Logarithmic energy distances and Gini covariance for Hilbert-valued random elements

Technical Summary: Logarithmic Energy Distances and Gini Covariance for Hilbert-valued Random Elements

Problem Statement
The paper investigates the boundary regime of generalized energy distances and Gini covariance statistics for random elements taking values in a real separable Hilbert space $H$. These statistics are traditionally defined using power-distance kernels of the form $(x, y) \mapsto \|x - y\|^\alpha$ for $\alpha \in (0, 2)$. While these methods are well-established for nonparametric inference and goodness-of-fit testing, the behavior of these statistics as the parameter approaches the lower boundary, $\alpha \downarrow 0$, had not been fully characterized in infinite-dimensional settings. The authors aim to determine the limiting object of this family of kernels and investigate its statistical properties, specifically for the two-sample and $k$-sample problems.

Methodology
The core methodological approach relies on the elementary expansion of the power function near zero: $\|x - y\|^\alpha = 1 + \alpha \log \|x - y\| + o(\alpha)$ as $\alpha \downarrow 0$. By applying this expansion to the definition of the generalized energy distance $E_\alpha(X, Y)$, the authors show that after canceling constant terms and suitable normalization (dividing by $\alpha$), the distance converges to a logarithmic energy distance:
$$E_0(X, Y) = 2\mathbb{E}[\log \|X - Y\|] - \mathbb{E}[\log \|X - X'\|] - \mathbb{E}[\log \|Y - Y'\|],$$
where $X'$ and $Y'$ are independent copies of $X$ and $Y$.

The authors extend this concept to the $k$-sample problem by defining a logarithmic Gini covariance, $c_{\log}$, which serves as a weighted sum of pairwise logarithmic energy distances. The theoretical development proceeds in several steps:

1. Characterization: The paper establishes that $E_0(X, Y) = 0$ if and only if the distributions of $X$ and $Y$ are identical. The proof utilizes Frullani's formula to represent the logarithmic distance as an integral of squared Maximum Mean Discrepancies (MMD) associated with Gaussian kernels $k_z(x, y) = \exp(-z\|x - y\|^2)$. This representation leverages Schoenberg's theorem and the fact that probability measures on a separable Hilbert space are determined by their finite-dimensional marginals.
2. Structural Properties: The logarithmic Gini covariance is shown to be invariant under common translations and scale changes, a property that distinguishes it from the standard power-kernel Gini covariance (which scales by $b^\alpha$).
3. Asymptotic Theory:
   • Under the Null Hypothesis ($H_0$): For fixed $k$, the empirical logarithmic Gini covariance is asymptotically degenerate. Its limiting distribution is derived as an infinite weighted sum of centered chi-square random variables, where the weights are the eigenvalues of an integral operator associated with the kernel $K_0(x, y)$.
   • Under Alternatives: When distributions differ, the statistic is non-degenerate. The paper proves asymptotic normality for the empirical statistic under fixed alternatives, establishing consistency.
4. Implementation: Due to the dependence of the null distribution on unknown eigenvalues, the authors propose a permutation-based procedure for calibrating the test statistic.

Key Contributions
The paper makes three primary contributions:

1. Introduction of Logarithmic Analogues: It formally introduces logarithmic energy distances and logarithmic Gini covariance as the natural boundary objects of the power-distance family when $\alpha \downarrow 0$.
2. Characterization in Hilbert Spaces: It proves that the logarithmic energy distance retains the fundamental characterization property ($E_0 = 0 \iff P=Q$) in separable Hilbert spaces, linking it to Gaussian-kernel MMDs. This extends the applicability of energy statistics to functional data without requiring modification of the methodology.
3. Asymptotic and Practical Framework: It derives the asymptotic null and alternative theories for the empirical logarithmic Gini covariance in a fixed-$k$ framework and provides a practical permutation-based implementation strategy.

Results
The authors present numerical results from a simulation study and real-data applications:

• Simulation: The study compares the proposed logarithmic tests ($E_0$ and a variant $E_{\log}$) against the standard energy-distance test ($E_1$) and another competitor. The results indicate that while all tests maintain the nominal significance level, the logarithmic procedures exhibit substantially higher power against scale and shape alternatives, particularly those involving changes in covariance structure, departures from normality, or differences in relative dispersion. The gains are noted to be more pronounced in higher dimensions and functional data settings.
• Real Data Applications:
  • Iris and Wine Datasets: The tests successfully reject the null hypothesis of equal distributions for the Fisher Iris and Wine datasets, consistent with known differences in these multivariate samples.
  • Berkeley Growth Data: Applied to functional data (growth curves of boys and girls), the logarithmic tests detect distributional differences in specific age intervals ([1, 4), [13, 18), and [1, 18]) where mean functions also differ, while correctly failing to reject equality in the interval [4, 13) where mean functions are similar.

Significance and Claims
The paper claims that the logarithmic framework reveals a "new boundary phenomenon" within the family of energy-type statistics. Its significance lies in:

• Functional Data Analysis: The Hilbert-space formulation allows for the direct treatment of functional data, avoiding the need for dimension reduction or specific modifications to the methodology.
• Sensitivity to Geometric Structure: Unlike ordinary energy kernels which emphasize global separation, the logarithmic kernel is described as being more sensitive to relative geometric structure and local interactions. This provides a complementary tool to existing energy statistics, particularly for detecting non-location alternatives.
• Connection to Kernel Methods: The derivation establishes a theoretical link between logarithmic energy distances and Gaussian-kernel maximum mean discrepancies, bridging different areas of nonparametric inference.

The authors conclude modestly, noting that the work opens several avenues for future research, including extensions to general Banach spaces, high-dimensional asymptotic regimes ($d_N \gg N$), and adaptive procedures combining multiple values of $\alpha$.