Assessing uncertainty in Gaussian mixtures-based entropy estimation

Entropy estimation plays a crucial role in various fields, such as information theory, statistical data science, and machine learning. However, traditional entropy estimation methods often struggle with complex data distributions. Mixture-based estimation of entropy has been recently proposed and gained attention due to its ease of use and accuracy. This paper presents a novel approach to quantify the uncertainty associated with this mixture-based entropy estimation method using weighted likelihood bootstrap. Unlike standard methods, our approach leverages the underlying mixture structure by assigning random weights to observations in a weighted likelihood bootstrap procedure, leading to more accurate uncertainty estimation. The generation of weights is also investigated, leading to the proposal of using weights obtained from a Dirichlet distribution with parameter $\alpha = 0.8137$ instead of the usual $\alpha = 1$. Furthermore, the use of centered percentile intervals emerges as the preferred choice to ensure empirical coverage close to the nominal level. Extensive simulation studies comparing different resampling strategies are presented and results discussed. The proposed approach is illustrated by analyzing the log-returns of daily Gold prices at COMEX for the years 2014--2022, and the Net Rating scores, an advanced statistic used in basketball analytics, for NBA teams with reference to the 2022/23 regular season.