Assessing goodness-of-fit for sparse categories using Rényi divergence

We present the Rényi divergence as a statistic for assessing goodness-of-fit in sparse frequency tables, where small expected counts can undermine the reliability of the traditional chi-square test. The Rényi divergence with index in (0,1) is a natural choice because it circumvents division-related issues by small frequencies. Our main result demonstrates that the Rényi statistic asymptotically follows a chi-square distribution. Through theoretical insights and Monte Carlo simulations, we evaluate the performance of the Rényi statistic across various values of the divergence index. We find that smaller index values improve the alignment of the Rényi statistic with the chi-square distribution and enhance its performance in sparse data settings. Additionally, the Rényi statistic exhibits good power properties in detecting deviations from the null hypothesis under these conditions. To illustrate its practical applicability, we present two real-world data analyses, highlighting the robustness of the Rényi divergence in scenarios involving sparse categories.