Could you help me to understand the definition of one-sample U statistic? I'm following the notation in van der Vaart "Asymptotic statistic" p. 161 here
In a particular I have a doubt regarding the following point (at the end of page 161):
Why can we write $U$ as a conditional expectation, i.e. $U=E(h(X_1,...,X_r)|X_{(1)},...,X_{(n)})$?
As an example suppose $r=1$. The conditional expectation now reads $E[h(X_1)|X_{(1)},\ldots,X_{(n)}]$ meaning that you take the expectation condition on your sample of size $n$. Since the sample is stripped of the order, according to the definition, this expectation is not just equal to $h(X_1)$. Instead it will be equal to th e sample average of the $h(X_i)$s.
Edit to clarify: if the set on which the expectation is conditioned on weren't "stripped of the order" (expression written in the source linked in the question), the definition wouldn't make sense. Suppose for example we have two observations, $X_1=-1,X_2=1$ and if $h$ is the identity function then $E[X_1|X_1,X_2]=X_1=-1$ and it's questionable whether this is an interesting statistic. However, $E[X_1|X_{(1)},X_{(2)}]=(-1+1)/2=0$ (conditional on the empirical distribution but we don't know to which index, (1) or (2), the index 1 corresponds) which is the sample mean and which can be a useful statistic. The sample mean is further a U statistic. According to the linked source U statistics can be written as a conditional expectation and to make that definition meaningful the "strip of the order" is done.