ j (Θφ j ) 2 ≤ +∞ j=kn+1 j -a O .
Comparing the latter sum to an integral, we get

+∞ j=kn+1 ΘΓ 1/2 φ j 2 = O k -(a O +1) n = O p -(a O +1)/(a O +2) = O n -η 1 (a O +1)/(a O +2) .
The second term in the convergence rate is

p -(a O -1)/(2(a O +2)) ∼ n -η 1 (a O -1)/(2(a O +2)) ,
and the third term in the convergence rate is

σ 2 ε k n n ∼ σ 2 ε n η 1 /(a O +2) n = σ 2 ε n η 1 /(a O +2)-1 .
If we compare the different rates, with the condition η 1 ≤ 1/2, we get

E θ, X new -θ, X new 2 = O p n -η 1 (a O -1)/(2(a O +2)) .
Finally, we can write

E θ 0 + θ, X new -θ 0 -θ, X new 2 = E Y -θ 0 + θ, X new -θ, X new 2 ≤ 2E Y -E(Y ) 2 + 2E θ, X new -θ, X new 2 .
The first term of the right-hand side is given by E Y -E(Y ) 2 = O p (n -1 ) (with Bienaymé-Tchebychev inequality), and the second term of the right-hand side gives a convergence rate in probability of n -η 1 (a O -1)/(2(a O +2)) , which gives the desired result E θ 0 + θ, X new -θ 0 -θ, X new 2 = O p n -η 1 (a O -1)/(2(a O +2)) .

Proof of Theorem 3.1

This proof follows the same lines as the proof of Theorem 2.2. We write the decomposition E θ, X -θ, X Again with Lemma 5.1 in [START_REF] Crambes | Regression imputation in the functional linear model with missing values in the response[END_REF], we obtain

B 3 := 16E Θ Π kn X -ΘΠ kn X new 2 = O λ kn k 2 n n + k n n .
The next term is B 4 := 8E (ΘΠ kn X -ΘX ) 2 . With Lemma 5.3 in [START_REF] Crambes | Regression imputation in the functional linear model with missing values in the response[END_REF], we get

B 4 = 8 +∞ j=kn+1 ΘΓ 1/2 φ j 2 .
Then, using again (2.3), we can write B 5 := 4E (ΘX -ΘX ) 2 = O p p -(a O -1)/(2(a O +2)) .

Finally, the last term of the decomposition comes from Lemma 5.2 in [START_REF] Crambes | Regression imputation in the functional linear model with missing values in the response[END_REF] and gives

B 6 := 2E 1 n n i=1
X i , Π obs kn,rec Γ obs n,rec Π obs kn,rec

-1 X δ [Y ] i ε i 2 = 2σ 2 ε k n n -m [Y ] n +O k n n -m [Y ] n
.

We can now conclude the proof of Theorem 3.1. Coming back to the decomposition from the beginning, we get

E θ, X -θ, X 2 = O p +∞ j=kn+1 ΘΓ 1/2 φ j 2 + p -(a O -1)/(2(a O +2)) + σ 2 ε k n n -m [Y ] n + O λ kn k 2 n n -m [Y ] n + k n n -m [Y ] n
.

Comparing the convergence rates, we obtain

E θ, X -θ, X 2 = O p n -η 1 (a O -1)/(2(a O +2)) + n η 1 /(a O +2) n -m [Y ] n
.

Finally, we can get the desired result including the intercept. We follow the end of the proof of Theorem 2.2 to write

E θ 0 + θ, X -θ 0 -θ, X 2 = E Y obs -θ 0 + θ, X -θ, X 2 ≤ 2E Y obs -E(Y ) 2 + 2E θ, X -θ, X 2 .
The first term of the right-hand side is given by E Y obs -E(Y )

2 = O p (n -m [Y ] n ) -1
(with Bienaymé-Tchebychev inequality), and the second term of the right-hand side gives a convergence rate in probability of n -η 1 (a O -1)/(2(a O +2)) + n η 1 /(a O +2)

n-m

[Y ] n

, which gives

E θ 0 + θ, X -θ 0 -θ, X 2 = O p n -η 1 (a O -1)/(2(a O +2)) + n η 1 /(a O +2) n -m [Y ] n .
Proof of Theorem 3.3

Following the same lines of previous proofs but first we write the cross covariance operator as

∆ n,rec = 1 n n i=1 X i , . Y i = 1 n n i=1 X i , . Y i δ [Y ] i + Y i,imp (1 -δ [Y ] i ) = 1 n n i=1 X i , . δ [Y ] i Y i + 1 n n i=1 X i , . (1 -δ [Y ] i )Y i,imp .
Next, we observe that .

