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We develop large deviation results with Cramér's series and the best possible remainder term for bootstrapped U -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 U -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].

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We adapt the techniques in Stigler [Ann. Statist. 1 (1973) 472-477] to obtain a new, general asymptotic result for trimmed U -statistics via the generalized L-statistic representation introduced by Serfling [Ann. Statist. 12 (1984) 76-86]. Unlike existing results, we do not require continuity of an associated distribution at the truncation points. Our results are quite general and are expressed in terms of the quantile function associated with the distribution of the Ustatistic summands. This approach leads to improved conditions for the asymptotic normality of these trimmed U -statistics.

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