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Why the test function in Kolmogorov-Smirnov test takes the supremum of the set of differences, not maximum? When maximum (greatest element) of a set does not exist then supremum may exist, but if greatest element exist then supremum is same as greatest element. I want to know why here greatest element does not exist?

Thanks
Prasenjit

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1 Answer 1

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Suppose a difference is an increasing continuous function on open interval and zero everywhere else. Then the maximum will not exist, but supremum will.

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