We study distribution-free goodness-of-fit tests with the proposed Binary Expansion Approximation of UniformiTY (BEAUTY) approach. This method generalizes the renowned Euler's formula, and approximates the characteristic function of any copula through a linear combination of expectations of binary interactions from marginal binary expansions. This novel theory enables a unification of many important tests of independence via approximations from specific quadratic forms of symmetry statistics, where the deterministic weight matrix characterizes the power properties of each test. To achieve a robust power, we examine test statistics with data-adaptive weights, referred to as the Binary Expansion Adaptive Symmetry Test (BEAST). For any given alternative, we demonstrate that the Neyman-Pearson test can be approximated by an oracle weighted sum of symmetry statistics. The BEAST with this oracle provides a useful benchmark of feasible power. To approach this oracle power, we devise the BEAST through a regularized resampling approximation of the oracle test. The BEAST improves the empirical power of many existing tests against a wide spectrum of common alternatives and delivers a clear interpretation of dependency forms when significant.

翻译：我们采用提出的二元展开均匀性逼近（BEAUTY）方法研究无分布拟合优度检验。该方法推广了著名的欧拉公式，通过边缘二元展开中二元交互作用期望的线性组合来逼近任何联结函数的特征函数。这一新颖理论使得许多重要的独立性检验能够通过对称统计量的特定二次型逼近实现统一，其中确定性权重矩阵刻画了每个检验的势特性。为获得稳健的势，我们研究了具有数据自适应权重的检验统计量，称为二元展开自适应对称检验（BEAST）。对于任意给定备择假设，我们证明Neyman-Pearson检验可通过对称统计量的理想加权和逼近。采用该理想权重的BEAST为可行势提供了有效的基准。为逼近该理想势，我们通过理想检验的正则化重抽样逼近来构建BEAST。BEAST在广泛常见备择假设下提升了众多现有检验的经验势，并在结果显著时对依赖形式给出清晰解释。