Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0704.1379

U-max-Statistics

Wei Lao·Michael Mayer
View original source ↗Read source document ↗Export metadata ↓

Abstract

In 1948, W. Hoeffding introduced a large class of unbiased estimators called U-statistics, defined as the average value of a real-valued k-variate function h calculated at all possible sets of k points from a random sample. In the present paper we investigate the corresponding extreme value analogue, which we shall call U-max-statistics. We are concerned with the behavior of the largest value of such function h instead of its average. Examples of U-max-statistics are the diameter or the largest scalar product within a random sample. U-max-statistics of higher degrees are given by triameters and other metric invariants.

Explore related subjects

math.STmath.PRstat.TH

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Wei Lao, Michael Mayer. 2007-04-11. U-max-Statistics. https://arxiv.org/abs/0704.1379

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Sparse estimation for the drift of high-dimensional Ornstein--Uhlenbeck processes with i.i.d. paths

We study sparsity-regularized maximum likelihood estimation for the drift parameter of high-dimensional non-stationary Ornstein--Uhlenbeck processes given repeated measurements of i.i.d. paths. In particular, we show that Lasso and Slope estimators can achieve the minimax optimal rate of convergence. We exhibit numerical experiments for sparse estimation methods and show their performance.

math.ST↗
arXiv2026

Minimax Wasserstein Testing under Differential Privacy

We study Wasserstein-$p$ goodness-of-fit and two-sample testing under differential privacy. For general compact metric spaces, we establish upper and lower bounds on the minimax separation radii in terms of covering numbers, matching up to logarithmic factors. On the hypercube $[0,1]^d$, these bounds match up to constant factors for two-sample testing in all dimensions and for goodness-of-fit testing whenever $d\neq 4p$, thereby closing logarithmic gaps in existing non-private results. At the critical case $(d,p)=(4,1)$, we further show that a logarithmic factor is unavoidable by establishing matching minimax bounds for non-private Wasserstein-$1$ goodness-of-fit testing under a bounded-density assumption. Finally, for unbounded domains, we establish minimax-optimal separation rates up to logarithmic factors under an exponential tail condition.

math.ST↗
arXiv2026

Quantitative Stability and Sample Complexity for Regularized Gromov-Wasserstein Distance

The Gromov-Wasserstein (GW) distance provides a principled framework for comparing two probability distributions supported on different metric spaces by matching their individual intrinsic geometric structures. In practice, the GW distance is estimated from finite samples and often incorporates regularization for statistical and computational tractability. This motivates a quantitative understanding of the stability and estimation error of regularized GW distances. In this work, we study a general family of divergence regularized GW distances, including entropy and Tsallis regularizations over the full range of powers, from a nonasymptotic perspective. Specifically, we establish Lipschitz continuity of the value of regularized GW and H{ö}lder continuity of its optimal coupling under Wasserstein perturbations of the input probability measures. Moreover, we derive the parametric rate of convergence for the empirical regularized (2,2)-GW distance on compactly supported Euclidean spaces.

math.ST↗