Kolmogorov-Smirnov for Pareto distribution on sample I want to use the Kolmogorov-Smirnov test to test if a sample is drawn from a Pareto distribution. Unfortunately, the only way to estimate the distribution's parameters is from the sample.
Does anybody know of the existence of tables with modified critical values or any other way to perform the test in this case?
 A: A Kolmogorov-Smirnov test with estimated parameters is known as Liliiefors' test. The values of the test statistic tend to be smaller than with the KS test.
It is no longer nonparametric; you need to work out the distribution of the test statistic for each distribution type (and it differs again if you estimate a subset of the parameters rather than all of them).
Some packages offer Lilliefors test for the exponential distribution and the normal distribution (which are the cases Lilliefors discusses in his two papers).
Here's a way to use the exponential for the Pareto (I'll use the notation at that link):
Case 1: known $x_m$, unknown $\alpha$:


*

*If $x_m$ isn't $1$, divide through by $x_m$, so you have a Pareto with $x_m=1$.

*take logs, resulting in an exponential with (unknown) scale parameter $\alpha$.

*apply Lilliefors test for an exponential r.v.
Case 2: unknown $x_m$, unknown $\alpha$. While this case can be done by simulation, we can make use of an exponential Lilliefors test as follows:


*

*take logs, yielding a shifted exponential

*subtract the minimum observation from all observations, and discard the minimum, leaving $n-1$ observations

*test the reduced sample for exponentiality via the Lilliefors test as before.
