Editing Kolmogorov–Smirnov test (section)

===Test with estimated parameters===

If either the form or the parameters of ''F''(''x'') are determined from the data ''X''<sub>''i''</sub> the critical values determined in this way are invalid. In such cases, [[Monte Carlo method|Monte Carlo]] or other methods may be required, but tables have been prepared for some cases. Details for the required modifications to the test statistic and for the critical values for  the [[normal distribution]] and the [[exponential distribution]] have been published,<ref name="Pearson & Hartley">{{cite book |title= Biometrika Tables for Statisticians |editor=Pearson, E. S. |editor2=Hartley, H. O. |year=1972 |volume=2 |publisher=Cambridge University Press |isbn=978-0-521-06937-3 |pages=117–123, Tables 54, 55}}</ref> and later publications also include the [[Gumbel distribution]].<ref name="Shorak & Wellner">{{cite book |title=Empirical Processes with Applications to Statistics |first1=Galen R. |last1=Shorack |first2=Jon A. |last2=Wellner |year=1986 |isbn=978-0-471-86725-8 |publisher=Wiley |page=239}}</ref> The [[Lilliefors test]] represents a special case of this for the normal distribution. The logarithm transformation may help to overcome cases where the Kolmogorov test data does not seem to fit the assumption that it came from the normal distribution.

Using estimated parameters, the question arises which estimation method should be used. Usually this would be the [[Maximum likelihood estimation|maximum likelihood method]], but e.g. for the normal distribution MLE has a large bias error on sigma. Using a moment fit or KS minimization instead has a large impact on the critical values, and also some impact on test power. If we need to decide for Student-T data with df&nbsp;=&nbsp;2 via KS test whether the data could be normal or not, then a ML estimate based on H<sub>0</sub> (data is normal, so using the standard deviation for scale) would give much larger KS distance, than a fit with minimum KS. In this case we should reject H<sub>0</sub>, which is often the case with MLE, because the sample standard deviation might be very large for T-2 data, but with KS minimization we may get still a too low KS to reject&nbsp;H<sub>0</sub>. In the Student-T case, a modified KS test with KS estimate instead of MLE, makes the KS test indeed slightly worse. However, in other cases, such a modified KS test leads to slightly better test power.{{Citation needed|date=May 2022}}