Split empirical likelihood via universal inference for bounded means

Standard empirical likelihood (EL) inference relies on Wilks’ theorem for asymptotic calibration, which often performs poorly under small samples or skewed data and can lead to substantial under-coverage. To address this issue, we propose a split-dual empirical likelihood method for constructing finite-sample valid confidence intervals. Our approach partitions the data into training and test sets: the training set is used to optimize the dual parameter, while the test set evaluates the corresponding betting score. By constraining the dual parameter to a global safe region determined by the data bounds, we ensure nonnegative e-values and hence strict validity without asymptotic approximations. Simulations on bounded distributions show that the proposed method maintains nominal coverage across sample sizes and provides a robust alternative to standard EL and existing betting-based methods.