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The Kolmogorov Smirnov test is based on the maximum vertical distance between the ECDFs of two provided samples.

Is there a variant that checks the maximum horizontal distance?

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  • $\begingroup$ What would the $H_{0}$ of this test mean substantively speaking? $\endgroup$
    – Alexis
    Jan 26, 2015 at 16:24
  • $\begingroup$ The same as in Kolmogorov-Smirnov: the samples are drawn from the same distribution. $\endgroup$ Jan 26, 2015 at 17:00
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    $\begingroup$ There's an issue of scale here - the vertical scale is a probability between 0 and 1, the horizontal distance depends upon choice of units. $\endgroup$
    – Silverfish
    Jan 26, 2015 at 17:05
  • $\begingroup$ Ricky, it is not at all clear to me that the null hypothesis would be "the same as" the K-S test, particularly when the math of what you propose diverges. (Of course whether something is "clear to me" may not matter... I'm just some jerk on the Internet. :) $\endgroup$
    – Alexis
    Jan 26, 2015 at 18:44

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My guess as to why people don't use it: if you have a test that looks at horizontal distance it won't be distribution-free (at least not without some modification*).

Consider a continuous cdf and an ecdf plotted on the same axes with the KS-statistic marked in at the point of greatest vertical distance between them.

Note that a completely monotonic transformation of the x-axis will change the horizontal scale but leave the vertical distances unchanged. It's this feature that in essence makes the K-S test work the same for every fully-specified continuous distribution (since you can convert between them by exactly such a transformation).

But if you measure horizontal distance, every non-identity transform is going to change the distribution of horizontal distances.

* It might work if you convert everything back to standard uniform and then measure horizontal distance, but I don't think that's what you're asking about.

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