{ "id": "eprxtykgt5", "slug": "u-statistic", "title": "U-statistic", "updated": "2026-09-30", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics" }, { "id": "physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.stats_statistical_inference" }, { "id": "physical.math.statistics.stats_statistical_inference.estimation", "label": "Estimation theory and estimator families", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.stats_statistical_inference.estimation" } ], "geo": [ { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.estimation", "label": "Non-geographic · 1946 to 2000: Estimation theory and estimator families", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.estimation", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math" }, { "id": "geo.nongeo.t1946.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics" }, { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference" }, { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.estimation", "label": "Estimation theory and estimator families", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.estimation" } ] } ], "excerpt": "A U-statistic is an estimator averaging a kernel function over all subsamples of fixed size m from the data, introduced by Wassily Hoeffding in 1948 and underlying the Mann–Whitney and Wilcoxon tests.", "snippet": "A U-statistic is an estimator averaging a kernel function over all subsamples of fixed size m from the data, introduced by Wassily Hoeffding in 1948 and underlying the Mann–Whitney and Wilcoxon tests.", "node": "physical.math.statistics.stats_statistical_inference.estimation", "markdown": "# U-statistic\n\nA U-statistic is an estimator built by averaging a kernel function over all subsamples of fixed size m from the data, giving an unbiased estimate of \\( \\theta = E\\,h(X_{1},...,X_{m}) \\). The class was introduced by [Wassily Hoeffding](https://www.edgepedia.org/wassily-hoeffding) in his 1948 Annals of Mathematical Statistics paper [1](https://doi.org/10.1214/aoms/1177730196), and it underlies many standard procedures: the sample variance, Gini's mean difference, the Mann–Whitney and Wilcoxon tests, and Kendall's tau are all U-statistics.[2](https://www.math.ucla.edu/~tom/Stat200C/Ustat.pdf)\n\n| Key fact | Statement |\n|---|---|\n| Definition | \\( U_{n} = \\binom{n}{m}^{-1} \\sum_{\\|\\beta\\|=m} h(X_{\\beta}) \\), unbiased for \\( \\theta = E\\,h(X_{1},...,X_{m}) \\) [2](https://www.math.ucla.edu/~tom/Stat200C/Ustat.pdf) |\n| Optimality | Minimum variance among all unbiased estimators based on \\( X_{1},...,X_{n} \\) [2](https://www.math.ucla.edu/~tom/Stat200C/Ustat.pdf) |\n| Variance | \\( m^{2}\\zeta_{1}/n \\le \\mathrm{Var}(U_{n}) \\le m\\zeta_{m}/n \\), with \\( \\mathrm{Var}(U_{n}) = (m^{2}/n)\\zeta_{1} + O(n^{-2}) \\) [3](https://pages.stat.wisc.edu/~shao/stat709/stat709-18.pdf) • [4](https://web.stanford.edu/class/stats300b/Slides/16-u-statistics.pdf) |\n| Limit law | \\( \\sqrt{n}(U_{n}-\\theta) \\to N(0, m^{2}\\zeta_{1}) \\) if \\( \\zeta_{1}>0 \\); if \\( \\zeta_{1}=0 \\), \\( n(U_{n}-\\theta) \\to \\binom{m}{2}\\sum_{j}\\lambda_{j}(\\chi^{2}_{1j}-1) \\) [3](https://pages.stat.wisc.edu/~shao/stat709/stat709-18.pdf) |\n| Cost | A complete degree-\\( m \\) statistic needs \\( O(n^{m}) \\) kernel evaluations; incomplete versions with \\( O(n) \\) terms keep the \\( O_{P}(n^{-1/2}) \\) rate [5](https://arxiv.org/html/1712.06160v3) • [6](https://researchers.lille.inria.fr/abellet/papers/jml16.pdf) |\n| U vs V | Estimating \\( \\mu^{2} \\) (with \\( \\mu=0 \\)), the asymptotic mean squared error is \\( 2\\sigma^{4}/n^{2} \\) for the U-statistic versus \\( 3\\sigma^{4}/n^{2} \\) for the biased V-statistic [3](https://pages.stat.wisc.edu/~shao/stat709/stat709-18.pdf) |\n\n## How it works\n\nLet \\( X_{1},...,X_{n} \\) be independent with common distribution P, and let h be a measurable function of m arguments, symmetric in them, called the kernel; m is the degree or order. Hoeffding's form averages over all ordered m-tuples of distinct indices, \\( U_{n}^{m}(\\Phi) = n^{-[m]} \\sum_{1 \\le j_{1} \\neq \\cdots \\neq j_{m} \\le n} \\Phi(X_{j_{1}},...,X_{j_{m}}) \\); when the kernel is symmetric this equals the average over the \\( \\binom{n}{m} \\) subsets.