arXiv subjects

Luis-Alberto Rodríguez

Publications and source records attributed to Luis-Alberto Rodríguez.

5 recordsLinked to original sources

arXiv2026

A Functional Central Limit Theorem for Locally Stationary Time Series in Banach Spaces

A functional central limit theorem for locally stationary time series taking values in a separable Banach space $B$ is established. The result does not require type-2 or cotype assumptions and therefore covers spaces central to functional data analysis, including $C([0,1])$ and $L^p([0,1])$. Under moment conditions, summable physical dependence coefficients, and a bracketing entropy condition controlling the infinite-dimensional tails, the centered and rescaled partial sum process converges weakly in $D([0,1],B)$. The limit is a centered $B$-valued Gaussian process whose covariance is given by the integral of the local long-run covariance, interpreted as an element of the projective tensor product. We also obtain a stochastic integral representation with respect to a cylindrical Brownian motion, connecting the Banach-space limit to the familiar scalar locally stationary structure. As an application, we derive a self-normalized CUSUM procedure for detecting changes in the mean of linear projections of Banach-valued observations, yielding a pivotal asymptotic null distribution. The finite-sample behavior is illustrated through Monte Carlo experiments and exploratory applications to EEG recordings and daily temperature curves. Examples based on the Faber-Schauder system in $C([0,1])$ and on a $p$-Laplacian model in $W^{1,p}_0([0,1])$ demonstrate how the entropy condition can be verified.

math.PR↗

arXiv2026

Gromov-Wasserstein Barycenter Surrogates: Statistical Methodology, Distributional Limits and Applications

We introduce statistical theory for the matching of finitely many objects, represented as metric measure spaces (mm-spaces). The approach is based on the second lower bound (SLB) of the Gromov-Wasserstein distance and thus is able to identify deviations in the distributions of the (pairwise) distances within each mm-space. We introduce a surrogate of the SLB barycenter which can be easily computed and expressed explicitly in terms of the distance distributions of each object. When comparing $m$ mm-spaces for $n$ randomly drawn samples in each space, the resulting statistic then can be calculated efficiently in $O(m \cdot n^2 \log(n))$ basic operations. We derive the asymptotic distribution and finite-sample bounds of the proposed test statistic, which serves as a basis for a variety of tools for statistical inference, specifically an asymptotic test for pose-invariant object discrimination and a classification method (based on the SLB barycenter) with controlled error rates. These methods are investigated in simulations and applied to the structural comparison of protein domains.

math.ST↗

arXiv2026

Distributional Convergence of Empirical Entropic Optimal Transport and Statistical Applications

Recently, the statistical properties of empirical Entropic Optimal Transport (EOT) have attracted great interest, as this quantity has been shown to be useful for complex data analysis, among other reasons due to its computational efficiency. In several applications, it has been observed that the EOT plan provides valuable information beyond just the optimal value. For example, in cell biology, colocalization analysis based on the EOT plan has been introduced as a measure for quantification of spatial proximity of different protein assemblies. Despite recent progress in the analysis of its risk properties, a precise understanding of its statistical fluctuations to make it accessible for inference remains elusive to a large extent. In this paper, we derive asymptotic weak convergence result for a large class of functionals of the EOT plan, in which the colocalization process is included. The proof is based on Hadamard differentiability and the extended delta method. As an application, we obtain uniform confidence bands for colocalization curves and bootstrap consistency. Our theory is supported by simulation studies and is illustrated by real world data analysis from mitochondrial protein colocalization.

math.ST↗

arXiv2024

A uniform kernel trick for high-dimensional two-sample problems

We use a suitable version of the so-called "kernel trick" to devise two-sample (homogeneity) tests, especially focussed on high-dimensional and functional data. Our proposal entails a simplification related to the important practical problem of selecting an appropriate kernel function. Specifically, we apply a uniform variant of the kernel trick which involves the supremum within a class of kernel-based distances. We obtain the asymptotic distribution (under the null and alternative hypotheses) of the test statistic. The proofs rely on empirical processes theory, combined with the delta method and Hadamard (directional) differentiability techniques, and functional Karhunen-Loève-type expansions of the underlying processes. This methodology has some advantages over other standard approaches in the literature. We also give some experimental insight into the performance of our proposal compared to the original kernel-based approach \cite{Gretton2007} and the test based on energy distances \cite{Szekely-Rizzo-2017}.

math.ST↗

arXiv2019

Directional differentiability for supremum-type functionals: statistical applications

We show that various functionals related to the supremum of a real function defined on an arbitrary set or a measure space are Hadamard directionally differentiable. We specifically consider the supremum norm, the supremum, the infimum, and the amplitude of a function. The (usually non-linear) derivatives of these maps adopt simple expressions under suitable assumptions on the underlying space. As an application, we improve and extend to the multidimensional case the results in \cite{Raghavachari} regarding the limiting distributions of Kolmogorov-Smirnov type statistics under the alternative hypothesis. Similar results are obtained for analogous statistics associated with copulas. We additionally solve an open problem about the Berk-Jones statistic proposed by \cite{Jager-Wellner-2004}. Finally, the asymptotic distribution of maximum mean discrepancies over Donsker classes of functions is derived.

math.ST↗