Mann-Whitney Normal Approximation process help

Question

Let $X_{1}, X_{2}, ..., X_{n}$ is i.i.d sample from $X$ and $Y_{1}, Y_{2}, ..., Y_{m}$ is i.i.d sample from $Y$. And both samples are independent each other.

Trying Mann-Whitney U-test then, $U =$ $\sum_{i=1}^{n}\sum_{j=1}^{m}S(X_{i}, Y_{j})$

While r.v. $S(X, Y)$ get value $1$ for case $X > Y$, $0.5$ for case $X = Y$ and $0$ for case $X < Y$.

With a quite a large number of samples, we can try normal approximation. So, $z = \frac{U - m_{U}} {\sigma_{U}}$

And $U = \sum_{i=1}^{n}T_{i}$ and $T_{1}, T_{2}, ..., T_{n}$ must be i.i.d from some kind of distribution to apply central limit theorem, as far as I know. But, I failed to find what $T$ is.

My first assumption is $ T_{i} = \sum_{j=1}^{m}S(X_{i}, Y_{j}) $. But I am not sure that $T_{i}$ s' are independent.

My intuition said it's not independent. Because if $T_{1} = \sum_{j=1}^m S(X_{1}, Y_{j}) = 5$, $X_{1} = 500$, and if we know that $X_{2} = 600$, anyhow we can find out directly that $T_{2}$ is bigger or same as $T_{1}$.

$T_{1}$ gives some information about $T_{2}$ so I think $T_{i}$ are not i.i.d.

So my question is,

(1) Are $T_{i}$ i.i.d or not? If they are independent, How they are independent and what is my mistake above?

(2) $T_{i}$ they are not independent, how can I represent $U$ as sum of some kind of r.v., which are i.i.d?

Answer 1

There are several ways to prove this.

1. Hájek developed a proof based on projections. You can approximate $U$ by $$m\sum_i E[S(X_i,Y_j)\mid X_i] +n\sum_j E[S(X_i,Y_j)\mid Y_j]$$ and each of these sums is iid. The summands here are projections of $S(\cdot,\cdot)$ on to functions of one $X$ or one $Y$, and this fact lets you prove that the approximation has the same limit as $U$ does. The details are in, eg, van der Vaart's Asymptotic Statistics, Chapter 16. This is the easiest if you want to start from scratch and perhaps the prettiest proof.

2. For this specific $U$ statistic, you can take advantage of its relationship to the Wilcoxon statistic. It can be written as a scaled version of $$\int \mathbb{F}(x)\,d\mathbb{G}(x)$$ where $\mathbb{G}$ and $\mathbb{F}$ are the empirical CDFs of the two distributions. Since $(\mathbb{F},\,\mathbb{G})$ converges to a Gaussian process and $(F,\,G)\mapsto \int F\,dG$ is a (Hadamard) differentiable function, an infinite-dimensional version of the delta method gives you the result. This approach has the advantage that it can be extended to non-iid data. It's also the easiest if you are willing to just cite to all the hard stuff (the convergence of CDFs to Gaussian processes, functional delta method). Details are in Chapter 20 of van der Vaart's Asymptotic Statistics

3. Or you can do it bare-handed as Mann and Whitney did, but the proof is no fun at all.

Updated to add

There's a fourth approach, 'decoupling', where $X_i$ gets replaced each time it is reused by an independent copy, so you end up with $mn$ $X$s and $mn$ $Y$s. There are techniques to bound how much this changes the result, and you get a law of large numbers and central limit theorem. This approach is overkill for real-valued $U$-statistics, but it generalises nicely to $U$-processes (families of $U$-statistics) and to infinite-dimensional problems. The reference is Decoupling: From Dependence to Independence by de la Pena and Giné

Answer 2

(1) You are correct, the $T_i$ you defined are not independent.

(2) I don't think it's possible to write $U$ as a sum of iid random variables.

As $U$ is not a sum of iid random variables, showing that $U$ is approximately normally distributed for large sample sizes is not an easy task. I suggest you look at the book Elements of large-sample theory by Lehmann. Equation (3.2.9) states that $$\frac{U-\frac{1}{2}mn}{\sqrt{mn(n+m+1)/12}}\xrightarrow{d}N(0,1)$$

This result is obtained using theory developed in Section 2.8 of the book, which is about central limit theorems for dependent variable. However, the key result - Theorem 2.8.2 - is not given a complete proof.

For a full proof of what you originally wanted to prove, you would probably have to go back to the original paper by Mann and Whitney from 1947, On a test of whether one of two random variables is stochastically larger than the other. The proof is not easy going. Apparently Hájek has written a better version - see this answer.