Let $F$ and $G$ are two distributions with a support, say $[0,1]$. Assume they are continuous and increasing. When we use the Kolmogorov distance $$\operatorname{dist}(F,G)=\sup_{x\in[0,1]}|F(x)-G(x)|,$$ What kind of relationship can we find between the distance and inverse distance? $$\operatorname{dist}(F,G) \text{ and } \operatorname{dist}(F^{-1},G^{-1})$$ Can there be an example that shows inverses are a lot closer than the just the distance of $F$ and $G$?
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I have tried to improve the readability of your question by changing some $\rm \LaTeX$ code of your post. It is possible that I unintentionally changed the meaning of your question. Please proofread the question to ensure this has not happened. – GNUSupporter 8964民主女神 地下教會 Feb 08 '18 at 16:10
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Thanks for the editing. It looks better :) – Andeanlll Feb 08 '18 at 17:23
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Do you want to modify the distance $\mathrm{dist}(F^{-1},G^{-1})$ to us the intersection of the domains of $F^{-1}$ and $G^{-1}$. If not, your written definition of distance only inspect the part that is in $[0,1]$. Also, what is the distance if $\mathrm{supp}(F^{-1}) \cap \mathrm{supp}(G^{-1}) = \varnothing$? – Eric Towers May 03 '18 at 13:37
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Thanks, Eric. I'd say $F$ and $G$ are strictly increasing. – Andeanlll May 03 '18 at 13:39
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As you assume $F$ and $G$ are both continuous functions from $\left[0,1\right]$ to $\left[0,1\right]$, there is no guaranteed order relation between $\text{dist}(F,G)$ and $\text{dist}(F^{-1},G^{-1})$.
Consider a simply example, where $$ F(x)=\left\{ \begin{array}{ll} kx,&x\in\left[0,\alpha\right]\\ \frac{1-k\alpha}{1-\alpha}\left(x-\alpha\right),&x\in\left(\alpha,1\right] \end{array} \right.\quad\text{and}\quad G(x)=\left\{ \begin{array}{ll} kx,&x\in\left[0,\beta\right]\\ \frac{1-k\beta}{1-\beta}\left(x-\beta\right),&x\in\left(\beta,1\right] \end{array} \right.. $$ Here the parameters $k>0$, $0<\alpha,\beta<1$ are chosen such that both $F$ and $G$ are increasing on $\left[0,1\right]$.
With this example, take $\alpha$ and $\beta$ close to $1$, and $k$ close to $0$. The image of $F$ and $G$ would then be similar to the following figure, where $\text{dist}(F,G)$ is obviously very large but $\text{dist}(F^{-1},G^{-1})$ is kind of small.
likewise, take $\alpha$ and $\beta$ close to $0$, and $k$ much larger than $1$. In this case, the image of $F$ and $G$ would be close to the following figure, where $\text{dist}(F,G)$ is kind of small but $\text{dist}(F^{-1},G^{-1})$ is obviously very large.
hypernova
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