Unfortunately lack of rigor leads to a complete kaos in literature about the test statistics. Lilliefors Test has a formulation only: The maximum of the differences (along all n sample points): _____dj = Max |j/n, Phi(z)|, |(j-1)/n, Phi(z-)| where z is the *step´s* end abcisse and z- the step,s beggining ( j=1, 2, ... , n)
The shortcuts text´s show are the main cause of so much misunderstanding. The above definition is the only way (I had checked) to attain rigorously the critical values of the test statistics Literature shows. Furthermore is unequivocal.