[1](https://doi.org/10.1214/aoms/1177730196) • [7](https://encyclopediaofmath.org/wiki/U-statistic) Because every subsample has the same distribution, \\( E\\,U_{n} = \\theta \\) exactly, for every n.\n\nThe variance structure is the analytical core. Define \\( h_{c} \\) as the conditional expectation of the kernel given c observations and \\( \\zeta_{c} = \\mathrm{Var}(h_{c}) \\). Hoeffding's theorem gives \\( \\mathrm{Var}(U_{n}) = \\sum_{c=1}^{m} \\binom{m}{c}\\binom{n-m}{m-c}/\\binom{n}{m} \\, \\zeta_{c} \\), which is nonincreasing in n; the bound \\( m^{2}\\zeta_{1}/n \\le \\mathrm{Var}(U_{n}) \\) is valid for finite n, and \\( m^{2}\\zeta_{1}/n \\) should be read as the leading asymptotic term.[3](https://pages.stat.wisc.edu/~shao/stat709/stat709-18.pdf) • [30](https://stats.stackexchange.com/questions/318539/showing-a-corollary-of-hoeffdings-theorem) The Hoeffding decomposition writes \\( U_{n} = \\theta + \\sum_{c=1}^{m} \\binom{m}{c} U_{n}(g_{c}) \\) with completely degenerate, pairwise orthogonal components; the first term \\( U_{n}(g_{1}) \\) is a sum of centered i.i.d. variables, which yields the central limit theorem directly, and the decomposition also gives the variance immediately.[7](https://encyclopediaofmath.org/wiki/U-statistic) • [8](https://encyclopediaofmath.org/wiki/Hoeffding_decomposition) Equivalently, the Hájek projection of \\( U_{n} \\) onto sums of univariate functions is \\( (m/n)\\sum_{i} h_{1}(X_{i}) \\) with \\( h_{1}(x) = E[h(x, X_{2},...,X_{m})] - \\theta \\), and \\( \\sqrt{n}(U_{n}-\\theta) \\to N(0, m^{2}\\zeta_{1}) \\).[4](https://web.stanford.edu/class/stats300b/Slides/16-u-statistics.pdf)\n\n## How it is done\n\nChoosing a kernel means writing the target functional as an expectation over m distinct observations. The sample variance uses \\( h(x_{1},x_{2}) = (x_{1}-x_{2})^{2}/2 \\) [7](https://encyclopediaofmath.org/wiki/U-statistic); Gini's mean difference uses \\( h(x_{1},x_{2}) = |x_{1}-x_{2}| \\) [3](https://pages.stat.wisc.edu/~shao/stat709/stat709-18.pdf); the one-sample Wilcoxon statistic uses \\( h = I_{(-\\infty,0]}(x_{1}+x_{2}) \\) for \\( \\theta = P(X_{1}+X_{2} \\le 0) \\).[3](https://pages.stat.wisc.edu/~shao/stat709/stat709-18.pdf)\n\nInference needs a variance estimate. The exact variance follows from the \\( \\zeta_{c} \\) formula above; Wang and Lindsay construct an unbiased estimator \\( \\hat{V}_{u} = Q(k) - Q(0) \\), valid whenever the kernel size k satisfies \\( k \\le n/2 \\), which is best unbiased because it is a function of the order statistics and admits an ANOVA-type partition-resampling computation competitive with the bootstrap and jackknife.[9](https://www3.stat.sinica.edu.tw/ss_newpaper/SS-12-215_na.pdf)\n\nTests follow from the limit laws. For the two-sample Mann–Whitney statistic with kernel \\( h(x,y) = 1\\{x \\le y\\} \\), under the null \\( \\sqrt{12mn/N}\\,(U_{N} - 1/2) \\to N(0,1) \\), where \\( N = m + n \\).[4](https://web.stanford.edu/class/stats300b/Slides/16-u-statistics.pdf) A Kendall's-tau-type independence test has \\( \\delta_{1}^{2} = 1/9 \\) under independence, so \\( \\sqrt{N} \\cdot K_{N} \\to N(0, 4/9) \\) and one rejects when \\( \\sqrt{9N/4}\\,|K_{N}| > z_{\\alpha/2} \\).[10](https://eml.berkeley.edu/~bgraham/Teaching/CEMFI_August2015/Lecture_4_U_Statistics.pdf)\n\n## Origin\n\nThe subject begins with Paul R. Halmos's 1946 paper \"The Theory of Unbiased Estimation,\" which studied unbiased estimators of minimum variance and first asked when distribution-free unbiased estimators of a functional \\( \\theta(F) \\) exist.