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I'm reading Kolmogorov–Smirnov test from a lecture note from MIT OpenCourseWare Statistics for Applications. In the lecture note, there are two important theorems:

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and

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It's unfortunate that the lecture does not prove Theorem 2. From this question, I got that Kolmogorov-Smirnov test is not applicable for non-continuous distributions. As such, I feel that Theorem 2 is not correct because it does not mention the condition of continuity.

Could you please verify if my observation is correct?

Akira
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  • Re your penultimate paragraph - It rather depends on what you intend by "not applicable". A test using the Komogorov distance will still work (one can in principle evaluate the permutation distribution); you just won't be able to use any of the results that assume continuity (in typical cases it would produce a very conservative test). More detail is in my answer at the post you linked to. – Glen_b Mar 14 '20 at 06:07

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