Many high dimensional hypothesis tests examine the moments of the distributions that are of interest, such as testing of mean vectors and covariance matrices. We propose a general framework that constructs a family of U statistics as unbiased estimators of those moments. The usage of the framework is illustrated by testing off-diagonal elements of a covariance matrix. We show that under null hypothesis, the U statistics of different finite orders are asymptotically independent and normally distributed. Moreover, they are also asymptotically independent with the max-type test statistic. Based on the asymptotic independence property, we further construct an adaptive testing procedure that maintains high power across a wide range of alternatives.