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

Area of Science:

• Statistics
• Nonparametric Regression
• Computational Statistics

Background:

• Regression analysis on the simplex is crucial for modeling compositional data.
• Existing methods often struggle with boundary properties.
• Local polynomial smoothing offers improved performance over local constant methods.

Purpose of the Study:

• To introduce a novel local linear smoother for regression on the simplex.
• To analyze the asymptotic properties of the proposed estimator.
• To compare its performance against existing estimators.

Main Methods:

• Developing a local linear smoother using a weighted least-squares approach.
• Employing a locally adaptive Dirichlet kernel for weighting.
• Deriving asymptotic results for bias, variance, and mean squared error.
• Conducting simulation studies for performance evaluation.

Main Results:

• The proposed local linear smoother exhibits favorable boundary properties.
• Asymptotic properties (bias, variance, MSE, MISE) are theoretically established.
• Simulation results indicate the new estimator outperforms the Nadaraya-Watson estimator with a Dirichlet kernel.

Conclusions:

• The local linear smoother with a Dirichlet kernel is an effective method for regression on the simplex.
• The theoretical and simulation results support its practical applicability.
• This work extends univariate smoothing results to the multivariate simplex domain.