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Problem setup: Suppose $X_1, \ldots, X_n$ is an i.i.d. sample from $F_X$ (CDF), and $Y_1, \ldots, Y_n$ is another i.i.d. sample from $F_Y$ (also CDF). In addition, $h(z_1, \ldots, z_n)$ is a real-valued symmetric function. Let $H_X = h(X_1, \ldots, X_n)$ and $H_Y = h(Y_1, \ldots, Y_n)$. Denote $D_{KS}(X, Y)$ as the Kolmogorov-Smirnov distance between $X \sim F_X$ and $Y \sim F_Y$, that is,

$$D_{KS}(X, Y) = \sup_{z \in \mathbb R} |F_X(z) - F_Y(z)|.$$

Question: How should I use $D_{KS}(X, Y)$ to bound $D_{KS}(H_X, H_Y)$, as in finding a function $g$ such that $D_{KS}(H_X, H_Y) \leq g(D_{KS}(X, Y))$? One can make assumptions on $F_X$, $F_Y$, and $h$, but should be as minimal as possible. (for example, compact support of $F_X$, $h$ being Lipschitz, and etc.).

My intuition is that if $X$ and $Y$ are from two close distributions (in terms of KS distance), so should $H_X$ and $H_Y$, but I couldn't find a way to formally make a statement.

Any help or ideas would be much appreciated!

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1 Answer 1

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If you think of $h$ as a test statistic, then you should be able to get substantial differences between the distributions of $h({\mathbf X})$ and $h({\mathbf X})$ when the K-S distance between the distributions of $X$ and $Y$ is of order $n^{-1/2}$.

For example, take $X\sim N(0,1)$ and $Y\sim N(\mu,1)$ and $h({\mathbf X})= n^{-1/2}\sum_i X_i$. Then $h(X)\sim N(0,1)$ and $h(Y)\sim N(n^{1/2}\mu,1)$, which is not small if $\mu$ is of order $n^{-1/2}$ or larger.

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