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Conditions ensuring the asymptotic normality ofU-statistics based on either trimmed samples or Winsorized samples are well known [P. Janssen, R. Serfling, and N. Veraverbeke,Asymptotic normality of U-statistics based on trimmed samples, J. Statist. Plann. Inference 16 (1987), pp. 63–74;U-statistics on Winsorized and trimmed samples, Statist. Probab. Lett. 9 (1990), pp. 439–447]. However, the class ofU-statistics has a much richer family of limiting distributions. This paper complements known results by providing general limit theorems forU-statistics based on trimmed or Winsorized samples where the limiting distribution is given in terms of multiple Ito–Wiener stochastic integrals. [ABSTRACT FROM AUTHOR]

Abstract: We develop large deviation results with Cramér’s series and the best possible remainder term for bootstrapped -statistics with non-degenerate bounded kernels. The method of the proof is based on the contraction technique of Keener, Robinson and Weber [R.W. Keener, J. Robinson, N.C. Weber, Tail probability approximations for -statistics, Statist. Probab. Lett. 37 (1) (1998) 59–65], which is a natural generalization of the classical conjugate distribution technique due to Cramér [H. Cramér, Sur un nouveau théoréme-limite de la theorie des probabilites, Actual. Sci. Indust. 736 (1938) 5–23]. [Copyright &y& Elsevier]

Abstract: Trimmed U-statistics can be constructed in two different ways: by basing the statistic on a trimmed sample or by averaging the trimmed set of kernel values. Mild conditions are given to ensure the rate of convergence to normality is in both cases. [Copyright &y& Elsevier]

Let {Un}, n = 1,2, ..., be Hubert space H-valued U-statistics with kernel Φ(·,·), corresponding to a sequence of observations (random variables) X1, X2The rate of convergence on balls in the central limit theorem for {Un} is investigated. The obtained estimate is of order n-1/2 and depends explicitly on E‖Φ((X1,X2)‖³ and on the trace and the first nine eigenvalues of the covariance operator of E(Φ(X1,X2)|X1). [ABSTRACT FROM AUTHOR]