The Estimating of the Conditional Density with Application to the Mode Function in Scalar-On-Function Regression Structure: Local Linear Approach with Missing at Random *

: The aim of this research was to study a nonparametric estimator of the density and mode function of a scalar response variable given a functional variable, when the observations are i


Introduction
It is very well recognised that robust regression in statistics is an attractive research method.It is used to overcome some of the weaknesses of classical regression, namely when outliers contain heteroscedastic data.
The study of the connection between a random variable  and set of covariates  is a common problem in statistics.In the literature, these variables are generally known as functional variables.Remember that the robustification method is an old statistical issue, investigated first by Hurber (1964) who studied an estimation of alocation parameter (see also (Collomb & Härdale, 1979;Laib & Ouled-Said 2000)) for some results containing the multivariate time series case under a mixing or an ergodic condition).
Note that many writers have tackled the functional local linear estimating (LLE) problem since the appearance of the monograph by Ferraty and Vieu (2006) in NFDA, which relates to some responses to question number five given there, local linear estimation technique has several advantages in the case of finite-dimensional data over the kernel method, such as bias reduction and the adaptation of edge effects.Ballo and Grané (2009) provided the first response to this question, demonstrating the  2 -consistency of an LLE of the Hilbertian regression operator, whereas Barrientos et al. (2010) presented a different solution that may be used to a more general functional regressor.In turn, Demongeot et al. (2013) developed the first results on the LLE of the conditional density using functional variables.In this, the (a.co.) almost--complete consistency of an LLE of the conditional density was proved (see, p. 230, definition A.3 in Ferraty and Vieu (2006)).In Rachdi, Laksaci, Demongeot et al. (2014), the leading term of the mean quadratic error of the LLE of the conditional density was then explicitly mentioned.Additionally, Zhou and Lin (2016) established the asymptotic normality of the LLE of the regression operator, and recently Almanjahie et al. (2022) treated the k Nearest Neighbour kNN LLE of the conditional density in a scalar-On-Function regression structure, for additional research on the LLE in NFDA (Nonparametric Functional Data Analysis), see, Chahad et al. (2017); Attouch et al. (2017).
Note that the LLE's significant advantages over the traditional kernel technique function as the primary inspiration for all of the cited works on functional LLE.Specifically, the kernel method's bias error is reduced using the LLE approach (see, for instance (Fan & Gijbels, 1996;Rachdi et al., 2014) for more discussions on advantages of this method).
The primary innovation in this study is the estimation of the conditional density mode using the LLE approach weighted by MAR procedure for the functional and i.i.d data, while many studies in this field use the kernel estimation approach to estimate nonparametric functional models.
There are many references on the estimation of the conditional density, and there has been an increasing interest in the study of functional variables, the prime results about the conditional mode estimation obtained by Ferraty and Vieu (2006), about the almost complete convergence of the kernel estimator, and in the both cases i.i.d. and strong mixing, Ezzahrioui and Ould-Sa𝑖𝑖d (2006) established the asymptotic normality of the kernel estimator of the conditional mode.Lately the nonparametric estimation of the conditional mode using kNN method for independent and dependent functional data was examined by Attouch and Bouabsa (2013) and Attouch, Bouabsa and Chiker el mozoaur (2018).One can also mention for example Amiri and Dabo-Niang (2016) who investigated the recursive version of the kernel estimator of the density function.for spatial data, they stated the asymptotic qualities of the built estimator.The author referred to Giraldo et al. (2018), for a thorough discussion of spatio-functional data analysis and its viability as an attractive method in spatial big data modeling.Taking the ergodicity structure into account, Chaouch and La𝑖𝑖b (2017) demonstrated the uniform consistency of the nonparametric estimator of functional modal regression.Maillot and Chesneau (2018) established the asymptotic properties of the conditional density for continuous time processes valued in functional space for current results.They also used this to calculate the convergence of the conditional mode and conditional expectation.Kirkby et al. (2021) investigated an alternate estimate of the conditional density, and employed the Galerkin method was to create an effective cross-validation technique for bandwidth parameter choice.This study's innovation lies in its combination of the LLE technique with MAR data to provide a new estimator of functional conditional density.The rest of the paper is organized as follows.Section 2, presents for functional data the local linear estimation of the conditional mode.The principal hypotheses and notations are given in Section 3, All the results and proof are determined in Section 4. Section 5 presents a simulation study to prove the effectiveness of this study.The conclusions and prospects for future research are provided in the last section.

The model
(  ,   ),  = 1, . .,  is n copies of random vector identically distributed as (, ),  is valued in infinite dimensional semimetric vector space (, d), and  is valued in ℝ,  in handy applications is a normed space that can be of infinite dimension, for example Banach or Hilbert space with norm ∥. ∥ that (, ′) =∥  − ′ ∥.
Recall that   (resp () ) isconditional density (resp its  ℎ order derivative) of .The article presents the almost complete convergence results and rates for nonparametric estimates of  () .Thus when   =  (0) , the convergence of conditional density estimate is deduced instantly from the general results concerning  () .
Moreover, density   is continuous with respect to Lebesgue's measure over ℝ.In the local linear modeling the estimation of conditional mode function  �  (()) is based on the approximation of conditional density function   () by a linear function.To expand the local linear ideas to the infinite dimensional framework the author used the technique of Baìllo and Grané (2009), (see Barrientos-Marin et al. (2010) for some examples and references).Function   (. ) as marked by Fan and Gijbels (1996) can be viewed as a nonparametric regression model with response variable ℎ  −  ()   �ℎ  −1 (. −  )� where  is some cumulative distribution function and ℎ  is a sequence of positive real numbers.This estimation is motivated by the fact that: Here, let us adopted fast functional local modelling, i.e. conditional density function  �  is estimated by  � where the pair ( �;  �) is obtained by the optimisation rule: where, ℓ(. ; .) and ℘(. ; .) are locating functions defined from ℱ 2 into ℝ, such that: ∀ ∈ ℱ; ℓ(; ) = 0  (.; .) = |℘(.; .)| and where function : kernel function, : distribution function (df) and ℎ = ℎ  : = ℎ , and ℎ  = ℎ , are suites of positive real numbers, as  goes to infinity, and goes to zero.Clearly, estimator  �, given by (3.1), can be explicitly written as follows where,  (1) is the derivative of Q, with and ℓ  = ℓ(  , ) and convention 0/0 = 0; Suppose that   (. ) has an only mode, noted by () assumed uniquely defined in compact set  which is given by (2.3) () is a kernel estimator of the conditional mode which given as random variable  ̂() that maximises kernel estimator  �  (. ) of   (. ) (2.4)
The convergence of the estimator can be obtained under minimal assumption (A6).
(A7) and (A8) are classical hypotheses in the functional estimation in finite or infinite dimension spaces,while assumption (A9) is a supposition for missing at random, hence it is a technical condition for the concision of the proof of the main results.