[11](https://doi.org/10.1214/aoms/1177731020) Hoeffding's 1948 paper \"A Class of Statistics with Asymptotically Normal Distribution\" founded the theory of U-statistics proper, proving asymptotic normality of \\( \\sqrt{n}(U-\\theta) \\) under the sole condition that \\( E\\Phi^{2} \\) exists.[1](https://doi.org/10.1214/aoms/1177730196) E. L. Lehmann extended the theory to k-sample problems in 1951.[12](https://doi.org/10.1214/aoms/1177729639) The term \"Hoeffding decomposition\" itself goes back to Manfred Denker's 1985 lecture notes [8](https://encyclopediaofmath.org/wiki/Hoeffding_decomposition) • [13](https://doi.org/10.1007/978-3-663-14229-4), and detailed expositions appear in Denker (1985) and A. J. Lee's 1990 monograph.[2](https://www.math.ucla.edu/~tom/Stat200C/Ustat.pdf) The plug-in counterpart, obtained by replacing F with the empirical \\( F_{n} \\), is the biased von Mises statistic.[14](https://publish.illinois.edu/xiaohuichen/files/2018/11/stat05986_Chen_v3_nohighlight.pdf)\n\n## Variants\n\nA complete degree-m U-statistic needs \\( \\binom{n}{m} \\) kernel evaluations, which is often infeasible. Incomplete U-statistics average over a subsample of tuples instead; the concept was introduced by Gunnar Blom in 1976.[5](https://arxiv.org/html/1712.06160v3) • [15](https://doi.org/10.1093/biomet/63.3.573) Design-based choices matter: Alan J. Lee showed in 1982 that balanced incomplete block designs yield minimum-variance incomplete U-statistics [16](https://doi.org/10.1111/j.1467-842x.1982.tb00833.x), Brown and Kildea's 1978 work on reduced U-statistics is closely tied to the Hodges–Lehmann estimator [17](https://doi.org/10.1214/aos/1176344256), and Svante Janson established the asymptotic distributions of randomly sampled incomplete U-statistics (ICUR) in 1984.[18](https://doi.org/10.1007/bf00531887) The ICUDO scheme, based on data division and an orthogonal array, is asymptotically efficient with \\( m \\asymp n \\) terms, whereas ICUR needs m to grow faster than n.[19](https://www3.stat.sinica.edu.tw/statistica/oldpdf/A31n322.pdf) Randomized incomplete U-statistics with sparse weights, studied by Xiaohui Chen and Kengo Kato, have computational cost independent of the order of the statistic and carry non-asymptotic Gaussian approximation bounds even in the degenerate case.[20](https://doi.org/10.1214/18-aos1773) For estimation problems whose empirical risk is a U-statistic, distributed divide-and-conquer schemes (SU-ERM and OS-ERM) reduce complexity from \\( O(N^{2}) \\) to \\( O(n^{2}) \\) per machine while remaining asymptotically efficient relative to the centralized U-estimator.[21](https://jmlr.org/papers/volume24/21-0890/21-0890.pdf) When the kernel index set itself varies with n, the objects are U-processes, treated with empirical-process tools by Nolan and Pollard.[22](https://doi.org/10.1214/aos/1176350374)\n\n## Applications\n\nEach named procedure maps to a kernel. The two-sample Wilcoxon test of Frank Wilcoxon (1945) and the Mann–Whitney test of H. B. Mann and D. R. Whitney (1947) use \\( h(x,y) = I(y < x) \\) with \\( \\theta = P(Y < X) \\).[2](https://www.math.ucla.edu/~tom/Stat200C/Ustat.pdf) • [23](https://doi.org/10.2307/3001968) • [24](https://doi.org/10.1214/aoms/1177730491) Kendall's measure of rank correlation (1938) is a degree-2 U-statistic.[25](https://doi.org/10.1093/biomet/30.1-2.81) The squared distance covariance estimator is a U-statistic of order 4 [26](https://nchenderson.github.io/elements-nonpar-stat/ustat.html), and the Cramér–von Mises goodness-of-fit statistic is a degenerate U-statistic under the null.[14](https://publish.illinois.edu/xiaohuichen/files/2018/11/stat05986_Chen_v3_nohighlight.pdf)\n\n## Limitations and alternatives\n\nDegeneracy changes everything about the limit law. If \\( \\zeta_{1} = 0 \\) but \\( \\zeta_{2} > 0 \\), the \\( \\sqrt{n} \\) scale collapses: \\( n(U_{n}-\\theta) \\) converges to a weighted sum of independent chi-squares, and a rank-r degenerate U-statistic converges at rate \\( n^{-r/2} \\) to a weighted sum of order-r polynomials of a standard normal variable.