Annals of Probability

On normal approximations to U-statistics

Abstract

Let X1, …, Xn be i.i.d. random observations. Let ${\mathbb{S}=\mathbb{L}+\mathbb{T}}$ be a U-statistic of order k≥2 where $\mathbb{L}$ is a linear statistic having asymptotic normal distribution, and $\mathbb {T}$ is a stochastically smaller statistic. We show that the rate of convergence to normality for $\mathbb{S}$ can be simply expressed as the rate of convergence to normality for the linear part $\mathbb{L}$ plus a correction term, $(\operatorname{var}\mathbb{T})\ln^{2}(\operatorname{var}\mathbb{T})$, under the condition ${\mathbb{E}\mathbb{T}^{2}<\infty}$. An optimal bound without this log factor is obtained under a lower moment assumption ${\mathbb{E}|\mathbb{T}|^{\alpha}< \infty}$ for ${\alpha < 2}$. Some other related results are also obtained in the paper. Our results extend, refine and yield a number of related-known results in the literature.

Article information

Source
Ann. Probab., Volume 37, Number 6 (2009), 2174-2199.

Dates
First available in Project Euclid: 16 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1258380786

Digital Object Identifier
doi:10.1214/09-AOP474

Mathematical Reviews number (MathSciNet)
MR2573555

Zentralblatt MATH identifier
1186.62025

Subjects
Primary: 62E20: Asymptotic distribution theory

Citation

Bentkus, Vidmantas; Jing, Bing-Yi; Zhou, Wang. On normal approximations to U -statistics. Ann. Probab. 37 (2009), no. 6, 2174--2199. doi:10.1214/09-AOP474. https://projecteuclid.org/euclid.aop/1258380786

References

• [1] Alberink, I. B. (2000). A Berry–Esseen bound for U-statistics in the non-i.i.d. case. J. Theoret. Probab. 13 519–533.
• [2] Alberink, I. B. and Bentkus, V. (2001). Lyapunov type bounds for U-statistics. Theory Probab. Appl. 46 724–743.
• [3] Alberink, I. B. and Bentkus, V. (2001). Berry–Esseen bounds for von Mises and U-statistics. Lithuanian Math. J. 41 1–20.
• [4] Bentkus, V., Götze, F. and Zitikis, R. (1994). Lower estimates of the convergence rate for U-statistics. Ann. Probab. 22 1707–1714.
• [5] Bentkus, V., Götze, F., Paulauskas, V. and Račkauskas, A. (2000). The accuracy of Gaussian approximation in Banach spaces. In Limit Theorems of Probability Theory (Yu. V. Prokhorov and V. A. Statulevičius, eds.) 25–111. Springer, Berlin.
• [6] Bloznelis, M. and Götze, F. (2001). Orthogonal decomposition of finite population statistics and its applications to distributional asymptotics. Ann. Statist. 29 899–917.
• [7] Chen, L. H. Y. and Shao, Q.-M. (2007). Normal approximation for nonlinear statistics using a concentration inequality approach. Bernoulli 13 581–599.
• [8] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II, 2nd ed. Wiley, New York.
• [9] Friedrich, K. O. (1989). A Berry–Esseen bound for functions of independent random variables. Ann. Statist. 17 170–183.
• [10] Giné, E., Latała, R. and Zinn, J. (2000). Exponential and moment inequalities for U-statistics. In High Dimensional Probability, II (Seattle, WA, 1999). Progress in Probability 47 13–38. Birkhäuser, Boston, MA.
• [11] Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19 293–325.
• [12] Jing, B.-Y. and Wang, Q. (2003). Edgeworth expansion for U-statistics under minimal conditions. Ann. Statist. 31 1376–1391.
• [13] Karlin, S. and Rinott, Y. (1982). Applications of ANOVA type decompositions for comparisons of conditional variance statistics including jackknife estimates. Ann. Statist. 10 485–501.
• [14] Korolyuk, V. S. and Borovskikh, Yu. V. (1985). Approximation of nondegenerate U-statistics. Theory Probab. Appl. 30 439–450.
• [15] Koroljuk, V. S. and Borovskich, Y. V. (1994). Theory of U-Statistics. Mathematics and Its Applications 273. Kluwer, Dordrecht.
• [16] Peccati, G. (2004). Hoeffding-ANOVA decompositions for symmetric statistics of exchangeable observations. Ann. Probab. 32 1796–1829.
• [17] van Zwet, W. R. (1984). A Berry–Esseen bound for symmetric statistics. Z. Wahrsch. Verw. Gebiete 66 425–440.