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Central limit theorem and the bootstrap for \(U\)-statistics of strongly mixing data. (English) Zbl 1177.62056
Summary: The asymptotic normality of \(U\)-statistics has so far been proved for iid data and under various mixing conditions, such as absolute regularity, but not for strong mixing. We use a coupling technique introduced by R. C. Bradley [Approximation theorems for strongly mixing random variables. Mich. Math. J. 30, 69–81 (1983; Zbl 0531.60033)] to prove a new generalized covariance inequality similar to K. Yoshihara’s [Limiting behavior of \(U\)-statistics for stationary, absolutely regular processes. Z. Wahrschlichkeitstheor. Verw. Geb. 35, 237–252 (1976; Zbl 0314.60028)]. It follows from the Hoeffding-decomposition and this inequality that \(U\)-statistics of strongly mixing observations converge to a normal limit if the kernel of the \(U\)-statistic fulfills some moment and continuity conditions.
The validity of the bootstrap for \(U\)-statistics has until now only been established in the case of iid data [see P. J. Bickel and D. Freedman, Some asymptotic theory for the bootstrap. Ann. Stat. 9, 1196-1217 (1981; Zbl 0449.62034)]. For mixing data, D. N. Politis and J. P. Romano [A circular block resampling procedure for stationary data. R. LePage et al. (eds.), Exploring the Limits of Bootstrap, NY: Wiley, 263–270 (1992; Zbl 0845.62036)] proposed the circular block bootstrap, which leads to consistent estimation of the sample mean distribution.
We extend these results to \(U\)-statistics of weakly dependent data and prove a CLT for the circular block bootstrap version of \(U\)-statistics under absolute regularity and strong mixing. We also calculate the rate of convergence for the bootstrap variance estimator of a \(U\)-statistic and give some simulation results.

MSC:
62G09 Nonparametric statistical resampling methods
60F05 Central limit and other weak theorems
62G20 Asymptotic properties of nonparametric inference
60F15 Strong limit theorems
60E15 Inequalities; stochastic orderings
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