[3](https://pages.stat.wisc.edu/~shao/stat709/stat709-18.pdf) • [27](https://xblk.ecnu.edu.cn/starf/Assets/userfiles/files/I-4-V-9/Asymptotic%20distributions%20of%20degenerated%20U-statistics.pdf) Degeneracy is common precisely under the null hypotheses of interest, which is why testing requires permutation or bootstrap methods.[27](https://xblk.ecnu.edu.cn/starf/Assets/userfiles/files/I-4-V-9/Asymptotic%20distributions%20of%20degenerated%20U-statistics.pdf) In high dimensions the classical dichotomy itself fails: for degree-2 U-statistics the limit undergoes a phase transition governed by the moment ratio \\( \\rho_{d} = \\sigma_{\\mathrm{full}}/\\sigma_{\\mathrm{cond}} \\), independent of classical degeneracy, with dimension-independent finite-sample error bounds.[28](https://proceedings.mlr.press/v195/huang23a/huang23a.pdf)\n\nAgainst alternatives: V-statistics average over all \\( n^{m} \\) ordered tuples including repeats and are biased by \\( O(n^{-1}) \\); when \\( \\zeta_{1} > 0 \\) the two have the same asymptotic mean squared error, but in the degenerate case the U-statistic is asymptotically more efficient unless \\( \\sum_{j}\\lambda_{j} = 0 \\).[3](https://pages.stat.wisc.edu/~shao/stat709/stat709-18.pdf) Computational cost is the practical limit: computing the empirical MMD costs \\( O(n^{2}) \\) because it requires all pairwise kernel evaluations [29](https://arxiv.org/pdf/2502.13570), and no minimum sample sizes for the asymptotic approximations are stated beyond conditions such as \\( k \\le n/2 \\) for the variance estimator.[9](https://www3.stat.sinica.edu.tw/ss_newpaper/SS-12-215_na.pdf)\n\n## References\n\n1. [Wassily Hoeffding (1948). A Class of Statistics with Asymptotically Normal Distribution. The Annals of Mathematical Statistics.](https://doi.org/10.1214/aoms/1177730196)\n2. [U-statistics (lecture notes by Thomas Ferguson, UCLA Stat 200C)](https://www.math.ucla.edu/~tom/Stat200C/Ustat.pdf)\n3. [Stat 709: Mathematical Statistics, Lecture 18 (U-statistics), Jun Shao, UW–Madison](https://pages.stat.wisc.edu/~shao/stat709/stat709-18.pdf)\n4. [U Statistics, Stats 300b slides, John Duchi, Stanford](https://web.stanford.edu/class/stats300b/Slides/16-u-statistics.pdf)\n5. [Concentration Inequalities for U-Statistics: A Survey with Explicit Constants](https://arxiv.org/html/1712.06160v3)\n6. [Scaling-up Empirical Risk Minimization: Optimization of Incomplete U-statistics (JMLR 2016)](https://researchers.lille.inria.fr/abellet/papers/jml16.pdf)\n7. [U-statistic, Encyclopedia of Mathematics (Korolyuk)](https://encyclopediaofmath.org/wiki/U-statistic)\n8. [Hoeffding decomposition, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Hoeffding_decomposition)\n9. [Variance Estimation of a General U-statistic with Application to Cross-Validation (Wang & Lindsay, Statistica Sinica)](https://www3.stat.sinica.edu.tw/ss_newpaper/SS-12-215_na.pdf)\n10. [Lecture 4: U-Statistics & U-Process Minimizers (Bryan Graham, Berkeley, CEMFI 2015)](https://eml.berkeley.edu/~bgraham/Teaching/CEMFI_August2015/Lecture_4_U_Statistics.pdf)\n11. [Paul R. Halmos (1946). The Theory of Unbiased Estimation. The Annals of Mathematical Statistics.](https://doi.org/10.1214/aoms/1177731020)\n12. [E. L. Lehmann (1951). Consistency and Unbiasedness of Certain Nonparametric Tests. The Annals of Mathematical Statistics.](https://doi.org/10.1214/aoms/1177729639)\n13. [Manfred Denker (1985). Asymptotic Distribution Theory in Nonparametric Statistics. Advanced lectures in mathematics.](https://doi.org/10.1007/978-3-663-14229-4)\n14. [Chen, U-statistics (review, Statistical Theory and Related Fields / author-hosted