An only descriptive item on the Lilliefors statistics
Result: with growing normal sample sizes N the Kolmogorov-Smirnov- Lilliefors one-sample test statistics tends to be inside the interval [0.10, 0.20): approximately 95 % for sizes 350, 98% to 99% for 600 and beyond, using 10´000 samples each size. ____________
TABLE. Distribution of Lilliefors statistics among intervals for normal data
Note low statistics, 10´000 samples, is not an adequate choice. In consequence the second decimal place of the [0.10, 0.20) target interval should be taken with reserve, even the first varies from 5 to 8, n=500 onwards. One can see that the firth interval contents becomes smaller whereas the third grows.