Estimating the Conditional Density in Scalar-On-Function Regression Structure: k-N-N Local Linear Approach

Ibrahim M Almanjahie,Ali Laksaci,Zoulikha Kaid,Mustapha Rachdi

doi:10.3390/math10060902

Ibrahim M Almanjahie, Ali Laksaci + Show 2 more

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https://doi.org/10.3390/math10060902

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Journal: Mathematics	Publication Date: Mar 11, 2022
Citations: 3	License type: CC BY 4.0

Affiliation: King Khalid University, Université Grenoble Alpes

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.

Highlights

Functional data analysis (FDA) has acquired a great deal of consideration in the last few years
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)
Concerning the functional L-L estimator (L-L-E) of the conditional density, the literature is limited; only the authors of [34] have proposed an estimator based on the Hilbertian version of the local linear approach introduced by [33]

Summary

Introduction

Functional data analysis (FDA) has acquired a great deal of consideration in the last few years. 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 in intensive development (see, for examples, [6,7], among others). 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. 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

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