A test for the two-sample problem based on empirical characteristic functions

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Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, background): “Another popular approach for representing a probability distribution is through the empirical characteristic function (ECF) (Feuerverger and Mureika 1977, Alba Fernàndez et al. 2008).”
Fast Two-Sample Testing with Analytic Representations of Probability Measures (2015, methodology): “This same weakness was used by [1] in justifying a test that integrates over the entire frequency domain (albeit at cost quadratic in the sample size), for which the quadratic-time MMD is a generalization [8].”
On Wasserstein Two-Sample Testing and Related Families of Nonparametric Tests (2015, background): “Another popular class of tests that has emerged over the last decade are kernel-based tests introduced independently by [37,38], and expanded on in [11].”
Learning Deep Kernels for Non-Parametric Two-Sample Tests (2020, methodology): “This general method (Dwass, 1957; Alba Fern ´ andez et al., 2008) observes that under H 0 , the samples from P and Q are inter-changeable; we can therefore estimate the null distribution of our test statistic by repeatedly re-computing it with the samples randomly re-assigned to S or S .”
Generative Models and Model Criticism via Optimized Maximum Mean Discrepancy (2016, unknown): “The second step follows by (3) and the convergence of ˆ c α → c α (Alba Fernández et al., 2008).”