E
The first term is given by the result of Theorem 2.2. For the second term

E Π kn,rec 1 n n i=1 X i , . Y i,imp (1 -δ [Y ] i ) Π kn,rec Γ n,rec Π kn,rec -1 X new -ΘX new 2 ≤ 2E 1 n n i=1 X i , Γ n,rec Π kn,rec -1 X new (Y i,imp -Y i )(1 -δ [Y ] i ) + 2E 1 n n i=1 X i , Γ n,rec Π kn,rec -1 X new Y i (1 -δ [Y ] i ) -ΘX new 2 .
We notice that the first term above is exactly the same as in Theorem 3.1 and the second term is directly the result of the Theorem 2.2. So, comparing the convergence rates, we get

  Case 2: REC X IMP Y: X is partially observed and Y is affected with missing values, the missing parts of X are reconstructed and the missing values of Y are imputed, according to the method presented in this paper, • Case 3: REC X MEAN IMP Y: X is partially observed and Y is affected with missing values, the missing parts of X are reconstructed and the missing values of Y are imputed by the mean of the response observed values, • Case 4: REC X RAND IMP Y: X is partially observed and Y is affected with missing values, the missing parts of X are reconstructed and the missing values of Y are imputed by a random response observed value, • Case 5: REC X ZERO IMP Y: X is partially observed and Y is affected with missing values, the missing parts of X are reconstructed and the missing values of Y are imputed by a value equal to zero, • Case 6: REC X DEL Y: X is partially observed and Y is affected with missing values, the missing parts of X are reconstructed and the missing values of Y are removed from the sample, • Case 7: DEL X DEL Y: X is partially observed and Y is affected with missing values, the individuals presenting either a partially observed curve or a missing response are removed from the sample. Our results are presented in Tables 2, 3 and 4. Other intermediate cases have been examinated (when X is fully observed and Y is affected by missing values, or when X is partially observed and Y is not affected by missing values). Complete results are available on demand to the authors. As it can be expected, the errors increase as the model error increases. The main point we want to discuss is related to the level of missing data in the sample. Our method (REC X IMP Y) always behaves better than the other methods, specially with respect to the imputation with the value zero (REC X ZERO IMP Y) or the more naive methods where we delete missing data on the response (REC X DEL Y) or where we delete all missing data (DEL X DEL Y). The other imputation methods with the mean (REC X MEAN IMP Y) or with a random value drawn in the observed values (REC X RAND IMP Y) behave better than (REC X DEL Y)

Figure 1 :

 1 Figure 1: Daily electricity price curves in function of the residual demand.

Figure 2 :

 2 Figure 2: Reconstructed daily electricity price curves in function of the residual de-

2≤.

  32E Θ Π obs kn,rec X -Θ Π kn X 2 + 32E Θ Π obs kn X -Θ Π kn X 2 + 16E Θ Π kn X -ΘΠ kn X new 2 + 8E (ΘΠ kn X -ΘX ) 2 + 4E (ΘX -ΘX )The first term in the above decomposition B 1 := 32E Θ Π obs kn,rec X -Θ Π kn X use (2.3) to obtainB 2 := 32E Θ Π obs kn X -Θ Π kn X 2 = O p p -(a O -1)/(2(a O +2)) .

  , n and j = 1, . . . , p, where t ij ∈ O i . In order to estimate the curve X O i and the covariance function γ s , a nonparametric curve estimation by local polynomials smoothers is used. The latter is similar to the procedure in[START_REF] Yao | Functional data analysis for sparse longitudinal data[END_REF] or Hall

	et al. (2006). For the curve X O i , we use a kernel κ 1 and a bandwidth h X , the local
	linear smoother of the curve X O i being denoted X O i (t; h X ). Similarly for the covari-
	ance function γ s , we use a kernel κ 2 and a bandwidth h γ , the local linear smoother
	of the covariance function γ being denoted γ(t, s; h γ ).
	For estimating the eigenvalues λ O k and the eigenfunctions φ O k , we use the Fredholm
	integral equation
	O

  and φ 1,rec , . . . , φ kn,rec and λ 1,rec , . . . , λ kn,rec represent respectively the k n first eigenfunctions and eigenvalues of the operator Γ n,rec . We use this estimation to predict a new value of the response Y when a new explanatory curve X new is given

