Kolmogorov-Smirnov Test gives result different from max(abs(difference(x, y)))

Question

I'm using ks.test function in r to perform Kolmogorov-Smirnov test. I found that Kolmogorov-Smirnov test gives result different from

max(abs(difference(x, y)))

According to the definition of Kolmogorov-Smirnov Test in Wikipedia, the results should be equivalent.

Does any one know why?

Answer 1

The KS statistic is not supposed to be equal to max(|x-y|). It is applied to the cumulative distribution function(s) (CDF). Thus, it represents rather the proportion of observations different between a sample and a reference distribution.

See the two examples below executed in MATLAB (although I expect the results to be identical in R):

    x = [1 2 3 4 5 6 7 8 9 10];
    y = [1 2 3 4 5 6 7 8 9 11];

    [~, ~, ks2s] = kstest2(x,y)

    ks2s =

        0.1000                       (1)



    x = [1 2 3 4 5 6 7 8 9 10];
    y = [1 2 3 4 5 6 7 8 9 12];

    [~, ~, ks2s] = kstest2(x,y)

    ks2s =

        0.1000                        (2)

Thus, although the maximum absolute magnitude difference between x and y is larger in (2), the KS statistic is the same because the proportion of samples that are different is the same.

If y has an extra sample, for example, the result changes:

    x = [1 2 3 4 5 6 7 8 9 10];
    y = [1 2 3 4 5 6 7 8 9 10 11];

    [h, p, ks2s] = kstest2(x,y)

    ks2s =

        0.0909