BEAUTY Powered BEAST

Abstract

We study inference about the uniform distribution with the novel binary expansion approximation of uniformity (BEAUTY) approach. Through an extension of the celebrated Euler's formula, we approximate the characteristic function of any copula distribution with a linear combination of means of binary interactions from marginal binary expansions. This novel characterization enables a unification of many important existing tests through an approximation from some quadratic form of symmetry statistics, where the deterministic weight matrix characterizes the power properties of each test. To achieve a uniformly high power, we study test statistics with data-adaptive weights through an oracle approach, referred to as the binary expansion adaptive symmetry test (BEAST). By utilizing the properties of the binary expansion filtration, we show that the Neyman-Pearson test of uniformity can be approximated by an oracle weighted sum of symmetry statistics. The BEAST with this oracle leads all existing tests we considered in empirical power against all complex forms of alternatives. By approximating this oracle with data-adaptive weights, we develop the BEAST that improves the empirical power of many existing tests against a wide spectrum of common alternatives while providing clear interpretation of the form of non-uniformity upon rejection.

The AUGUST Two-Sample Test: Powerful, Interpretable, and Fast

“…In practice, d = 3 should be sufficient for global distributional differences, which existing ECDF-based methods can detect. In Zhang [15], it is shown that a depth of d = 3 is sufficient for a symmetry statistic-based test of independence to outperform both distance correlation and F -test, which are known to be optimal, in detecting correlation in bivariate normal distributions.…”

Data-Driven Representations for Testing Independence: A Connection with Mutual Information Estimation

“…The authors in [34] have recently introduced a non-parametric test with a similar oracle-base design principle.…”

“…That strategy has an interesting connection with our design approach in Section IV because they adapt key parameters of their method to approximate a best (oracle) statistic. On the differences, the authors in [34] addressed the independence test by performing a binary expansion filtration of the observations and adapting the weights matrix of a quadratic form. Instead, our work addresses the independence test by estimating the mutual information (our statistics) through an adaptive tree-structured partition of the observation space.…”

“…This work proposed a binary expansion filtration of the space and deeply studied some symmetric properties to create a novel test that is uniformly consistent (on the power loss) and minimax for the power (in the sample size) assuming a large class of alternative distributions. Zhang et al [34] presented a novel extension of the binary expansion filtration strategy [22] to a multidimensional setting by cleverly approximating an oracle Neyman-Pearson (NP) statistic. They propose a data-adaptive weight strategy (for the binary expansion) to approximate the ideal weight of the NP test.…”

Comments on: A Gibbs sampler for a class of random convex polytopes

“…One general approach in nonparametric tests of independence is the multi-resolution approach. See some recent work by Ma and Mao (2019); Zhang (2019); Lee et al (2019); Gorsky and Ma (2018); Zhang et al (2021). Some advantages of this approach include uniform consistency, minimax optimal power, clear interpretability and efficient computation.…”