Table 2 :

 2 Mean and standard deviation errors for the predicted values based on 400 simulation replications with different levels of missing data and a sample size N =

	1400. Partially observed curves are fully observed on [3/8, 6/8] and the error ε ∼
	N (0, σ 2 ε ) with σ ε = 0.2.
	Rate of missing	26.97	26.99	44.99	45.01	55.07	54.91
	data in Y in %	(1.41)	(1.22)	(1.47)	(1.58)	(1.50)	(1.43)
	Rate of missing	30.03	45.01	30.07	44.81	29.89	44.94
	data in X in %	(1.16)	(1.36)	(1.19)	(1.32)	(1.22)	(1.36)
	(FULL) M SE × 10 3	18.65	17.07	18.45	18.29	18.41	18.61
		(16.48) (14.23) (15.87) (16.30) (15.96)	(16.94)
	RT	1.47	1.44	1.47	1.46	1.47	1.47
		(0.41)	(0.39)	(0.42)	(0.42)	(0.40)	(0.44)
	(REC X IMP Y) M SE × 10 3	31.46	30.92	49.33	52.54	68.51	67.95
		(28.39) (27.65) (39.08) (48.53) (59.54)	(59.67)
	RT	1.79	1.79	2.24	2.31	2.72	2.72
		(0.72)	(0.74)	(0.97)	(1.21)	(1.46)	(1.53)
	(REC X MEAN IMP Y) M SE × 10 3	31.58	31.44	52.82	56.15	72.59	70.48
		(27.40) (25.46) (36.49) (40.56) (42.02)	(44.07)
	RT	1.79	1.80	2.33	2.40	2.83	2.78
		(0.70)	(0.68)	(0.92)	(1.01)	(1.08)	(1.12)
	(REC X RAND IMP Y) M SE × 10 3	31.81	31.26	52.31	56.00	72.26	70.49
		(27.68) (25.19) (36.01) (40.90) (41.86)	(44.39)
	RT	1.80	1.79	2.31	2.40	2.83	2.78
		(0.71)	(0.68)	(0.91)	(1.02)	(1.07)	(1.12)
	(REC X ZERO IMP Y) M SE × 10 2	72.31	72.96	194.18	194.74	287.29	286.35
		(8.53)	(8.23)	(15.01) (14.51) (16.93)	(17.04)
	RT	19.27	19.38	49.63	49.96	72.94	73.22
		(2.62)	(2.67)	(5.57)	(5.23)	(7.47)	(7.68)
	(REC X DEL Y) M SE × 10 3	39.55	42.69	72.04	78.83	96.71	96.87
		(32.79) (36.91) (53.85) (59.65) (70.89)	(75.14)
	RT	1.99	2.08	2.81	2.98	3.42	3.45
		(0.84)	(0.95)	(1.35)	(1.50)	(1.77)	(1.95)
	(DEL X DEL Y) M SE × 10 3	48.66	57.22	84.33	103.34	112.44	123.62
		(46.92) (53.12) (69.73) (91.15) (89.72) (107.01)
	RT	2.21	2.46	3.14	3.61	3.81	4.14
		(1.17)	( 1.43)	(1.78)	(2.32)	(2.28)	(2.68)

Table 3 :

 3 Mean and standard deviation errors for the predicted values based on 400 simulation replications with different levels of missing data and a sample size N =

