I understand that the Kolmogorov–Smirnov test statistic for a given cumulative distribution function $F(x)$ is $D_n = \sup_x |F_n(x) - F(x)|$. However, if I have to rank its sensitivity to location, scale and shape, what would be the rank and why? Is it:

  1. Location
  2. Shape
  3. Scale?

EDIT: I am in interested in two sample KS test.


closed as unclear what you're asking by kjetil b halvorsen, Reinstate Monica, Peter Flom - Reinstate Monica Sep 4 at 12:46

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  • 2
    $\begingroup$ Definitely most sensitive to location, as a small shift in the CDF caused the KS test statistic to spike. Unsure about the other two, but I'd guess shape is second and scale is third, Maybe someone more knowledgable can answer this properly. $\endgroup$ – Xiaomi Oct 3 '18 at 3:43
  • $\begingroup$ @Xiaomi Could you explain your first conclusion? It seems implicitly to suppose that $F$ is not an absolutely continuous distribution or else has some points of infinite slope, for those are the only mathematically possible ways that small shifts in location of $F$ can lead to arbitrarily large changes in $D_n.$ ML_Pro: in order to answer this question, we need you to tell us how you intend to quantify "sensitivity." Otherwise how can we know what you are asking or how to compare (1), (2), and (3)? And why do you describe a one-sample test and then change your question at the end? $\endgroup$ – whuber Oct 3 '18 at 15:04
  • $\begingroup$ For this to be answerable, I think you'd have to specify how you are comparing location, shape and scale. What changes in each are equivalent? $\endgroup$ – Peter Flom - Reinstate Monica Sep 4 at 12:46