RANDOM QUADRATIC-FORMS AND THE BOOTSTRAP FOR U-STATISTICS

[摘要] We study the bootstrap distribution for U-statistics with special emphasis on the degenerate case. For the Efron bootstrap we give a short proof of the consistency using Mallows' metrics. We also study the i.i.d. weighted bootstrap [GRAPHICS] where (X(i)) and (xi(i)) are two i.i.d. sequences, independent of each other and where E xi(i)=0, Var(xi(i))=1. It turns out that, conditionally given (X(i)), this random quadratic form converges weakly to a Wiener-Ito double stochastic integral integral(0)(1) integral(0)(1) h(F-1(x),F-1(y)) dW(x) dW(y). As a by-product we get an a.s. limit theorem for the eigenvalues of the matrix H-n=((1/n)h(X(i), X(j)))(1 less than or equal to i,j less than or equal to). (C) 1994 Academic Press, Inc.

[发布日期] 1994-11-01

[关键词] EFRON BOOTSTRAP;WEIGHTED BOOTSTRAP;U-STATISTICS;RANDOM QUADRATIC FORMS;MULTIPLE STOCHASTIC INTEGRALS