	1400. Partially observed curves are fully observed on [3/8, 6/8] and the error ε ∼
	N (0, σ 2 ε ) with σ ε = 1.5.
	Rate of missing	27.12	27.14	45.16	45.23	54.91	54.84
	data in Y in %	(1.35)	(1.28)	(1.57)	(1.43)	(1.49)	(1.46)
	Rate of missing	29.92	45.16	30.00	45.08	30.06	44.83
	data in X in %	(1.20)	(1.26)	(1.21)	(1.29)	(1.26)	(1.29)
	(FULL) M SE × 10 3	23.52	22.89	27.12	22.68	23.44	24.35
		(18.16) (19.36) (22.32) (19.22)	(18.06)	(21.13)
	RT	1.01	1.01	1.01	1.01	1.01	1.01
		(0.02)	(0.02)	(0.02)	(0.02)	(0.02)	(0.02)
	(REC X IMP Y) M SE × 10 3	37.45	34.22	62.68	57.56	76.55	76.61
		(29.46) (25.32) (42.74) (41.20)	(44.91)	(42.59)
	RT	1.02	1.02	1.03	1.03	1.03	1.04
		(0.02)	(0.02)	(0.03)	(0.03)	(0.03)	(0.03)
	(REC X MEAN IMP Y) M SE × 10 3	38.84	36.41	63.90	57.45	79.56	79.80
		(33.00) (31.36) (50.87) (52.04)	(64.31)	(64.38)
	RT	1.02	1.02	1.03	1.03	1.04	1.04
		(0.02)	(0.02)	(0.03)	(0.03)	(0.04)	(0.04)
	(REC X RAND IMP Y) M SE × 10 3	39.55	35.75	63.88	60.15	77.23	76.95
		(31.29) (27.00) (44.83) (44.00)	(47.60)	(44.50)
	RT	1.02	1.02	1.03	1.03	1.04	1.04
		(0.02)	(0.02)	(0.03)	(0.03)	(0.03)	(0.03)
	(REC X ZERO IMP Y) M SE × 10 2	73.55	73.31	195.88	195.79	286.40	285.60
		(11.01)	(9.76)	(17.75) (17.13)	(20.29)	(18.75)
	RT	1.33	1.34	1.88	1.88	2.30	2.29
		(0.09)	(0.09)	(0.16)	(0.15)	(0.20)	(0.20)
	(REC X DEL Y) M SE × 10 3	44.07	48.93	85.43	81.84	110.13	113.27
		(36.69) (41.30) (67.26) (64.54)	(85.67)	(82.88)
	RT	1.02	1.02	1.04	1.04	1.05	1.05
		(0.03)	(0.03)	(0.04)	(0.04)	(0.05)	(0.05)
	(DEL X DEL Y) M SE × 10 3	63.17	68.44	102.19	115.53	133.12	154.27
		(55.99) (61.75) (81.25) (99.83) (121.22) (125.97)
	RT	1.03	1.03	1.05	1.05	1.06	1.07
		(0.04)	(0.04)	(0.05)	(0.06)	(0.07)	(0.08)

Table 4 :

 4 Mean and standard deviation errors for the predicted values based on 400 simulation replications with different levels of missing data and a sample size N = of the demand). These data are provided from three different publicly available sources: The European Power Exange (www.epexspot.com), the European Network of Transmission System Operators for Electricity (www.entsoe.eu) and the European Energy Exchange (www.eex-transparency.com). The observation period corresponds to n = 241 working days from March 15, 2012 to March 14, 2013. The dataset consists in n = 241 daily electricity prices curves in Germany (measured every hour) in function of the residual electricity demand, which is the relevant value for considering electricity demand. It corresponds to germany's gross electricity demand minus infeeds from renewable energy sources plus net-imports from foreign countries. Some prices greater than 120 EUR/MWh have to be treated as outliers since they cannot

