Statistics & Probability Letters 16 (1993) 225-233 North-Holland 19 February 1993 Generalized bootstrap for studentized U-statistics: A rank statistic approach Marie HuSkovA zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Charles University, Prague, Czechoslovakia Paul Janssen * Limburgs Universitair Centrum, Diepenbeek, Belgium Received May 1992 Abstract: The notion of generalized bootstrapping is introduced in Mason and Newton (1992). They study the consistency of generalized bootstrapped means. We extend the validity of the generalized bootstrap to the case of U-statistics and studentized U-statistics. From the proofs it will become clear that the rank statistic methodology is a powerful tool that provides a unified approach to handle different resampling schemes. AMS Subject Classifications: Primary 62E20; Secondary 60F0.5. Keywords: Bootstrap consistency; studentized U-statistics; rank statistics. 1. Introduction and main results Let x,, . . .) X, be an i.i.d. sample with d.f. F defined on a probability space (0, .G?,PI. A U-statistic of degree two is obtained by averaging a symmetric kernel h(x, y) over the sample, i.e., u,= “2 ( 1 -’ CC h(Xi, xj). 1 gi<jCn Recall that U, is an unbiased estimator for B(F) = jjh(x, y) dF(x) dF(y) and that the projection of the kernel is g(x; F) = lh(x, y) dF(y) - B(F). From Hoeffding (1948) and Arvesen (1969) we have that (Ul) Eh2(X,, X,) < 03, (U2) a,‘=Eg2(X,; F) >O, are sufficient conditions to guarantee n”2(U, - 0( F))/25 + M-(0, 1)) TZ”~(U, - 0( F))/a,, 2 _N(O, l), (l-1) (1.2) 0167-7152/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved 225