A density based empirical likelihood approach for testing bivariate normality.

Area of Science:

• Statistics
• Nonparametric Statistics
• Biostatistics

Background:

• One-dimensional normality tests like sample entropy, sieves, and Grenander estimation are established nonparametric tools.
• Density-based empirical likelihood (EL) offers a standardized, distribution-free method for approximating optimal parametric likelihood ratio tests.
• Extending these methods to bivariate settings presents significant challenges.

Purpose of the Study:

• To address the difficulties in constructing density-based empirical likelihood ratio techniques for bivariate normality testing.
• To propose and validate a novel approach for bivariate normality assessment.

Main Methods:

• Development of a novel bivariate sample entropy expression.
• Demonstration that the new expression aligns with bivariate histogram density estimation principles.
• Application of Monte Carlo simulations to evaluate the proposed density-based empirical likelihood ratio tests.

Main Results:

• The newly derived bivariate sample entropy satisfies established density estimation concepts.
• The proposed density-based empirical likelihood ratio tests for bivariate normality exhibit excellent performance in finite sample sizes.
• Successful application of the method to real-world data concerning biomarkers for myocardial infarction.

Conclusions:

• The novel density-based empirical likelihood approach provides a robust solution for bivariate normality testing.
• The method demonstrates practical utility and effectiveness, as shown by simulation and real data analysis.
• This work extends powerful nonparametric testing methodologies to a crucial bivariate context.