Are these estimators of $P(XI have two continuous random variables, $X$ and $Y$, whose distributions are unknown. I can draw samples from them and in particular, I am interested in estimating $P(X<Y)$ based on those samples. Let's assume that each sample has $n$ elements: $x_1, x_2, \ldots, x_n$ are drawn from $X$ and $y_1, y_2, \ldots, y_n$ from $Y$.
Can I use $$T=\frac{1}{n^2}\sum_{i=1}^{n}\sum_{j=1}^{n}I(x_i < y_j)$$
as the estimate of $P(X<Y)$ if I assume $X$ and $Y$ are independent and is such an estimator asymptotically normally distributed? If I knew
Also, if I assume that $X$ and $Y$ are dependent, can I use $$S=\frac{1}{n}\sum_{i=1}^{n}I(x_i < y_i)$$ as the estimator of said probability and is it asymptotically normally distributed?
I'm asking about asymptotic normality because I would like to determine lower and upper bounds of $P(X<Y)$ and think that confidence intervals can act as such. 
P.S. $I(E)=1$ if $E$ is true, and $0$ otherwise.
 A: The statistic $T$ is an example of a general $U$-statistic, introduced
by Hoeffding in his 1948 paper A class of statistics with asymptotically
normal distribution. Moreover, it is among the most famous of that class,
namely the Mann-Whitney U test. It has lower mean
squared error than $S$, for it equals $E\left(S\mid X_{\left(1\right)},X_{\left(2\right)},\ldots,Y_{\left(1\right)},Y_{\left(2\right)},\ldots\right)$,
and since the order statistics are sufficient, the Rao-Blackwell theorem
can be applied. Furthermore, it is normally distributed, a fact that
follows from a general theorem on $U$-statistics, see the chapters
on $U$-statistics in e.g. Lehmann's Elements of Large Sample
Theory
or Serfling's Approximation Theorems of Mathematical Statistics.
Its limiting distribution, with a small sample correction, can be
found on wikipedia., and it is implemented in R through the function $\mathtt{wilcox.test}$ (with the option $\mathtt{paired=FALSE}$).
Note that the definition of the Mann-Whitney U test is ambiguous. Sometimes it is defined as the $T$ above, but sometimes it counts "victories for $x_i$" as positive and "victories for $y_i$" as negative. This is the approach taken in e.g. the R function $\mathtt{wilcox.test}$.
A: Here is a fairly straightforward and intuitive argument for both estimators being asymptotically normal.
Call $$T(X|y_1)=\frac{1}{n} \sum_{i=1}^{n} I(x_i>y_1)$$
As with $S$, central limit theorem implies this is asymptotically normal.
Furthermore, take $$T = \frac{1}{n}\sum_{i=1}^{n} T(X|y_i)$$
Since each $T(X|y_i)$ is asymptotically normal, and any linear combination of normal variables is normal, $T$ must too be asymptotically normal.
