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Let $x_1, \dots, x_n$ be a random sample from a distribution $D$. Say, I want to test whether $F(z)$, the cdf of $D$, is Lipschitz continuous, i.e. there exists $L$ such that $F(z + \delta) - F(z) \leq L\delta$ for $z \in \mathbb{R}$ and $\delta \geq 0$.

The above formulation is quite general and seems to be unsuitable for testing.

Hopefully, it might be possible to test for other properties implying Lipschitz continuity or non-Lipschitz-continuity. A trivial example: if $\exists~i \neq j$ such that $x_i = x_j$, then $F$ must be discontinuous.

I have searched for different literature resources (e.g. Anirban DasGupta: Asymptotic Theory of Statistics and Probability) with no success.

I realize the question is very general (I wish I knew how to make it more specific). Any literature or test suggestions would be highly appreciated.

Tomas
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    Without making some restrictive assumptions about $F,$ the "trivial example" you give is the only possible one. – whuber May 13 '19 at 19:41
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    Good point --- is it possible to be more precise pls --- (i) is there a proof method for negative results, to formalize the intuition "nothing better than the trivial example can be done", (ii) would you have an idea how to make such a restriction on $F$? M. – Michal May 13 '19 at 21:44
  • @whuber, thank you for your comment,.. can you, please, elaborate on that - along the lines of comment above (by @Michal)? – Tomas May 15 '19 at 19:08
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    I've been thinking a lot about this. I'm unable to develop a rigorous argument. My thinking is somewhat similar to the response I posted at https://stats.stackexchange.com/questions/2504. – whuber May 15 '19 at 20:47
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    You can test if a function is "locally" Lipschitz continuous if the local neighborhood is small enough to convincingly sample from. – hipHopMetropolisHastings Nov 19 '20 at 01:33
  • @hipHopMetropolisHastings , thank you for the comment. Please, can you elaborate on that? Do you have a particular testing approach in mind? – Tomas Nov 19 '20 at 08:18
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    @Tomas Start at some point, $x_0$, draw a large number, $N$, points, $x_1$ close to $x_0$, and test if the maximum ratio of the distance between $d_y(f(x_1),f(x_0))$ and $d_x(x_1,x_0)$ is finite. Also you can increase the size of the neighborhood and make sure that the maxima Lipschitz constant converges and does not explode to infinity. – hipHopMetropolisHastings Mar 10 '23 at 22:44

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