Local linear smoothing for regression surfaces on the simplex using Dirichlet kernels

TL;DR

This paper develops a local linear smoothing method for regression surfaces on the simplex using Dirichlet kernels, providing asymptotic analysis and demonstrating superior performance over existing estimators through simulations.

Contribution

It introduces a novel local linear estimator on the simplex with Dirichlet kernels, extending univariate results and improving upon existing methods.

Findings

The estimator has favorable bias and variance properties.

Asymptotic results generalize previous univariate findings.

Simulation shows improved accuracy over existing estimators.

Abstract

This paper introduces a local linear smoother for regression surfaces on the simplex. The estimator solves a least-squares regression problem weighted by a locally adaptive Dirichlet kernel, ensuring good boundary properties. Asymptotic results for the bias, variance, mean squared error, and mean integrated squared error are derived, generalizing the univariate results of Chen [Ann. Inst. Statist. Math., 54(2) (2002), pp. 312-323]. A simulation study shows that the proposed local linear estimator with Dirichlet kernel outperforms its only direct competitor in the literature, the Nadaraya-Watson estimator with Dirichlet kernel due to Bouzebda, Nezzal and Elhattab [AIMS Math., 9(9) (2024), pp. 26195-26282].