Bootstrapping parameter estimated degenerate U and V statistics

Let X1,X2,...,Xn be independent random vectors with common distribution function F and let be a parametric family of distributions. Let Tn([theta])=Tn(X1,X2,...,Xn;[theta]) be a degree-2 V statistic and let be a consistent estimator of [theta]. Several test statistics for testing the composite null hypothesis has the form . Typically, the null distribution of depends on the unknown value of [theta]. The purpose of this paper is to show that the bootstrap can be used to approximate the null distribution of this type of statistics. We also give similar results for statistics , with Wn([theta]) a degree-2 U statistic.