  Citation: Almanjahie, I.M.; Kaid, Z.; Laksaci, A.; Rachdi, M. Estimating the Conditional Density in Scalar-On-Function Regression Structure: k-N-N Local Linear Approach. Mathematics 2022, 10, 902. https://doi.org/10.3390/math 10060902 Academic Editor: Ana M. Aguilera Received: 4 February 2022 Accepted: 5 March 2022 Published: 11 March 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). mathematics Article Estimating the Conditional Density in Scalar-On-Function Regression Structure: k-N-N Local Linear Approach Ibrahim M. Almanjahie 1,2, * , Zoulikha Kaid 1,2 , Ali Laksaci 1,2 and Mustapha Rachdi 3 1 Department of Mathematics, College of Science, King Khalid University, Abha 62529, Saudi Arabia; kaedzoulekha@yahoo.fr (Z.K.); alilak@yahoo.fr (A.L.) 2 Statistical Research and Studies Support Unit, King Khalid University, Abha 62529, Saudi Arabia 3 Laboratoire AGEIS, Université Grenoble Alpes (France), EA 7407, AGIM Team, UFR SHS, BP. 47, CEDEX 09, F38040 Grenoble, France; mustapha.rachdi@univ-grenoble-alpes.fr * Correspondence: imalmanjahi@kku.edu.sa Abstract: In this study, the problem of conditional density estimation of a scalar response variable, given a functional covariable, is considered. A new estimator is proposed by combining the k-nearest neighbors (k-N-N) procedure with the local linear approach. Then, the uniform consistency in the number of neighbors (UNN) of the proposed estimator is established. Such result is useful in the study of some data-driven rules. As a direct application and consequence of the conditional density estimation, we derive the UNN consistency of the conditional mode function estimator. Finally, to highlight the efficiency and superiority of the obtained results, we applied our new estimator to real data and compare it to its existing competitive estimator. Keywords: functional mixing data; complete convergence (a.co.); local linear method; distribution function; kernel weighting; conditional predictive region; k nearest neighbors smoothing (k-N-N) 1. Introduction Functional data analysis (FDA) has acquired a great deal of consideration in the last few years. We cite, for instance, some precursor contributions developed by [1–3], as well as some recent advances published in the special issues [4,5]. In this data analysis area, the nonparametric fitting is actually in intensive development (see, for examples, [6,7], among others). While many of the works in this field consider nonparametric estimation of the functional models by employing the kernel estimation method, we are interested in estimating the conditional density using the method of local linear (L-L) estimation smoothed by the k-nearest neighbors procedure (k-N-N). The subject of this paper is an intersection between three emergent fields in statistical mathematics: functional statistics, local linear modeling, and the nearest number neighbor- hood. The first axis regarding the functional statistics analysis is in a colossal development. Indeed, to cite a few recent advances in this year, we refer to [8], who proposed to estimate the mean and the covariance function using the full quasi-likelihood and the kernel method. Ref. [9] used the functional Horvitz–Thompson estimator to approximate the confidence bands for the mean of curves. As a real-life application, they used their approach to fit the tourists’ daily expenditure in the Majella National Park in Italy. We refer to [10] for the quantile estimation in multivariate functional data. They constructed a new estimator obtained by combining nonparametric techniques, the time-varying coefficient model, and basic functions. Such an estimator is employed in air pollution forecasting and is used for estimation and prediction. For practical purposes, [11] provided a new clustering algo- rithm, allowing them to proceed without the normality assumption of the random function. Concerning the local linear analysis, a recent advance can be found in [12]. They studied an estimation of the volume under an ROC surface using a semiparametric regression model. Mathematics 2022, 10, 902. https://doi.org/10.3390/math10060902 https://www.mdpi.com/journal/mathematics