Approximating high-dimensional infinite-order $U$-statistics: Statistical and computational guarantees

Yanglei Song; Xiaohui Chen; Kengo Kato

doi:10.1214/19-ejs1643

journal article Jan 01, 2019

Approximating high-dimensional infinite-order $U$-statistics: Statistical and computational guarantees

Yanglei Song Xiaohui Chen

Kengo Kato

Electronic Journal of Statistics Vol. 13 No. 2 · Institute of Mathematical Statistics

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References

31

[1]

[7] Xiaohui Chen and Kengo Kato. Jackknife multiplier bootstrap: finite sample approximations to the $U$-process supremum with applications. 2017., <a href="arXiv:1708.02705">arXiv:1708.02705</a>.

[2]

[8] Xiaohui Chen and Kengo Kato. Randomized incomplete $u$-statistics in high dimensions., The Annals of Statistics, accepted (available at arXiv:1712.00771), 2018+. 10.1214/18-aos1773

[3]

[1] Gunnar Blom. Some properties of incomplete $U$-statistics., Biometrika, 63(3):573–580, 1976. 10.1093/biomet/63.3.573

[4]

[2] Yu. V. Borovskikh., U-Statistics in Banach Spaces. V.S.P. Intl Science, 1996.

[5]

Bagging predictors

Leo Breiman

Machine Learning 10.1007/bf00058655

[6]

Random Forests

Leo Breiman

Machine Learning 10.1023/a:1010933404324

[7]

[5] B. M. Brown and D. G. Kildea. Reduced $U$-statistics and the Hodges-Lehmann estimator., Annals of Statistics, 6:828–835, 1978. 10.1214/aos/1176344256

[8]

[6] Xiaohui Chen. Gaussian and bootstrap approximations for high-dimensional u-statistics and their applications., The Annals of Statistics, 46(2):642–678, 2018. 10.1214/17-aos1563

[9]

[9] Victor Chernozhukov, Denis Chetverikov, and Kengo Kato. Comparison and anti-concentration bounds for maxima of gaussian random vectors., Probability Theory and Related Fields, 162(1-2):47–70, 2015. 10.1007/s00440-014-0565-9

[10]

[10] Victor Chernozhukov, Denis Chetverikov, and Kengo Kato. Central limit theorems and bootstrap in high dimensions., Ann. Probab., 45(4) :2309–2352, 07 2017. 10.1214/16-aop1113

[11]

[11] Victor Chernozhukov, Denis Chetverikov, Kengo Kato, et al. Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors., The Annals of Statistics, 41(6) :2786–2819, 2013. 10.1214/13-aos1161

[12]

[12] Stéphan Clémençon, Gábor Lugosi, and Nicolas Vayatis. Ranking and empirical minimization of $u$-statistics., Annals of Statistics, 36(2), 844-874. 10.1214/009052607000000910

[13]

[13] Victor De la Pena and Evarist Giné., Decoupling: from dependence to independence. Springer Science & Business Media, 2012.

[14]

[14] Edward W. Frees. Infinite order u-statistics., Scandinavian Journal of Statistics, 16(1):29–45, 1989.

[15]

[15] Edward W. Frees. Estimating densities of functions of observations., Journal of the American Statistical Association, 89(426):517–525, 1994. 10.1080/01621459.1994.10476776

[16]

[16] Karl O. Friedrich. A berry-esseen bound for functions of independent random variables., The Annals of Statistics, pages 170–183, 1989. 10.1214/aos/1176347009

[17]

[17] Evarist Giné, David M. Mason, et al. On local u-statistic processes and the estimation of densities of functions of several sample variables., The Annals of Statistics, 35(3) :1105–1145, 2007. 10.1214/009053607000000154

[18]

[18] Charles Heilig and Deborah Nolan. Limit theorems for the infinite-degree $u$-process., Statistica Sinica, 11:289–302, 2001.

[19]

[19] Wassily Hoeffding. A class of statistics with asymptotically normal distribution., The Annals of Mathematical Statistics, 19(3):293–325, 1948. 10.1214/aoms/1177730196

[20]

[20] Svante Janson. The asymptotic distributions of incomplete $U$-statistics., Z, Wahrscheinlichkeitstheorie verw. Gebiete, 66:495–505, 1984. 10.1007/bf00531887

[21]

[21] Alan J. Lee., U-Statistics: Theory and Practice. Statistics: A Series of Textbooks and Monographs (Book 110). CRC Press, 1990.

[22]

[22] P. Major. Asymptotic distributions for weighted U-statistics., Annals of Probability, 21(2) :1514–1535, 1994. 10.1214/aop/1176988610

[23]

[23] Lucas Mentch and Giles Hooker. Quantifying uncertainty in random forests via confidence intervals and hypothesis tests., The Journal of Machine Learning Research, 17(1):841–881, 2016.

[24]

[24] K. A. O’Neil and R. A. Redner. Asymptotic distributions of weighted $U$-statistics of degree 2., Annals of Probability, 21(2) :1159–1169, 1993. 10.1214/aop/1176989286

[25]

[25] M. Rifi and F. Utzet. On the asymptotic behavior of weighted U-statistics., Journal of Theoretical Probability, 13(1):141–167, 2000. 10.1023/a:1007786926882

[26]

[26] C. P. Shapiro and L. Hubert. Asymptotic normality of permutation statistics derived from weighted sums of bivariate functions., Annals of Statistics, 7(4):788–794, 1979. 10.1214/aos/1176344728

[27]

[27] Robert P. Sherman. Maximal inequalities for degenerate u-processes with applications to optimization estimators., The Annals of Statistics, 439–459, 1994. 10.1214/aos/1176325377

[28]

[28] Grace S. Shieh. Infinite-order $v$-statistics., Statistics & Probability Letters, 20:75–80, 1994. 10.1016/0167-7152(94)90237-2

[29]

[29] Aad W. Van Der Vaart and Jon A. Wellner. Weak convergence. In, Weak convergence and empirical processes, pages 16–28. Springer, 1996. 10.1007/978-1-4757-2545-2_3

[30]

[30] A. J. van Es and R. Helmers. Elementary symmetric polynomials of increasing order., Probability Theory and Related Fields, 80(1):21–35, Dec 1988. 10.1007/bf00348750

[31]

[31] van W. R. Zwet. A Berry-Esseen bound for symmetric statistics., Z. Wahrscheinlichkeitstheorie verw. Gebiete, 66:425–440, 1984. 10.1007/bf00533707

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Published: Jan 01, 2019
Vol/Issue: 13(2)

Authors

Department of Otolaryngology–Head and Neck Surgery Tohoku University School of Medicine Sendai Japan

Cite This Article

Yanglei Song, Xiaohui Chen, Kengo Kato (2019). Approximating high-dimensional infinite-order $U$-statistics: Statistical and computational guarantees. Electronic Journal of Statistics, 13(2). https://doi.org/10.1214/19-ejs1643

Approximating high-dimensional infinite-order $U$-statistics: Statistical and computational guarantees

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