It seems that lognormal and Burr distributions are the best to fit my data. Can I compare their goodness of fit using a Kolmogorov-Smirnov test? The lognormal has 2 parameters whereas Burr has 3. To compare the values of the KS test, must the distributions have the same number of parameters?

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    $\begingroup$ I have tried to edit the question to correct a few minor grammatical things but mostly to make the title more closely reflect the intent of the question. If you don't like my changes, feel free to revert them. $\endgroup$ – Silverfish Jun 4 '16 at 3:34

The Kolmogorov-Smirnov test finds the maximum difference between the CDFs of the two distributions (or ECDFs, if you're using empirical data rather than distributions) so using the Kolmogorov-Smirnov test should be no problem for you.

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    $\begingroup$ 1. If the test is being used (rather than merely comparing test statistics), the parameters are being estimated from the data, so this wouldn't be a Kolmogorov-Smirnov test, it would be a form of Lilliefors test - and no longer nonparametric. $\;$ 2. If we're just comparing KS-statistics, and if the three parameter Burr is flexible enough to approximate the lognormal fairly well, it will essentially always fit at least as well as the lognormal, since it will also be free to not approximate the lognormal ... ctd $\endgroup$ – Glen_b May 19 '16 at 22:31
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    $\begingroup$ ctd... (something the lognormal can never do), so that extra degree of freedom will allow for it to adapt more closely to the data (leading to typically smaller KS statistics). This suggests that if you're comparing distributions with similar characteristics (e.g. continuous, right skew on the positive half line, able to fit similar shapes) with different numbers of parameters, then without some penalty for the extra parameter, you'd nearly always see the one with more parameters have a better fit. $\endgroup$ – Glen_b May 19 '16 at 22:34

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