	1400. Partially observed curves are fully observed on [3/8, 6/8] and the error ε equals
	η -0.5 with η ∼ Beta(2, 2).
	Rate of missing	26.98	26.90	45.04	45.06	54.96	54.95
	data in Y in %	(1.38)	(1.27)	(1.50)	(1.37)	(1.52)	(1.43)
	Rate of missing	29.92	45.14	29.89	45.01	30.08	44.92
	data in X in %	(1.22)	(1.34)	(1.26)	(1.31)	(1.15)	(1.23)
	(FULL) M SE × 10 3	19.31	18.89	18.33	18.69	19.26	18.35
		(18.28) (15.85) (16.81) (16.77) (17.99)	(16.03)
	RT	1.38	1.38	1.37	1.37	1.39	1.37
		(0.38)	(0.33)	(0.35)	(0.34)	(0.37)	(0.34)
	(REC X IMP Y) M SE × 10 3	32.16	33.62	48.88	52.85	69.84	68.75
		(29.96) (29.51) (44.84) (51.28) (44.40)	(59.57)
	RT	1.64	1.67	1.98	2.05	2.40	2.39
		(0.62)	(0.60)	(0.92)	(1.03)	(0.90)	(1.21)
	(REC X MEAN IMP Y) M SE × 10 3	32.24	34.30	54.38	56.56	70.74	70.03
		(26.73) (27.56) (39.92) (38.24) (44.25)	(42.84)
	RT	1.64	1.69	2.09	2.15	2.42	2.42
		(0.55)	(0.57)	(0.82)	(0.77)	(0.90)	(0.87)
	(REC X RAND IMP Y) M SE × 10 3	32.25	34.49	53.79	56.45	70.20	69.49
		(26.75) (27.73) (39.75) (38.38) (66.94)	(43.25)
	RT	1.64	1.69	2.08	2.14	2.41	2.41
		(0.55)	(0.57)	(0.82)	(0.78)	(1.38)	(0.88)
	(REC X ZERO IMP Y) M SE × 10 2	72.39	71.81	194.32	194.74	286.33	287.001
		(8.34)	(8.32)	(14.90) (14.08) (17.00)	(16.60)
	RT	15.57	15.47	40.15	39.98	58.39	58.61
		(1.96)	(1.98)	(4.11)	(3.78)	(5.24)	(4.83)
	(REC X DEL Y) M SE × 10 3	40.28	41.28	69.12	74.00	98.81	98.99
		(33.20) (35.94) (53.94) (62.68) (77.89)	(76.62)
	RT	1.81	1.83	2.39	2.48	2.99	3.00
		(0.69)	(0.73)	(1.10)	(1.26)	(1.60)	(1.56)
	(DEL X DEL Y) M SE × 10 3	49.27	53.20	79.46	95.38	110.30	126.28
		(43.41) (49.08) (66.09) (95.39) (90.27) (113.34)
	RT	1.99	2.07	2.60	2.92	3.24	3.58
		(0.91)	(1.00)	(1.37)	(1.92)	(1.89)	(2.39)

  241, where X i is the daily electricity price curve on day i (function of the residual demand), and Y i is the value of electricity production (in MWh) on

		100	Partially Observed Functions
			All Reconstructed Functions
	Electricity Prices (EUR/MWh)	50
		0
		20000	30000	40000	50000	60000	70000	80000
				Electricity Demand (MW)

  θ , X new -θ, X new 2 = E Π kn,rec ∆ n,rec Π kn,rec Γ n,rec Π kn,rec

							-1		2
								X new -ΘX new
	≤ 2E Π kn,rec	1 n	i=1 n	X i , . δ	[Y ]	-1	X new	2
	+ 2E Π kn,rec		1 n	n i=1	X

i Y i Π kn,rec Γ n,rec Π kn,rec i , . Y i,imp (1 -δ [Y ] i ) Π kn,rec Γ n,rec Π kn,rec -1 X new -ΘX new 2

day i. The production data come from https://www.agora-energiewende.de 1 . Only a graphic (with numerical values marked at the observation points) was available on this website to collect a data (neither a table nor an Excel file). It can be possible to use a software to get numerical values from a graphic (see https://automeris.io 2 ).

However, this software is not completely reliable and some numerical values, being not possible, can be considered as missing data for the response variable. In our case, the percentage of missing data is 13.26%.

We split the initial sample into a learning sample (the index set is denoted I L ) with size 181 and a test sample with size 60 (the index set is denoted I T ). Firstly, we reconstructed the missing parts of the differents curves (see Figure 2) and, on the learning sample, we imputed the missing values on the response. Then, on the test sample, we computed the prediction values for the response. In order to evaluate the quality of the prediction with our method (REC X IMP Y), we calculated the mean square error Again, our method performs better than the other ones, even if the differences are 1 https://www.agora-energiewende.de/en/service/recent-electricitydata/chart/power generation/15.03.2012/14.03.2013/ 2 https://automeris.io/WebPlotDigitizer/ sometimes slight. Notice finally that, in this situation, the method (DEL X DEL Y)

would not be possible since all the curves are partially observed and this would cause removing all individuals in the sample.

Proofs

Proof of Proposition 2.1

For any x ∈ H such that x = 1, we have

Using the Cauchy-Schwarz inequality, we get

from which we deduce with (2.3) that

We prove in the same way that

We start with the first term in the above decomposition

Applying Lemma 5.1 in [START_REF] Crambes | Regression imputation in the functional linear model with missing values in the response[END_REF], we obtain

With Lemma 2.2 in Horváth and Kokoszka (2012), we get

Now, we use (2.3) to obtain

Moreover, again with Lemma 5.1 in [START_REF] Crambes | Regression imputation in the functional linear model with missing values in the response[END_REF], we obtain

, which gives the desired result.