copy)](https://publish.illinois.edu/xiaohuichen/files/2018/11/stat05986_Chen_v3_nohighlight.pdf)\n15. [GUNNAR BLOM (1976). Some properties of incomplete U-statistics. Biometrika.](https://doi.org/10.1093/biomet/63.3.573)\n16. [Alan J. Lee (1982). On Incomplete U‐Statistics Having Minimum Variance. Australian Journal of Statistics.](https://doi.org/10.1111/j.1467-842x.1982.tb00833.x)\n17. [B. M. Brown, D. G. Kildea (1978). Reduced $U$-Statistics and the Hodges-Lehmann Estimator. The Annals of Statistics.](https://doi.org/10.1214/aos/1176344256)\n18. [Svante Janson (1984). The asymptotic distributions of incomplete U-statistics. Probability Theory and Related Fields.](https://doi.org/10.1007/bf00531887)\n19. [Design Based Incomplete U-Statistics (ICUDO), Statistica Sinica](https://www3.stat.sinica.edu.tw/statistica/oldpdf/A31n322.pdf)\n20. [Xiaohui Chen, Kengo Kato (2019). Randomized incomplete $U$-statistics in high dimensions. The Annals of Statistics.](https://doi.org/10.1214/18-aos1773)\n21. [Distributed Algorithms for U-statistics-based Empirical Risk Minimization (JMLR, Volume 24)](https://jmlr.org/papers/volume24/21-0890/21-0890.pdf)\n22. [Deborah Nolan, David Pollard (1987). $U$-Processes: Rates of Convergence. The Annals of Statistics.](https://doi.org/10.1214/aos/1176350374)\n23. [Frank Wilcoxon (1945). Individual Comparisons by Ranking Methods. Biometrics Bulletin.](https://doi.org/10.2307/3001968)\n24. [H. B. Mann, D. R. Whitney (1947). On a Test of Whether one of Two Random Variables is Stochastically Larger than the Other. The Annals of Mathematical Statistics.](https://doi.org/10.1214/aoms/1177730491)\n25. [M. G. KENDALL (1938). A NEW MEASURE OF RANK CORRELATION. Biometrika.](https://doi.org/10.1093/biomet/30.1-2.81)\n26. [Chapter 6: U-Statistics, Elements of Nonparametric Statistics (N. Henderson)](https://nchenderson.github.io/elements-nonpar-stat/ustat.html)\n27. [Asymptotic distributions of degenerated U-statistics (J. Shao)](https://xblk.ecnu.edu.cn/starf/Assets/userfiles/files/I-4-V-9/Asymptotic%20distributions%20of%20degenerated%20U-statistics.pdf)\n28. [A High-dimensional Convergence Theorem for U-statistics with Applications to Kernel-based Testing (ICML 2023, PMLR v195)](https://proceedings.mlr.press/v195/huang23a/huang23a.pdf)\n29. [A Scalable Nyström-Based Kernel Two-Sample Test with Permutations, arXiv 2025](https://arxiv.org/pdf/2502.13570)\n30. [Showing a Corollary of Hoeffdings Theorem, stats.stackexchange.com](https://stats.stackexchange.com/questions/318539/showing-a-corollary-of-hoeffdings-theorem)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Statistical inference, estimation, sampling, and testing › Estimation theory and estimator families*\n\n*Initially written Sep 29, 2026 · Reviewed: Sep 30, 2026 · Edited: Sep 30, 2026 · Last review: Sep 30, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "https://www.math.ucla.edu/~tom/Stat200C/Ustat.pdf", "https://pages.stat.wisc.edu/~shao/stat709/stat709-18.pdf", "https://eml.berkeley.edu/~bgraham/Teaching/CEMFI_August2015/Lecture_4_U_Statistics.pdf" ], "url": "https://www.edgepedia.org/u-statistic", "markdown_url": "https://www.edgepedia.org/u-statistic.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"U-statistic\", Edgepedia (EdgeChat), https://www.edgepedia.org/u-statistic. Edgepedia Community License 1.0.", "credit_md": "\"[U-statistic](https://www.edgepedia.org/u-statistic)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/u-statistic](https://www.edgepedia.org/u-statistic). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"U-statistic\", Edgepedia (EdgeChat), https://www.edgepedia.org/u-statistic. Edgepedia Community License 1.0.", "speakable": "A U-statistic is an estimator averaging a kernel function over all subsamples of fixed size m from the data, introduced by Wassily Hoeffding in 1948 and underlying the Mann–Whitney and Wilcoxon tests." }