Result and proof
The following denotes by C and C' some strictly positive generic constants.and defines the quantities, for any  ∈ ℱ and for all  = 1, ⋯ , .  = (ℎ  −1 ℘(,   )) and (5.1) The uniform complete convergence of the conditional density estimate beyond the compact set | − ,  + | (see 1 below) can be used leading directly from both precedent inequalities to Finally, the following consistency result is checked as long as the following Lemmas could be verified.
for some  * between  and  ̂, use (5.3), as along as it is possible to check that (5.4) Hence so it suffices to check (5.4), and this is done directly by using the second part of (A8) together with (1).
The proof is a direct consequence of the following decomposition: where ∑ ≠   (), and and of the Lemmas below.Lemma 5.1.Under assumptions (A1), (A3), and (A4) one has

Proof
From assumption (A4), and when pairs (  ,   ) are identically distributed, one obtains By using classical change of variables  =

𝑏𝑏−𝑑𝑑 ℎ 𝑄𝑄
, one obtains and by supposition (A2) Thus we obtain the claimed result of this Lemma from the direct consequence ofassumption (A5), and  (1) is a probability density.

Proof
The demonstration of this lemma is given by a straightforward adaptation of the proof of Lemma 5.2 in (Barrientos-Marin et al., 2010) by writing where With the similar method utilised above, one can show that (5.6) (5.7) and (5.9) First, note that (5.7) was already set by Barrientos et al., (2010).Thus, it suffices to show (5.6), (5.8) and (5.9) to finish the proof of the Lemma.To show the result (5.6), set where By (A3), one has ] ≤ ′   (ℎ  ) .Thus, using the classical Bernstein's inequality (Uspensky, 1937, p. 205) allows to give for all  ∈ (0; and the relevant choice of  permits to subtract that In fact, one has to evaluate for both results �     ℓ   �  = 0; 1; 2;   = 0; 1.

Application: simulated data
The aim of this simulation was to prove the effectiveness of this study by comparing the classical regression and the conditional mode with MAR in the presence of or without outliers.
Functional variable  is taken as a function with support [0,1] based on the following observation where   are i.i.d.∼ (0,1) and   are i.i.d.∼ (0,1), and are independent from   and   .For simplicity, Figure 1   and the mode estimator, and  �  () defined in (6.1).
One splits the sample of size 200 into a learning sub-sample (  ,   ),  = 1, … , 150 and a testing sub-sample �  ,   �,  = 151, … , 200.For the missing mechanism, the author adopted the following expression where, exp () =   /(1 +   ) for ∀ ∈ ℝ.To control the quantity (), compute ℘ � = 1 − 1 150 ∑ 150 =1 ℘  .For optimal bandwidth ℎ  , choose the automatic selection 2) Incomplete case response MAR: the obtained results in Figure 3 show that the mode estimation is better than the classical kernel regression; i.e. the classical kernel method with MAR.Now, compare the performance of both estimators(classic and mode) in the presence of outliers.To do this, the author introduced artificial outliers by multiplying some values of responses  by 100 with a fixed degree of dependence ().The mode estimator has a better performance than the classical one, even if the MMSE of both estimators increases substantially relative to the number of outliers, but it remains very low for the mode method, as shown in Table 2.The mode estimator with MAR has a better performance than the classical MAR, even if the MMSE of both estimators increases substantially relative to the number of outliers, but it remains very low for the mode MAR method, as shown in Table 1.

Conclusion
In this paper, the author studied the problem of the nonparametric estimation of the conditional density with application to the mode function using the local linear approach.The feature of this study is the possibility to cover the incomplete data situation characterised by the missing phenomena.Empirical analysis showed the excellent performance of the proposed methodology, which varied with respect to the missing level.In addition to these features, the presented study opened up some significant avenues for the future research.In particular, it will be interesting to investigate other types of incomplete functional data, such as censored or truncated data.Another possible direction is to study the asymptotic property of kNN local linear estimator in the functional times series case (complete or incomplete cases).In addition, the asymptotic distribution of the proposed estimators is an interesting open question.Such asymptotic property is essential as preliminary statistical analyses, including the confidence interval or hypotheses testing.Moreover, extending this kind of estimation to other nonparametric models, such as the conditional hazard function or the conditional distribution function, is also a natural prospect of this study.

Table 1 .
MMSE comparison between both methods for the combinations of parameters ()

Table 2 .
MMSE for the Classical Kernel Estimator and the mode Estimator according to the numbers of the introduced artificial outliers