A weighted empirical likelihood ratio test for normality

We propose a weighted empirical likelihood ratio (ELR) test for normality based on moment constraints, which is designed to achieve high power against symmetric short-tailed alternatives. To achieve this, the study exploits the characterization of the normal distribution, which incorporates the idea of symmetry, asymptotic tails, and the concept of the empirical rule through observation-weighted empirical likelihood ratios, which enhance sensitivity to symmetric deviations while maintaining robustness. The resulting test statistic has good control of the Type I error rate and is shown to have a limiting weighted χ2 distribution for the overall test statistic. In addition, the divergence of the weighted ELR test statistic under a wide range of fixed alternatives provides the theoretical foundation for the test's consistency. Monte Carlo simulations revealed that the proposed testing procedure outperformed the studied traditional tests under symmetric short-tailed alternatives. The findings of some empirical studies further reveal the superiority, robustness, and applicability of the proposed test statistic in practice.