On the Dirichlet-kernel Gasser--M\"uller estimator and its competitors for fixed design regression on the simplex

TL;DR

This paper introduces a Dirichlet-kernel Gasser-M"uller estimator for fixed design regression on the simplex, analyzing its statistical properties and comparing its performance with alternative estimators through simulations and real data analysis.

Contribution

It extends the Gasser-M"uller estimator to the simplex, providing theoretical analysis and empirical comparison with recent competitors.

Findings

The D-LL estimator outperforms D-GM and D-NW in simulations.

D-GM estimator has the worst small-sample performance among the three.

Real data analysis demonstrates the practical application of the estimators.

Abstract

A Dirichlet-kernel Gasser-M\"uller (D-GM) estimator is introduced for fixed design regression on the simplex, extending the univariate analog due to Chen [Statist. Sinica, vol. 10(1) (2000), pp. 73-91]. Its pointwise bias and variance, asymptotic normality, and mean integrated squared error are investigated. Some simulation experiments are conducted to compare its small-sample performance with that of two recently proposed alternatives: the Dirichlet-kernel Nadaraya-Watson (D-NW) and local linear (D-LL) estimators. The simulation results reveal that the D-LL estimator is best among the D-LL, D-NW, and D-GM estimators and that the proposed D-GM estimator is worst. A real data analysis is also reported for the GEMAS dataset to analyze the relationship between soil composition and pH levels across various agricultural and grazing lands in Europe.