A Dirichlet-kernel Gasser-Müller (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.

翻译：针对单形上的固定设计回归问题，引入Dirichlet核Gasser-Müller（D-GM）估计量，将Chen [Statist. Sinica, vol. 10(1) (2000), pp. 73-91]提出的单变量类似方法进行推广。本文研究了该估计量的逐点偏差与方差、渐近正态性以及积分均方误差。通过仿真实验，将其小样本性能与两种近期提出的替代方法——Dirichlet核Nadaraya-Watson（D-NW）估计量和局部线性（D-LL）估计量进行对比。仿真结果表明，D-LL估计量在D-LL、D-NW与D-GM三类估计量中表现最优，而本文提出的D-GM估计量表现最差。此外，基于GEMAS数据集开展实际数据分析，探究欧洲各类农业用地和牧场上土壤成分与pH值之间的关系。