Gap-Entropy Testing (GET): A Framework for Structural and Distributional Differences

The classical two-sample problem is framed as the detection of differences in marginal distributions, in terms of location, scale, or overall shape. In this paper, Gap–Entropy Testing (GET) is proposed, a novel nonparametric framework that extends the two-sample paradigm by adding a spacing-sensitive component to conventional distributional analysis. The approach is based on order statistics, log1p-transformed adjacent spacings normalized by their median gap, and entropy estimation using a Vasicek-type estimator. Inference is performed by permutation testing under a common continuous i.i.d. null. Three complementary statistics were developed: GET-V compares spacing-entropy estimates; GET-Plus combines spacing and median-location components; and GET-Ω additionally incorporates energy distance. Under the common i.i.d. null, GET-V has a type-I error close to the nominal 0.05 level and is insensitive to a pure location shift. For the lattice and micro-cluster example, GET-V has large rejection rateσ, which is related to the spacing regularity and the organization of the local gap. GET-V is also sensitive to marginal shape, and individual-label permutation is anti-conservative under matched lattice and shifted-lattice nulls. Application to public benchmark datasets allows us to use GET-V for continuous variables without ties. GET is a reproducible and interpretable framework that adds spacing information to standard two-sample comparisons.